OE- 


REESE;  LIBRA 


UNIVERSITY   OF   CALIFORNIA. 


/fo.3 


*.She/f 
+ 


-I- 


A  MANUAL 


OF 


RULES,  TABLES,  AND  DATA 


FOR 


MECHANICAL    ENGINEERS. 


A    MANUAL 


OF 


RULES,  TABLES,  AND  DATA 


FOR 


MECHANICAL    ENGINEERS 


BASED    ON    THE  MOST    RECENT    INVESTIGATIONS: 

OF    CONSTANT    USE 
IN  CALCULATIONS   AND   ESTIMATES   RELATING  TO 

STRENGTH  OF  MATERIALS  AND  OF  ELEMENTARY  CONSTRUCTIONS;  LABOUR; 

HEAT  AND  ITS  APPLICATIONS,  STEAM  AND  ITS  PROPERTIES,  COMBUSTION  AND  FUELS, 

STEAM  BOILERS,  STEAM  ENGINES,  HOT-AIR  ENGINES,  GAS-ENGINES  ;  FLOW  OF  AIR  AND  OF 

WATER;  AIR  MACHINES;  HYDRAULIC  MACHINES;   MILL-GEARING;   FRICTION  AND  THE  RESISTANCE  OF 

MACHINERY,  &c. ;  WEIGHTS,  MEASURES,  AND  MONIES,  BRITISH  AND  FOREIGN,  WITH  THE  RECIPROCAL 

EQUIVALENTS  FOR  THE  CONVERSION  OF  BRITISH  AND  FRENCH  COMPOUND  UNITS  OF 

WEIGHT,  PRESSURE,  TIME,  SPACE,  AND  MONEY;  SPECIFIC  GRAVITY  AND 

THE  WEIGHT  OF  BODIES  ;  WEIGHT  OF  METALS,  &c. 

WITH 

TABLES  OF  LOGARITHMS,  CIRCLES,  SQUARES,  CUBES,  SQUARE-ROOTS,  AND  CUBE-ROOTS; 
AND  MANY  OTHER  USEFUL  MATHEMATICAL  TABLES. 

BY 

DANIEL    KINNEAR    CLARK, 

MEMBER  OF  THE   INSTITUTION   OF  CIVIL  ENGINEERS ; 

AUTHOR  OF  "RAILWAY  MACHINERY,"  "EXHIBITED  MACHINERY  OF  1862,"  ETC. 


FOURTH  EDITION. 


LONDON: 
BLACKIE   &    SON:    OLD    BAILEY,  E.G. 

GLASGOW  AND  EDINBURGH. 

1889. 
All  Rights  Reserved. 


PREFACE. 


THIS  Work  is  designed  as  a  book  of  general  reference  for  Engineers : 
— to  give  within  a  moderate  compass  the  leading  rules  and  data, 
with  numerous  tables,  of  constant  use  in  calculations  and  estimates 
relating  to  Practical  Mechanics.  The  Author  has  endeavoured  to 
concentrate  the  results  of  the  latest  investigations  of  others  as  well 
as  his  own,  and  to  present  the  best  information,  with  perspicuity, 
conciseness,  ana  scientific  accuracy. 

Amongst  the  new  and  original  features  of  this  Work,  the  follow- 
ing may  be  named : — 

In  the  section  on  Weights  and  Measures,  the  weight,  volume, 
and  relations  of  water  and  air  as  standards  of  measure,  are  concisely 
set  forth.  The  various  English  measures,  abstract  and  technical, 
are  given  in  full  detail,  with  tables  of  various  wire-gauges  in  use: 
and  equivalent  values  of  compound  units  of  weight,  power,  and 
measure — as,  for  example,  miles  per  hour  and  feet  per  second. 
The  French  Metric  Standards  are  defined,  according  to  the  latest 
determinations,  with  tables  of  metric  weights  and  measures,  equi- 
valents of  British  and  French  weights  and  measures,  and  a 
number  of  convenient  approximate  equivalents.  There  is,  in  addi- 
tion, a  full  table  of  equivalents  of  French  and  English  compound 
units  of  weight,  pressure,  time,  space,  and  money — as,  for  example, 
pounds  per  yard  and  kilogrammes  per  metre;  which  will  be  found 
of  great  utility  for  the  reciprocal  conversion  of  English  and  French 
units. 

The  tables  of  the  Weight  of  bars,  tubes,  pipes,  cylinders,  plates, 
sheets,  wires,  &c.,  of  iron  and  other  metals,  have  been  calculated 
expressly  for  this  Work,  and  they  contain  several  new  features 
designed  to  add  to  their  usefulness.  They  are  accompanied  by  a 
summary  of  the  various  units  of  weight  of  wrought  iron,  cast  iron, 
and  steel,  with  plain  rules  for  the  weight. 

In  the  section  on  Heat  and  its  Applications,  the  received  mechan- 
ical theory  is  defined  and  illustrated  by  examples.  The  relations 
of  the  pressure,  volume,  and  temperature  of  air  and  other  gases, 


VI  PREFACE. 

with  their  specific  heat,  are  investigated  in  detail.  The  transmission 
of  heat  through  plates  and  pipes,  between  water  and  water,  steam 
and  air,  &c.,  for  purposes  of  heating  or  cooling,  is  verified  by  many 
experimental  data,  which  are  reduced  to  units  of  performance. 

The  physical  properties  of  steam  are  deduced  from  the  results 
of  Regnault's  experiments,  with  the  aid  of  the  mechanical  theory 
of  heat.  A  very  full  table  of  the  Properties  of  Saturated  Steam  is 
given.  The  table  is,  for  the  most  part,  reproduced  from  the  article 
"  Steam,"  contributed  by  the  Author  to  the  Encyclopedia  Britannica, 
8th  edition,  and  it  was  the  first  published  table  of  the  same  extent, 
in  the  English  language,  based  on  Regnault's  data.  An  original 
table  of  the  properties  of  saturated  mixtures  of  air  and  aqueous 
vapour  is  added. 

In  the  section  on  Combustion,  new  and  simple  formulas  and  data 
are  given  for  the  quantity  of  air  consumed  in  combustion,  and  of 
the  gaseous  products  of  combustion,  the  heat  evolved  by  combus- 
tion, the  heating  power  of  combustibles,  and  the  temperature  of 
combustion ;  with  several  tables. 

On  Coal  as  a  Fuel,  both  English  and  Foreign,  its  composition, 
with  the  results  of  many  series  of  experiments  on  its  combustion, 
are  collected  and  arranged.  The  quantity  of  air  consumed  in  its  com- 
bustion, and  of  the  gaseous  products,  with  the  total  heat  generated, 
are  calculated  in  detail.  Coke,  lignite,  asphalte,  wood,  charcoal, 
peat,  and  peat-charcoal,  are  similarly  treated;  whilst  the  combus- 
tible properties  of  tan,  straw,  liquid-fuels,  and  coal-gas,  are  shortly 
treated. 

The  section  on  Strength  of  Materials  is  wholly  new.  The  great 
accumulation  of  experimental  data  has  been  explored,  and  the  most 
important  results  have  been  abstracted  and  tabulated.  The  results 
of  the  experiments  of  Mr.  David  Kirkaldy  occupy  the  greater  por- 
tion of  the  space,  since  he  has  contributed  more,  probably,  than  any 
other  experimentalist  to  our  knowledge  of  the  Strength  of  Materials. 
The  Author  has  investigated  afresh  the  theory  of  the  transverse 
strength  and  deflection  of  solid  beams,  and  has  deduced  a  new  and 
simple  series  of  formulas  from  these  investigations,  the  truth  of  which 
has  been  established  with  remarkable  force  by  the  evidence  of  experi- 
ment. These  investigations,  based  on  the  action  of  diagonal  stress, 
throw  light  upon  the  element  called  by  Mr.  W.  H.  Barlow,  "the 
resistance  of  flexure:"  revealing,  in  a  simple  manner,  the  nature 
of  that  hitherto  occult  entity;  and  showing  that  flexure  is  not  the 
cause,  but  the  effect  of  the  resistance.  In  addition  to  formulas 


PREFACE.  vii 

for  beams  of  the  ordinary  form,  special  formulas  have  been 
deduced  for  the  transverse  strength  and  deflection  of  railway 
rails,  double-headed  or  flanged,  of  iron  or  steel;  in  the  estab- 
lishment of  which  he  has  availed  himself  of  the  important 
experimental  data  published  by  Mr.  R.  Price  Williams,  and  by 
Mr.  B.  Baker.  To  our  knowledge  of  the  strength  of  timber, 
Mr.  Thomas  Laslett  has  recently  made  important  additions,  and  the 
results  of  his  experiments  have  been  somewhat  fully  abstracted  and 
analyzed.  But  woods,  by  their  extremely  variable  nature,  are  not 
amenable,  like  wrought-iron  and  steel,  to  the  unconditional  applica- 
tion of  formulas  for  transverse  strength.  The  Author  has,  never- 
theless, deduced  from  the  evidence,  certain  formulas  for  the  trans- 
verse strength  and  deflection  of  woods,  with  tables  of  constants, 
which,  if  applied  with  intelligence  and  a  knowledge  of  the  uncer- 
tainties, cannot  fail  to  prove  of  utility. 

The  Torsional  Strength  of  Solid  Bodies  has  also  been  investigated 
afresh,  and  reduced  to  new  formulas. 

In  dealing  with  the  Strength  of  Elementary  Constructions,  the 
Author  has  brought  together  many  important  experimental  results. 
In  treating  of  rivet-joints  and  their  employment  in  steam-boilers, 
he  has,  he  believes,  clearly  developed  the  elements  of  their  strength 
and  their  weakness.  By  a  close  comparison  of  the  results  of  tests  of 
cast-iron  flanged  beams,  it  is  plainly  shown  that  the  ultimate 
strength  of  a  cast-iron  beam  is  scarcely  affected  by  the  proportionate 
size  of  the  upper  flange,  and  that  the  lower  flange  and  the  web  are, 
practically,  the  only  elements  which  regulate  the  strength.  The 
tests  of  solid-rolled  and  rivetted  wrought-iron  joists  are  also  ana- 
lyzed ;  and  for  the  strength  and  deflection  of  these,  as  for  those  of 
cast-iron  flanged  beams,  new  and  simple  rules  and  formulas  are 
given.  A  new  investigation,  with  appropriate  formulas,  is  given 
for  the  bursting  strength  of  hollow  cylinders,  of  whatever  thickness. 
It  is  shown  that  the  variation  of  stress  throughout  the  thickness, 
follows  a  diminishing  hyperbolic  ratio  from  the  inner  surface  to- 
wards the  outer  surface.  The  resistance  of  tubes  and  cylindrical 
flues  to  collapsing  pressure  is  also  investigated,  and  formulas  based 
on  the  results  of  experience  are  given. 

On  the  subject  of  Mill-gearing,  a  new  and  compact  table  of  the 
pitch,  number  of  teeth,  and  diameter  of  toothed  wheels  is  given, 
with  new  formulas  and  tables  for  the  strength  and  horse-power  of 
the  teeth  of  wheels,  and  for  the  weight  of  toothed  wheels.  New 
formulas  and  tables  are  given  for  the  driving  power  of  leather 


Vlll  PREFACE. 

belts,  and  the  weight  of  cast-iron  pulleys.  For  the  strength  of 
Shafting, — cast-iron,  wrought-iron,  and  steel, — a  new  and  complete 
series  of  formulas  has  been  constructed,  comprising  its  resistance 
to  transverse  deflection  and  to  torsion,  with  very  full  tables  of  the 
weight,  strength,  power,  and  span  of  shafting. 

The  Evaporative  Performance  of  Steam-boilers  is  exhaustively 
investigated  with  respect  to  the  proportions  of  fuel,  water,  grate- 
area,  and  heating-surface,  and  the  relations  of  these  quantities  are 
reduced  to  simple  formulas  for  different  types  of  boilers.  The 
actual  evaporative  performances  of  boilers  are  abstracted  in  tabular 
form. 

The  Performance  of  Steam  worked  expansively,  in  single  and  in 
compound  cylinders,  is  exhaustively  analysed  by  the  aid  of  diagrams; 
the  similarity  and  the  dissimilarity  of  its  action  in  the  Woolf-engine 
and  the  Receiver-engine,  are  investigated;  and  the  principles  of 
calculation  to  be  applied  respectively  to  these,  the  leading  classes 
of  compound  engines,  are  explained.  The  best  working  ratios  of 
expansion  are  deduced  from  the  results  of  numerous  experiments 
and  observations  on  the  performance  of  steam-engines. 

The  principles  of  Air-compressing  Machines,  and  Compressed-air 
Engines  are  investigated,  and  convenient  formulas  and  tables  for 
use  are  deduced. 

The  whole  of  the  materials  for  the  preparation  of  this  work  have 
been  drawn  from  the  best  available  sources,  foreign  as  well  as 
English.  Vast  stores  of  the  results  of  experience  are  accumulated 
in  the  Proceedings  of  the  Institution  of  Civil  Engineers,  the 
Proceedings  of  the  Institution  of  Mechanical  Engineers,  and  other 
journals.  From  these  and  other  sources,  the  Author  has  drawn 

much  of  his  material. 

D.  K.  CLARK. 

8  Buckingham  Street,  Adelphi, 

LONDON,  zoth  March,  1877. 


NOTE   ON   THE   FOURTH    EDITION. 

I  have  thoroughly  revised  this  book,  and,  besides  correcting  it 
up  to  date,  I  have  introduced  much  new  matter,  which  will  render 
this  edition  even  more  valuable  than  the  last. 

D.  K.  CLARK. 

January r,  1889. 


CONTENTS. 


GEOMETRICAL   PROBLEMS. 

PAGE 

Straight  Lines — Straight  Lines  and  Circles — Circles  and  Rectilineal  Figures — The 
Ellipse — The  Parabola — The  Hyperbola — The  Cycloid  and  Epicycloid — 
The  Catenary — Circles — Plane  Trigonometry — Mensuration  of  Surfaces — 
Solids — Heights  and  Distances,  .........  I 

MATHEMATICAL   TABLES. 

Explanation  of  the  following  Tables  : — 32 

Logarithms  of  Numbers  from  i  to  10,000,          ........       38 

Hyperbolic  Logarithms  of  Numbers  from  1. 01  to  30, 60 

Numbers    or   Diameters  of   Circles,    Circumferences,   Areas,    Squares,    Cubes, 

Square  Roots,  and  Cube  Roots,        ........       66 

Circles  : — Diameter,  Circumference,  Area,  and  Side  of  Equal  Square,         .         .       87 

Lengths  of  Circular  Arcs  from  i°  to  180°, 95,  97 

Areas  of  Circular  Segments,      ..........     100 

Sines,  Cosines,  Tangents,  Cotangents,  Secants,  and  Cosecants  of  Angles,  .         .     103 
Logarithmic  Sines,  Cosines,  Tangents,  and  Cotangents  of  Angles,      .         .         .no 

Rhumbs,  or  Points  of  the  Compass, 117 

Reciprocals  of  Numbers  from  I  to  I ooo,    .         .         .         .         .         .         .         .118 

WEIGHTS   AND   MEASURES. 

WATER  as  a  Standard — Weight  and  Volume  of  pure  Water — The  Gallon  and  other 
Measures  of  Water — Pressure  of  Water — Sea- water — Ice  and  Snow — French 
and  English  Measures  of  Water,  .  .  .  .  .  .  .  .  .124 

AlR  as  a  Standard — Pressure  of  the  Atmosphere — Measures  of  Atmospheric  Pres- 
sure— Weight  of  Air — Volume — Specific  Heat,  ......  127 

GREAT  BRITAIN  AND  IRELAND — Imperial  Weights  and  Measures,  .  .  .  128 
Measures  of  Length  : — Lineal — Land — Nautical — Cloth,  .....  129 

Wire-gauges, 130 

Inches  and  their  Equivalent  Decimal  Values  in  parts  of  a  Foot — Fractional  Parts 

of  an  Inch,  and  their  Decimal  Equivalents,        ......     135 

Measures  of  Surface  : — Superficial — Builders'  Measurement — Land,    .         .         .     136 
Measures  of  Volume  : — Solid  or  Cubic — Builders' Measurement,         .         .         -137 
Table  of  Decimal  Parts  of  a  Square  Foot  in  Square  Inches,        ....     138 

Measures  of  Capacity : — Liquid— Dry — Definition  of  the  Standard  Bushel — Coal 

— Old  Wine  and  Spirit — Old  Ale  and  Beer — Apothecaries'  Fluid,  .  138 

Measures  of  Weight: — Avoirdupois  —  Troy — Diamond — Apothecaries' — Old 

Apothecaries' — Weights  of  Current  Coins — Coal — Wool — Hay  and  Straw 

— Corn  and  Flour,  ...........     140 

Miscellaneous  Tables : — Drawing  Papers — Commercial  Numbers — Stationery — 

Measures   relating    to    Building — Commercial    Measures — Measures   for 

Ships, 143 

Comparison  of  English  Compound  Units : — Measures  of  Velocity — Of  Volume 

and  Time — Of  Pressure  and  Weight — Of  Weight  and  Volume — Of  Power,     144 


X  CONTENTS. 

PAGE 

FRANCE — The  Metric  Standards  of  Weights  and  Measures — Metre — Kilogramme,  .     146 
Countries  where  the  system  is  legalized,    .         .         .         .         .         .         .         .146 

Measures  of  Length,         ...........     147 

Wire-gauges, 148 

Measures  of  Surface,         .         .         .         .         .         .         .         .         .         .         .149 

Measures  of  Volume : — Cubic — Wood, 149 

Measures  of  Capacity : — Liquid — Dry,       ........      149 

Measures  of  Weight,         .         .         .         .         .  .         .         .         .         .150 

EQUIVALENTS  of  British  Imperial  and  French  Metric  Weights  and  Measures,          .      150 
Measures  of  Length — Tables  of  Equivalent  Values  of  Millimetres  and  Inches — 
Square  Measures  or  Measures  of  Surface — Cubic  Measures — Wood  Mea- 
sure— Measures  of  Capacity— Measures  of  Weight, 150 

Approximate  Equivalents  of  English  and  French  Measures, 156 

Equivalents  of  French  and  English  Compound  Units  of  Measurement : — Weight, 
Pressure,  and  Measure — Volume,  Area,  and  Length — Work — Heat — Speed — 

Money,       .............  157 

GERMAN  EMPIRE  :  Weights  and  Measures  : — Length — Surface — Capacity — Weight,  160 

Values  of  the  German  Fuss  or  Foot  in  the  various  States,  .         .         .         .         .  161 

Old  Weights  and  Measures  in  Prussia  (Kingdom  of) — Bavaria  (Kingdom  of) — 
Wurtemberg  (Kingdom  of) — Saxony  (Kingdom  of) — Baden  (Grand-duchy  of) 
— Hanse  Towns: — Hamburg — Bremen — Lubec — Old  German  Customs 
Union — Oldenburg — Hanover,  &c.,  ........  162 

Austrian  Empire,  . 170 

Russia, 171 

Holland — Belgium — Norway  and  Denmark — Sweden     .         .         .         .         .         .173 

Switzerland — Spain — Portugal — Italy,    .         .         .         .         .         .         .         .  175 

Turkey — Greece  and  Ionian  Islands — Malta,  .         .         .         .         .         .         .         .178 

Egypt — Morocco — Tunis — Arabia — Cape  of  Good  Hope,        .         .         .         .  1 79 

Indian  Empire — Bengal — Madras — Bombay — Ceylon,     .         .         .         .         .         .180 

Burmah — China — Cochin-China — Persia — Japan — Java,          .         .         .         .         .183 

United  States  of  North  America,   . 186 

British  North  America,  .         .         .         .         .         .         .         .         .         .         .187 

Mexico — Central  America  and  West  Indies — West  Indies  (British) — Cuba — Guate- 
mala and  Honduras — British  Honduras — Costa  Rica — St.  Domingo,    .         .      187 
South  America — Colombia — Venezuela — Ecuador — Guiana — Brazil — Peru — Chili — 

Bolivia — Argentine  Confederation — Uruguay — Paraguay,      .         .         .         .188 

Australasia: — New  South  Wales — Queensland — Victoria — New  Zealand,  &c.,         .     189 

MONEY— BRITISH    AND    FOREIGN. 

Great  Britain  and  Ireland : — Value,  Material,  and  Weight  of  Coins — Mint  Price  of 

Standard  Gold,  &c.,  ..........     190 

France: — Material  and  Weight  of  French  Coins,  and  Value  in  English  Money,        .      190 
German  Empire  : — Names  and  Equivalent  Values  of  Coins,     .         .         .         .         .191 

North  and  South  Germany  (Old  Currency  of),  ......      191 

Hanse  Towns  (Old  Monetary  System  of): — Hamburg,  Bremen,  Lubec,  .         .         .      191 
Austria — Russia — Holland — Belgium — Denmark — Sweden — Norway,     .         .         .192 
Switzerland — Spain — Portugal — Italy — Turkey — Greece  and  Ionian  Islands — Malta,     1 93 
Egypt — Morocco — Tunis — Arabia — Cape  of  Good  Hope,      .         .         .         .  194 

Indian  Empire — China — Cochin-China — Persia — Japan — Java,       .         .         .  195 

United  States  of  North  America,  ..........      195 

Canada — British  North  America,  .         .         .         .         .         .         .         .         .         .196 

Mexico — West  Indies  (British)  —  Cuba — Guatemala — Honduras  —  Costa  Rica — 

St.  Domingo 196 


CONTENTS.  XI 

PAGE 

South  America — Colombia  — Venezuela — Ecuador — Guiana — Brazil — Peru — Chili 

— Bolivia — Argentine  Confederation— Uruguay— Paraguay,          .         .         .196 
Australasia, 197 

WEIGHT   AND    SPECIFIC    GRAVITY. 

Standard  Bodies  and  Temperatures  for  Comparative  Weight— Rules  for  Specific 

Gravity, 198 

General  Comparison  of  the  Weights  of  Bodies,        .         .         ...'..         .         .         .199 
Tables  of  the  Volume,  Weight,  and  Specific  Gravity  of  Metallic  Alloys— Metals — 

Stones,       .............     200 

Coal — Peat — Woods — Wood-Charcoal,    .         . 206 

Animal  Substances — Vegetable  Substances, 212 

Weight  and  Volume  of  various  Substances,  by  Tredgold,         .....     213 
Weight  and  Volume  of  Goods  carried  over  the  Bombay,  Baroda,  and  Central 

Indian  Railway,          . 213 

Weight  and  Specific  Gravity  of  Liquids,         .         .         .         .         .         .         .         .215 

Weight  and  Specific  Gravity  of  Gases  and  Vapours,        .         .         .         .         .         .216 

WEIGHT  OF  IRON   AND   OTHER  METALS. 

Data  for  Wrought  Iron — for  Steel — for  Cast  Iron,  .         .         .         .         .         .217 

Tables  of  Weights: — Weights  of  given  Volumes  of  Metals — Volumes  of  given  Weights 
of  Metals — Weight  of  One  Square  Foot  of  Metals — Weight  of  Metals  of  a 
given  Sectional  Area,  .  .  .  .  .  .  .  .  .  .  .218 

Special  Tables  for  the  Weight  of  Wrought  Iron:— 

Rules  for  the  Weight  of  Wrought  Iron — Cast  Iron — and  Steel,  .         .         .     223 

Rule  for  the  Length  of  I  cwt.  of  Wire  of  different  Metals,  of  a  given  thickness,       224 
Weight  of  French  Galvanized  Iron  Wire,          .......     225 

Special  Tables  of  the  Weight  of  Wrought-Iron  Bars,  Plates,  &c. ;  Multipliers 
for  other  Metals  : — Flat  Bar  Iron — Square  Iron — Round  Iron — Angle  Iron 
and  Tee  Iron — Wrought-Iron  Plates — Sheet  Iron — Black  and  Galvanized- 
Iron  Sheets — Hoop  Iron — Warrington  Iron  Wire — Wrought-Iron  Tubes, 
by  Internal  Diameter — Wrought-Iron  Tubes,  by  External  Diameter,  .  226 

Weight  of  Cast  Iron,  Steel,  Copper,  Brass,  Tin,  Lead,  and  Zinc— Special  Tables  :— 
Cast-Iron  Cylinders,  by  Internal  Diameter — Cast-Iron  Cylinders,  by  External 
Diameter — Volumes  and  Weight  of  Cast-Iron  Balls,  for  given  Diameters; 
Multipliers  for  other  Metals — Diameter  of  Cast-Iron  Balls  for  given  Weights,     253 
Weight  of  Flat-Bar  Steel— Square  and  Round  Steel — Chisel  Steel,    .         .         .     259 
Weight  of  One  Square  Foot  of  Sheet  Copper — Copper  Pipes  and  Cylinders,  by 
Internal  Diameter — Brass  Tubes,  by  External  Diameter — One  Square  Foot 

of  Sheet  Brass, 261 

Size  and  Weight  of  Tin  Plates— Weight  of  Tin  Pipes  and  Lead  Pipes— Dimen- 
sions and  Weight  of  Sheet  Zinc, 268 

FUNDAMENTAL  MECHANICAL  PRINCIPLES. 

FORCES  IN  EQUILIBRIUM  :— Solid   Bodies — Fluid  Bodies, 271 

MOTION  : — Uniform  Motion — Velocity — Accelerated  and  Retarded  Motion,    .         .  277 

GRAVITY  :— Relations  of  Height,  Velocity,  and  Time  of  Fall— Rules  and  Tables,  .  277 
ACCELERATED  AND  RETARDED  MOTION  IN  GENERAL: — General  Rules— Descent 

on  Inclined  Planes, 282 

MASS, 287 

MECHANICAL  CENTRES: — Centre  of  Gravity — Centre  of  Gyration — Radius  of 
Gyration — Moment  of  Inertia — Centre  of  Oscillation — The  Pendulum — 

Length  of  Seconds  Pendulum — Centre  of  Percussion 287 

CENTRAL  FORCES  : — Centripetal  Force— Centrifugal  Force, 294 


Xll  CONTENTS. 

PAGE 

MECHANICAL  ELEMENTS: — The  Lever — The  Pulley — The  Wheel  and  Axle — 
The  Inclined  Plane — Identity  of  the  Inclined  Plane  and  the  Lever — The 
Wedge— The  Screw, 296 

WORK:— English  and  French  Units  of  Work — Work  done  by  the  Mechanical  Ele- 
ments— By  Gravity — Work  accumulated  in  Moving  Bodies — Work  done  by 
Percussive  Force, 312 

HEAT. 

THERMOMETERS: — Table  of  Equivalent  Degrees  by  Centigrade  and  Fahrenheit — 

Pyrometers, 317,  967 

MOVEMENTS  OF  HEAT: — Radiation — Conduction — Convection,    ....     329 
THE  MECHANICAL  THEORY  OF  HEAT: — Mechanical  Equivalent  of  Heat — Joule's 

Equivalent  in  English  and  French  Units — Illustrations,         ....     332 

EXPANSION  BY  HEAT: — Linear  and  Cubical  Expansion, 335 

Table  of  Linear  Expansion  of  Solids,        ........     336 

Expansion  of  Liquids,      ...........     338 

Expansion  of  Gases — The  Absolute  Zero-point — Table  of  the  Compression  of 

Gases  by  Pressure  under  a  Constant  Temperature,    .....     342 

Relations  of  the  Pressure,  Volume,  and  Temperature  of  Air  and  other  Gases — 
General  Rules — Special  Rules  for  One  Pound  weight  of  a  Gas,  with  Table 
of  Coefficients — Table  of  the  Volume,  Density,  and  Pressure  of  Air  at 
various  Temperatures,  ..........  346 

SPECIFIC  HEAT:— Specific  Heat  of  Water,  with  Table— Specific  Heat  of  Air- 
Specific  Heat  of  Solids — Specific  Heat  of  Liquids — Specific  Heat  of  Gases,  .  352 

FUSIBILITY  OR  MELTING  POINTS  OF  SOLIDS: — Table, 363 

Latent  Heat  of  Fusion  of  Solid  Bodies,  with  Rule  and  Table,    ....  367 

BOILING  POINTS  OF  LIQUIDS, 368 

Latent  Heat  and  Total  Heat  of  Evaporation  of  Liquids,  .....  370 
Boiling  Points  of  Saturated  Vapours  under  various  Pressures,     .         .         .         '371 

Latent  Heat  and  Total  Heat  of  Evaporation  of  Liquids  under  One  Atmosphere,  372 

LIQUEFACTION  AND  SOLIDIFICATION  OF  GASES,        ......  372 

SOURCES  OF  COLD: — Siebe's  Ice-making  Machine — Carre's  Cooling  Apparatus — 

Frigorific  Mixtures,    .         .         .         .         .         .         .         .         .         .         •  373 

STEAM. 

Physical  Properties  of  Steam,         ..........  378 

Gaseous  Steam — Its  Expansion — Its  Total  Heat,       ......  383 

Specific  Heat  of  Steam — Specific  Density  of  Steam — Density  of  Gaseous  Steam,  384 

Properties  of  Saturated  Steam  from  32°  to  212°  F.,   .         .         .         .         .         .  386 

Properties  of  Saturated  Steam  for  Pressures  of  from  I  Ib.  to  400  Ibs.,          .         .  387 

Comparative  Density  and  Volume  of  Air  and  Saturated  Steam,           .         .         .  391 

MIXTURE   OF   GASES   AND  VAPOURS. 

Respective  Pressures  of  Gas  and  Vapours  in  Mixture, 392 

Hygrometry,          .............  392 

Properties  of  Saturated  Mixtures  of  Air  and  Aqueous  Vapour,  with  Table,         .  394 

COMBUSTION. 

Combustible  Elements  of  Fuel — Process  of  Combustion,         .....     398 
AIR  CONSUMED  IN  THE  COMBUSTION  OF  FUELS  :— Quantity  of  the  Gaseous  Pro- 
ducts of  the  Complete  Combustion  of  One  Pound  of  Fuel — Surplus  Air,        .     400 
HEAT  EVOLVED  BY  THE  COMBUSTION  OF  FUEL  : — Heat  of  Combustion  of  Simple 

and  Compound  Bodies — Heating  Powers  of  Combustibles,  ....     402 
Temperature  of  Combustion,     ..........     407 


CONTENTS.  Xlll 


FUELS. 

PAGE 

Fuels  or  Combustibles  generally  used, 409 

COAL: — Its  Varieties — Small    Coal: — Its    Utilization — Washing    Small  Coal — 

Deterioration  of  Coal  by  Exposure  to  Atmosphere, 409 

British  Coals — Composition  of  Bituminous  Coals — Dr.  Richardson's  Analysis,          .     412 
Weight   and   Composition  of  British   and   Foreign  Coals,  by  Delabeche   and 

Playfair, 413 

Weight  and  Bulk  of  British  Coals, 416 

Hygroscopic  Water  in  British  Coals, 416 

Torbanehill  or  Boghead  Coal,  with  Table  of  its  Composition,  .  .  .  .417 
American  and  Foreign  Coals : — Composition,  Weight  and  Bulk,  ....  418 
French  Coals: — Utilization  of  the  Small  Coal — Composition  of  French  Coals — 

Mean  Density,  Composition,  and  Heating  Power,         .....     420 

Indian  Coals : — Australian  and  Indian  Coals — Composition, 423 

COMBUSTION  OF  COAL  : — Process  of  Combustion — Gaseous  Products  of  the  Com- 
bustion of  Coal — Surplus  Air — Total  Heat  of  Combustion  of  British  Coals,  .     426 
COKE  : — Proportion  of  Coke  from  Coals — Anthracitic  Coke — Weight  and  Bulk  of 

Coke — Composition  of  Coke — Moisture  in  Coke — Heating  Power  of  Coke,  .     430 
LIGNITE  AND  ASPHALTE  : — Density,  Composition,  and  Heating  Power  of  Lignites 

and  Asphaltes, 436 

WOOD: — Moisture  in  Wood — Composition— Weight  and  Bulk  of  Wood,  with 
Table — Firewood—  Quantity  of  Air  Chemically  Consumed  in  the  Complete 
Combustion  of  Wood — Gaseous  Products — Total  Heat  of  Combustion — 

Temperature  of  Combustion, .  439 

WOOD-CHARCOAL  : — Yield  of  Charcoal — Composition,  with  Table  of  Composition 
at  various  Temperatures — Carbonization  of  Wood  in  Stacks,  and  Yield  of 
Charcoal — Manufacture  of  Brown  Charcoal — Distillation  of  Wood — Charbon 
de  Paris  (artificial  fuel) — Weight  and  Bulk  of  Wood-Charcoal  —  Absolute 
Density  of  Charcoal — Moisture  in  Charcoal — Air  Consumed  in  the  Combus- 
tion of  Charcoal — Gaseous  Products — Heat  of  Combustion,  .  .  .  444 
PEAT: — Nature  and  Composition — Condensed  Peat — Average  Composition — Pro- 
ducts of  Distillation — Heating  Power  of  Irish  Peat,  .....  452 

PEAT-CHARCOAL  :— Composition  and  Heating  Power, 455 

TAN  : — Composition  and  Heating  Power,       ........     455 

STRAW: — Composition, .     456 

LIQUID  FUELS  : — Petroleum,  Petroleum-Oils,  Schist  Oil,  and  Pine-wood  Oil ;  their 

Composition  and  Heating  Power,         .         .         .         .         .         .         .         .456 

COAL-GAS  :— Composition  and  Heating  Power,      .         .         .         .         .         .         -457 

APPLICATIONS   OF   HEAT. 

TRANSMISSION  OF  HEAT  THROUGH  SOLID  BODIES — FROM  WATER  TO  WATER 
THROUGH  SOLID  PLATES  AND  BEDS: — M.  Peclet's  Experiments — Mr.  James 
R.  Napier's  Experiments — Circumstances  which  affect  the  Ratio  of  Trans- 
mission— Mr.  Craddock's  Experiments,  .......  459 

HEATING  AND  EVAPORATION  OF  LIQUIDS  BY  STEAM  THROUGH  METALLIC 
SURFACES: — Experiments  by  Mr.  John  Graham,  by  M.  Clement,  by  M. 
Peclet,  by  MM.  Laurens  and  Thomas,  by  M.  Havrez,  by  Mr.  William 
Anderson,  by  Mr.  F.  J.  Bramwell — Table  of  Performance  of  Coiled  Pipes 
and  Boilers  in  Heating  and  Evaporating  Water  by  Steam,  with  Deductions,  461 

COOLING  OF  HOT  WATER  IN  PIPES: — Observations  of  M.  Darcy — Experiments 

by  Tredgold — Deductions, 469 

COOLING  OF  HOT  WORT  ON  METAL  PLATES  IN  AIR: — Results  of  Experiments 

at  Trueman's  Brewery, 47° 

COOLING  OF  HOT  WORT  BY  COLD  WATER  IN  METALLIC  REFRIGERATORS: — 

Table  of  Results  of  Performance,  and  Deductions,  .  .  .  .  •  471 


XIV  CONTENTS. 

PAGE 

CONDENSATION  OF  STEAM  IN  PIPES  EXPOSED  TO  AIR: — Experiments  by  Tred- 
gold,  and  by  M.  Burnat,  on  Pipes  with  various  Coverings,  with  Table — 
Experiments  by  Mr.  B.  G.  Nichol,  by  M.  Clement,  by  M.  Grouvelle — 
Condensation  of  Steam  in  a  Boiler  Exposed  in  Open  Air,  ....  472 

CONDENSATION  OF  VAPOURS  IN  PIPES  OR  TUBES  BY  WATER: — M.  Audenet's 
Experiments  on  Steam — Mr.  B.  G.  Nichol's  Experiments — Condensation  of 
other  Vapours,  ............  475 

WARMING  AND  VENTILATION: — Allowance  of  Air  for  Ventilation,      .        .         .        477 

VENTILATION  OF  MINES  BY  HEATED  COLUMNS  OF  AIR.— Furnace- Ventilation 

— Mr.  Mackworth's  Data,  ..........  479 

COOLING  ACTION  OF  WINDOW-GLASS: — Mr.  Hood's  Data,         ....     480 
HEATING  ROOMS  BY  HOT  WATER: — Mr.  Hood's  Estimates — Total  Quantity  of 
Air  to  be  Warmed  per  Minute — Table  of  the  Length  of  4-inch  Pipe  required 
to  Warm  any  Building — Boiler-power — French  Practice — Perkins'  System,  .     481 
HEATING  ROOMS  BY  STEAM: — Length  of  4-inch  Pipe  required — French  Practice,     486 
HEATING  BY  ORDINARY  OPEN  FIRES  AND  CHIMNEYS:— M.  Claudel's  Data,      .     488 
HEATING  BY  HOT  AIR  AND  STOVES: — Sylvester's  Cockle-Stove — French  Prac- 
tice— House- Stoves  placed  in  the  Rooms  to  be  Warmed — House- Stoves 

placed  outside  the  Rooms  to  be  Warmed,     . 488 

HEATING  OF  WATER  BY  STEAM  IN  DIRECT  CONTACT: — Mr.  D.  K.  Clark's 

Experiments,       ............     490 

EVAPORATION  (SPONTANEOUS)  IN  OPEN  AIR: — Dalton's  Experiments,  and  Deduc- 
tions— Rule  for  Spontaneous  Evaporation — Dr.  Pole's  Formula,  .         .         .     491 
DESICCATION  BY  DRY  WARM  AIR: — Design  of  a  Drying  Chamber — Results  of 
Experiments — Drying-house  for  Calico — Drying  Linen  and  Various  Stuffs— 
Drying  Stuffs  by  Contact  with  Heated  Metallic  Surfaces — Drying  Grain — 
Drying  Wood,    ............     493 

HEATING  OF  SOLIDS: — Cupola  Furnace— Plaster  Ovens — Metallurgical  Furnaces 

— Blast  Furnaces, 497 


STRENGTH   OF  MATERIALS. 

DEFINITIONS, 500 

WORK  OF  RESISTANCE  OF  MATERIAL, 501 

COEFFICIENT  OF  ELASTICITY, 503 

TRANSVERSE  STRENGTH  OF  HOMOGENEOUS  BEAMS, 503 

SYMMETRICAL  SOLID  BEAMS: — Investigation  and  Generalized  Formula,       .        .  503 
Formula  for  the  Transverse  Strength  of  Solid  Beams  of  Symmetrical  Section, 
without  Overhang,  and  Flanged  or  Hollow — For  Unsymmetrical  Flanged 

Beams — Neutral  Axis — Elastic  Strength,  ......  509 

FORMS  OF  BEAMS  OF  UNIFORM  STRENGTH:— Semi-Beams  Loaded  at  One  End 

— Uniformly  Loaded,       .         .         .         .         .         .         .         .         .         .  517 

Forms  of  Beams  of  Uniform  Strength,  Supported  at  Both  Ends — Under  a  Con- 
centrated Rolling  Load, 521 

SHEARING  STRESS  IN  BEAMS  AND  PLATE-GIRDERS 525 

DEFLECTION   OF  BEAMS  AND   GIRDERS  : — Investigation — Rectangular  Beams— 

Double-flanged — Uniform  Beams  Supported  at  Three  or  more  Points,  .         .  527 

TORSIONAL  STRENGTH  OF  SHAFTS: — Round— Hollow — Square— Deflection,       .  534 

STRENGTH  OF  TIMBER: — Results  of  Experiments, 537 

Transverse  Strength  of  Timber  of  Large  Scantling,  ......  542 

Elastic  Strength  and  Deflection  of  Timber: — Experiments  by  MM.  Chevandier 

and  Wertheim,  by  Mr.  Laslett,  by  Mr.  Kirkaldy,  by  Mr.  Barlow,    .         .  545 

Rules  for  the  Strength  and  Deflection  of  Timber, 548 

STRENGTH  OF  CAST  IRON:— Tensile  Strength  and  Compressive  Strength— Results 

of  Experiments, 553 

Shearing  Strength, 561 


CONTENTS.  XV 

Transverse  Strength: — Results  of  Experiments — Test  Bars — Transverse  Deflection 

and  Elastic  Strength, 561 

Torsional  Strength, 565 

STRENGTH  OP  WROUGHT  IRON: — Tensile  Strength,  &c. — Mr.  Kirkaldy's  Experi- 
ments,        .............  567 

Experiments  of  the  Steel  Committee  of  Civil  Engineers,  .....  579 

Hammered  Iron  Bars  (Swedish) — Krupp  and  Yorkshire  Plates — Prussian  Plates,  581 

Iron  Wire, 586 

Shearing  and  Punching  Strength,      .........  587 

Transverse  Strength — Deflection  and  Elastic  Strength, 588 

Torsional  Strength, 590 

STRENGTH  OF  STEEL: — Mr.   Kirkaldy's   Early  Experiments— Hematite  Steel — 

Krupp  Steel, 593 

Experiments  of  the  Steel  Committee, 596 

Experiments  at  H.M.  Gun  Factory,  Woolwich— Fagersta  Steel,  Mr.  Kirkaldy's 

Experiments,  in  seven  series, 604 

Siemens- Steel  Plates  and  Tyres — Mr.  Kirkaldy's  Experiments, .         .         .         .612 

Whitworth's  Fluid-compressed  Steel, 614 

Sir  Joseph  Whitworth's  Mode  of  Expressing  the  Value  of  Steel,        .         .         .615 

Chernoff  s  Experiments  on  Steel, .         .616 

Steel  Wire, 617 

Shearing  Strength  of  Steel,       .         .         .         .         .         .         .         .         .         .617 

Transverse  Strength  and  Deflection,          .         .         .         .         .         ..         .617 

Torsional  Strength, .         .         .         .619 

Strength  Relatively  to  the  Proportion  of  Constituent  Carbon,    .         .         .         .621 

Resistance  of  Steel  and  Iron  to  Explosive  Force,       ......  622 

RECAPITULATION  OF  DATA  ON  THE  DIRECT  STRENGTH  OF  IRON  AND  STEEL: — 
Tensile  and  Compressive  Strength  of  Cast  Iron,  Wrought  Iron,  and  Steel — 
Diagram  of  the  Relative  Elongation  of  Bars  of  Cast  Iron,  Wrought  Iron,  and 

Steel, 623 

WORKING  STRENGTH  OF  MATERIALS— FACTORS  OF  SAFETY: — Factors  of  Safety 
for  Cast  Iron,  Wrought  Iron,  Steel,  and  Timber — Load  on  Foundations, 

Mason-work — Ropes — Dead  Load — Live  Load, 625 

TENSILE  STRENGTH  OF  COPPER  AND  OTHER  METALS: — Tables  of  the  Strength 

of  Copper  and  its  Alloys:  Tin,  Lead,  Zinc,  Solder,     .....  626 

TENSILE  STRENGTH  OF  WIRE  OF  VARIOUS  METALS: — Tenacity  of  Metallic 

Wires  at  Various  Temperatures — Wires  of  Various  Metals,           .         .         .  628 

STRENGTH  OF  STONE,  BRICKS,  &c. : — Table  of  the  Tensile  Strength  of  Sandstones 
and  Grits,  Marbles,  Glass,  Mortar,  Plaster  of  Paris,  Portland  Cement,  Roman 
Cement,  Granites,  Whinstone,  Limestone,  Slates,  Bricks,  Brickwork  in 

Cement — Adhesion  of  Bricks,     .         .         .         .         .         .         .         .         .  629 

STRENGTH  OF  ELEMENTARY  CONSTRUCTIONS. 

RIVET-JOINTS  : — In  Iron  Plates, 633 

In  Steel  Plates, 640 

PILLARS  OR  COLUMNS  : — Compressive  Strength, 648 

CAST-IRON  FLANGED  BEAMS: — Transverse  Strength, 647 

Deflection  and  Elastic  Strength, 652 

WROUGHT-IRON    FLANGED   BEAMS   OR  JOISTS: — Solid  Wrought-iron  Joists — 

Transverse  Strength  and  Deflection, -653 

Rivetted  Wrought-iron  Joists, 657 

BUCKLED  IRON  PLATES, 660 


XVI  CONTENTS, 

PAGE 

RAILWAY  RAILS: — Transverse  Strength  of  Rails  of  Symmetrical  Section,     .         .661 

Rails  of  Unsymmetrical  Section,      . 665 

Deflection  of  Rails, 668 

STEEL  SPRINGS: — Laminated  and  Helical, 671 

ROPES: — Hemp  and  Wire, 673 

CHAINS, 677 

LEATHER  BELTING, 679 

BOLTS  AND  NUTS, 680 

SCREWED  STAY-BOLTS  AND  FLAT  SURFACES, 685 

HOLLOW  CYLINDERS— TUBES,  PIPES,  BOILERS,  &c.  .-—Resistance  to  Internal  or 

Bursting  Pressure — Transverse  Resistance,  .......  687 

Longitudinal  Resistance  to  Bursting  Pressure,  .......  692 

Wrought-iron  Tubes, 693 

Cast-iron  Pipe,         . 693 

Resistance  to  External  or  Collapsing  Pressure — Solid- drawn  Tubes — Large  Flue 

Tubes— Lead  Pipes, 694 

FRAMED   WORK — CRANES,   GIRDERS,   ROOFS,   £c.: — The  Triangle  the  Funda- 
mental Feature,  ............  697 

Warren- Girder  Loaded  at  the  Middle,  and  at  an  Intermediate  Point — Uniformly 

Loaded — Rolling  Load,  ,         =         .,......  699 

Parallel  Lattice- Girder, 708 

Parallel  Strut-Girder, 708 

Roofs, 713 

WORK,   OR  LABOUR. 

UNITS  OF  WORK  OR  LABOUR: — Horse-power — Mechanical  Equivalent  of  Heat 

— Labour  of  Men,      ............  718 

Labour  of  Horses — Work  of  Animals  Carrying  Loads, 720 

FRICTION   OF  SOLID   BODIES. 

LAWS  OF  FRICTION: — Friction  of  Journals — Friction  of  Flat  Surfaces,          .         .  722 

FRICTION  ON  RAILS: — M.  Poiree's  Experiments, 724 

WORK  AND  HORSE-POWER  ABSORBED  BY  FRICTION: — Formulas,      .        .        .  725 

MILL-GEARING. 

TOOTHED    GEAR:— Pitch  of  the  Teeth  of  Wheels— Spur  Fly-wheels— Toothed 

Wheels  for  Mill  work — Rules,      .         . 727 

Form.of  the  Teeth  of  Wheels, .         .         .731 

Proportions  of  the  Teeth  of  Wheels, 734 

Transverse  Strength  of  the  Teeth  of  Wheels — Working  Strength,     .         .         .  735 

Breadth  of  the  Teeth  of  Wheels, 737 

Horse-power  Transmitted  by  Toothed  Wheels, 737 

Weight  of  Toothed  Wheels, 739 

FRICTIONAL  WHEEL-GEARING, 741 

BELT-PULLEYS  AND  BELTS:— Tensile  Strength, 742 

Horse-power  Transmitted  by  Belts, 743 

Adhesion  and  Power  of  Belts — Examples  of  very  wide  Belts,    ....  744 

India-rubber  Belting, 750 

Weight  of  Belt- Pulleys, 750 

ROPE  GEARING: — Transmission  of  Power  by  Ropes  to  Great  Distances,         .         .  753 

Cotton  Ropes, 755 


CONTENTS.  Xvii 

SHAFTING: — Transverse  Deflection  of  Shafts, 756 

Ultimate  Torsional  Strength  of  Round  Shafts, 758 

Torsional  Deflection  of  Round  Shafts, 759 

Power  Transmitted  by  Shafting, 760 

Weight  of  Shafting,          .         . 761 

Strength  and  Horse-power  of  Round  Wrought-iron  Shafting,    ....  762 

Frictional  Resistance  of  Shafting, 763 

Ordinary  Data  for  the  Resistance  of  Shafting,  ...  ...  763 

Journals  of  Shafts, 766 

EVAPORATIVE  PERFORMANCE   OF   STEAM-BOILERS. 

NORMAL  STANDARDS, 768 

HEATING  POWER  OF  FUELS:— Table  of  Heating  Power, 769 

EVAPORATIVE  PERFORMANCE  OF  STATIONARY  AND  MARINE  STEAM-BOILERS, 

WITH  COAL: — Surplus  Air  Admitted  to  the  Furnace,  ....     770 

Experiments  on  the  Evaporative  Power  of  British   Coals,  by  Delabeche  and 

Playfair, 770 

Evaporative  Performance  of   Lancashire  Stationary  Boilers  at  Wigan — With 

Economizer   and  Without   Economizer  —  Water-tubes  —  Temperature  of 

the  Products  of   Combustion,  and  of  the  Feed-water — Trials  of  D.  K. 

Clark's  Steam-Induction  Apparatus — Of  Vicars'  Self- feeding  Fire-grate,    .     771 
Evaporative  Performance  of  South  Lancashire  and  Cheshire  Coals  in  a  Marine 

Boiler,  at  Wigan, 781 

Trials  of  Newcastle  and  Welsh  Coals  in  the  Wigan  Marine  Boiler,  .  .  .  784 
Evaporative  Performance  of  Newcastle  Coals  in  a  Marine  Boiler,  at  Newcastle- 

on-Tyne, 785 

Trials  of  Newcastle  and  Welsh  Coals  in  the  Marine  Boiler  at  Newcastle,  for  the 

Board  of  Admiralty,        ..........     787 

Trials  of  Welsh  and  Newcastle  Coals  in  a  Marine  Boiler  at  Keyham  Factory,  .  790 
Evaporative  Performance  of  American  Coals  in  a  Stationary  Boiler,  .  .  -791 
Evaporative  Performance  of  an  Experimental  Marine  Boiler,  Navy  Yard,  New 

York, 795 

Evaporative  Performance  of  Stationary  Boilers  in  France,          ....     796 

Evaporative  Performance  of  Locomotive  Boilers,       ......     798 

Evaporative  Performance  of  Portable-Engine  Boilers,        .         .  .         .     801 

RELATIONS  OF  GRATE-AREA  AND  HEATING  SURFACE  TO  EVAPORATIVE  PER- 
FORMANCE:— Mr.  Graham's  Experiments — Experiments  by  Messrs.  Woods 
and  Dewrance — Experimental  Deductions  of  M.  Paul  Havrez,  .  .  .  802 

FORMULAS  FOR  THE  RELATIONS  OF  GRATE- AREA,  HEATING  SURFACE,  WATER, 

AND  FUEL: — General  Equations, .     804 

Formulas  for  the  Experimental  Boilers,     .         .         .         .         .         .         ...     807 

General  Formulas  for  Practical  Use,          .         .         .         .         .         .         .         .819 

Table  of  the  Equivalent  Weights  of  Best  Coal  and  Inferior  Fuels,     .         .         .     820 

STEAM-ENGINE. 

ACTION  OF  STEAM  IN  A  SINGLE  CYLINDER:— The  Work  of  Steam  by  Expan- 
sion— Clearance — Formulas  for  the  Work  of  Steam — Initial  Pressure  in  the 
Cylinder — Average  Total  Pressure  in  the  Cylinder — Average  Effective  Pres- 
sure— Period  of  Admission  and  the  Actual  Ratio  of  Expansion — Relative 
Performance  of  Equal  Weights  of  Steam  Worked  Expansively — Proportional 
Work  Done  by  Admission  and  by  Expansion — Influence  of  Clearance  in 
Reducing  the  Performance  of  Steam,  ........     822 

Table  of  Ratios  of  Expansion  of  Steam,  with  Relative  Periods  of  Admission, 

/YY,       Pressures,  and  Total  Performance, .     835 

b 


XV111  CONTENTS. 

Total  Work  Done  by  One  Pound  of  Steam  Expanded  in  a  Cylinder,  .  .  838 
Consumption  of  Steam  Worked  Expansively  per  Horse-power  of  Net  Work 

per  Hour,        ............     840 

Table  of  the  Work  Done  by  One  Pound  of  Steam  of  loo-lbs.  Pressure  per 

Square  Inch,  ............     841 

Net  Cylinder-Capacity  Relative  to  the  Steam  Expanded  and  Work  Done  in 

One  Stroke, 843 

Table  of  ditto,  . 844 

COMPOUND  STEAM-ENGINE: — Woolf  Engine — Receiver-Engine — Ideal  Diagrams, 
without  Clearance — Work  of  Steam  as  Affected  by  Intermediate  Expansion 
— Intermediate  Expansion — Work,  with  Clearance— Comparative  Work  of 
Steam  in  the  Woolf  Engine  and  the  Receiver- Engine,  .....     849 
Formulas  and  Rules  for  Calculating  the  Expansion  and  the  Work  of  Steam,      .     869 

COMPRESSION  OF  STEAM  IN  THE  CYLINDER, 878 

PRACTICE  OF  THE  EXPANSIVE  WORKING  OF  STEAM: — Actual  Performance- 
Data — Deductions — Conclusions,  ........  879 

FLOW   OF  AIR  AND   OTHER  GASES. 

Discharge  of  Air  through  Orifices — Anemometer,  .         .         .         .         .         .         .891 

Outflow  of  Steam  through  an  Orifice,    .........     893 

Flow  of  Air  through  Pipes  and  Other  Conduits,     .......     894 

Resistance  of  Air  to  the  Motion  of  Flat- Surfaces,  .......     897 

Ascension  of  Air  by  Difference  of  Temperature,    .......     897 

WORK   OF   DRY   AIR   OR   OTHER  GAS,    COMPRESSED   OR 
EXPANDED. 

WORK  AT  CONSTANT  TEMPERATURES: — Isothermal  Compression  or  Expansion,       899 
WORK  IN  A  NON-CONDUCTING  CYLINDER,  ADIABATICALLY,     .        .        .        .901 

EFFICIENCY  OF  COMPRESSED-AIR  ENGINES, 909 

Compression  and  Expansion  of  Moist  Air,        .         .         .         .         .         .         .912 

AIR  MACHINERY. 

MACHINERY  FOR  COMPRESSING  AIR  AND  FOR  WORKING  BY  COMPRESSED 
AIR: — Compression  of  Air  by  Water  at  Mont  Cenis  Tunnel  Works — By 
Direct-action  Steam-pumps — Compressed-air  Machinery  at  Powell  Duffryn 

Collieries,  .............  915 

HOT- AIR  ENGINES: — Rider's — Belou's, 917 

GAS-ENGINES: — Lenoir's— Otto  &  Langen's — Otto's — Clerk's,       ....  918 

GASEOUS  FUEL  : — Wilson  Gas  Producer — Dowson  Generator  Gas,         .         .         .  922 

FANS  OR  VENTILATORS  : — Common  Centrifugal  Fan — Mine- Ventilators,       .         .  924 

BLOWING  ENGINES,     .......        v        ....  926 

ROOT'S  ROTARY  PRESSURE-BLOWERS, 927 

FLOW  OF  WATER. 

FLOW  OF  WATER  THROUGH  ORIFICES: — Formulas — Mr.  Bateman's  Experi- 
ments, .............  929 

Mr.  Brownlee's  Experiments  with  a  Submerged  Nozzle, 931 

FLOW  OF  WATER  OVER  WASTE-BOARDS,  WEIRS,  &c., 932 

FLOW  OF  WATER  IN  CHANNELS,  PIPES,  AND  RIVERS,     .....  932 

CAST-IRON  WATER  PIPES, 934 

CAST-IRON  GAS  PIPES, 936 


CONTENTS.  XIX 


WATER-WHEELS. 

PAGE 

WHEELS  ON  A  HORIZONTAL  Axis: — Undershot- Wheels — Paddle- Wheels — Breast- 

Wheels— Overshot- Wheels, 937 

WHEELS  ON  A  VERTICAL  Axis: — Tub— Whitelaw's  Water-mill — Turbines — 

Tangential  Wheels, 939 

MACHINES   FOR  RAISING  WATER. 

PUMPS: — Reciprocating  Pumps — Centrifugal  Pumps — Chain  Pump — Noria,  .      944,  968 

Water-works  Pumping  Engines,        .........  948 

Hydraulic  Rams, 949 

HYDRAULIC  MOTORS. 

HYDRAULIC  PRESS, "  .    " 95° 

ARMSTRONG'S  HYDRAULIC  MACHINES, 950 

FRICTIONAL  RESISTANCES. 

STEAM  ENGINES, 951 

TOOLS: — Shearing  Machines— Plate-bending  Machines — Circular  Saws,          .         -951 

Work  of  Ordinary  Cutting  Tools,  in  Metal,      .......  952 

Screw-cutting  Machines — Wood-cutting  Machines — Grindstones,        .         .         .  954 

COLLIERY  WINDING  ENGINES, 956 

WAGGONS  IN  COAL  PITS,   ...........  956 

MACHINERY  OF  FLAX  MILLS: — M.  Cornut's  Experiments, 957 

Horse-power  Required, 959 

MACHINERY  OF  WOOLLEN  MILLS:— Dr.  Hartig's  Experiments,  ....  959 

MACHINERY  FOR  THE  CONVEYANCE  OF  GRAIN,         ......  960 

TRACTION  ON  COMMON  ROADS: — M.  Dupuit's  Experiments — M.  Debauve's  De- 
ductions— M.  Tresca's  Experiments,    ........  961 

CARTS  AND  WAGGONS  ON  ROADS  AND  ON  FIELDS, 962 

RESISTANCE  ON  RAILWAYS, 965 

RESISTANCE  ON  STREET  TRAMWAYS, 966 

APPENDIX. 

DR.  SIEMENS'  WATER  PYROMETER, 967 

ATMOSPHERIC  HAMMERS, 967 

BERNAYS'  CENTRIFUGAL  PUMPS,         .        . 968 

STEAM-VACUUM  PUMP, 969 

INDEX, 971 


AUTHORITIES   CONSULTED   OR   QUOTED. 


American,  United,  Railway  Master  Car- 
Builders'  Association,  Standard  Sizes  of 
Bolts  and  Nuts  by,  683. 

American  Society  of  Civil  Engineers,  Journal 
of: — Mr.  J.  F.  Flagg,  on  Steam-vacuum 
Pumps,  969. 

Anderson,  Dr.,  on  the  Strength  of  Cast  Iron, 

555- 
Anderson,  William,    on   Heating  Water  by 

Steam,  465,  466,  468  ;  Translation  of  Cher- 

noff 's  Paper  on  Steel,  616. 
Annales  des  Mines: — M.  Krest,  on  the  Slip 

of  Belts,  742. 
Annales  des  Fonts  et  Chaussdes: — M.  Hirn's 

Rope  Transmitter  of  Power,  754. 
Annales  du  Ge"nie  Civile: — M.  Paul  Havrez, 

on  Heating  Surface  of  Locomotives,  803. 
Annals  of  Philosophy  : — Mr.  Dunlop,  on  Tor- 

sional  Strength  of  Cast  Iron,  565. 
Annua  ire  de  FA  ssociation  des  Inge"n  ieurs  sortis 

de  V£cole  de  Lie"ge: — Rivetted  Joints,  641. 
Armengaud,     French    Standard     Bolts    and 

Nuts,  by,  683. 
Armstrong,  Sir  Wm.,  on  Evaporative  Power 

of  Coals,  785;   his  Hydraulic   Machinery, 

95°- 

Arson,  Anemometer  by,  892. 
Ashby  &  Co.,  Work  of  Steam  in  Portable 

Engine  by,  883. 
Audenet,  on  Surface-Condensers,  475. 


B 


Baker,   B.,  on  the  Strength  of  Beams,  512; 

of  Oak,  544,  549;  of  Columns,  645,  646; 

of  Rails,  662,  666. 
Barlow,  Peter,  on  Strength  of  Timber,  547 ; 

of  Cast  Iron,  561 ;  of  Wrought  Iron,  567, 

588,  590  ;  of  Iron  Wire,  586. 
Barlow,  W.  H.,  on  the  "  Resistance  of  Flex- 
ure," 507. 
Barnaby,  Mr.,  on  Strength  of  Punched  Steel 

Plate,  642. 
Barrow  Hematite  Steel  Company,  Strength 

of  Steel  made  by,  594,  618,  619,  620,  621. 
Bateman,  J.  F.,  on  Flow  of  Water  through 

Submerged  Openings,  930;   his  Cast-Iron 

Pipes,  934. 


Baudrimont,  on  Strength  of  Metallic  Wires, 
628. 

Beardmore,  on  the  Work  of  Horses,  720 ;  on 
Limits  of  Velocity  at  the  Bottom  of  a 
Channel,  934. 

Beaufoy,  Colonel,  on  Resistance  of  Air,  897. 

Bell,  J.  Lothian,  on  the  Heat  in  Blast  Fur- 
naces, 498. 

Berkley,  George,  on  the  Strength  of  Cast- 
iron  Beams,  647-650. 

Berkley,  J.,  Specific  Gravity  of  Indian  Woods, 
by,  209. 

Bernays,  Joseph,  on  Centrifugal  Pumps,  968. 

Bertram,  W.,  on  Rivetted  Joints,  634-637. 

Borsig,  Herr,  Strength  of  Wrought-Iron 
Plates,  586. 

Box,  Thomas,  on  the  Load  on  Journals,  766 ; 
Thickness  of  Gas  Pipes,  by,  936. 

Boyden,  Outflow  Turbine  by,  940. 

Bradford,  W.  A.,  on  Otto  and  Langen's  Gas- 
Engine,  924. 

Bramwell,  F.  J.,  on  Heating  Water  by  Steam, 
467,  468 ;  on  the  Strength  of  Cast  Iron, 
556  ;  on  Portable  Steam  Engines,  801,  883, 
886 ;  on  the  Expansive  Working  of  Steam, 
889. 

Brereton,  R.  P.,  on  Strength  of  Timber  Piles, 
646. 

Briggs,  Blowing  Engine  by,  927. 

British  Associatiom,  Transactions  of: — F.  W. 
Shields,  on  Strength  of  Cast-Iron  Columns, 

645. 
Brown  &  May,  Work  of  Steam  in  Portable 

Engine  by,  882. 
Brownlee,  J.,  on  Saturated  Steam,  382;  on 

the  Outflow  of  Steam,  893 ;  Flow  of  Water 

through  a  Submerged  Nozzle,  931. 
Bruce,  G.  B.,  on  the  Work  of  a  Labourer, 

719. 
Brunei,  on  the  Strength  of  Rivetted  Joints, 

638 ;  and  of  Bolts  and  Nuts,  680. 
Buchanan,  W.  M.,  on  Saturated  Steam,  379. 
Buckle,  W.,  on  Fans,  924. 
Buel,  R.  H.,  on  the  Slip  of  Belts,  742. 
Bulletin  de  la  Socidtd  Industriette  de  Mul- 

house: — M.   Leloutre  on  Steam   Engines, 

886. 
Burnat,  on  Condensation  of  Steam  in  Pipes, 

472,  474. 

Bury,  Wm.,  on  Strength  of  Flat  Stayed  Sur- 
faces, 686. 

b  2 


XX11 


AUTHORITIES  CONSULTED   OR   QUOTED. 


Cameron,  Dr. ,  Analysis  of  Peat  by,  454. 

Charie-Marsaines,  on  Flemish  Horses,  964. 

Chenot  Aine",  Atmospheric  Hammer  by,  967. 

Chernoft,  on  Steel,  616. 

Chevandier,  on  Composition  of  Wood,  440; 
on  its  Weight  and  Bulk,  442,  443. 

Chevandier  &  Wertheim,  on  Strength  of  Tim- 
ber, 538,  545,  546,  549. 

Clark,  D.  K.,  on  Properties  of  Saturated 
Steam,  387;  on  Locomotive  Boilers,  798; 
on  the  Work  of  Steam,  879,  880,  884 ;  on 
Resistance  on  Railways,  965 ;  Tramways,  966. 

Clark,  Edwin,  on  the  Strength  of  Beams,  512; 
of  Red  Pine,  543,  544,  549;  of  Cast  Iron, 
562 ;  of  Bar  Iron,  570,  588,  590,  623. 

Clark,  Latimer,  on  Wire  Gauges,  130. 

Claudel,  on  Fuels  and  Woods,  by,  207,  211, 
212 ;  tints  of  Heated  Iron,  328  ;  on  Heating 
Factories,  486 ;  on  Heating  Rooms,  488, 
489;  on  Belts,  743,  746;  on  Blowing  En- 
gines, 927;  on  Pumps,  944. 

Clement,  on  Transmission  of  Heat,  462,  468 ; 
on  Condensation  of  Steam  in  Pipes,  474; 
on  Drying  Stuffs,  496  ;  on  the  Heat  to  Melt 
Iron,  497. 

Cochrane,  J.,  on  Strength  of  Perforated  Bar 
Iron,  633. 

Cockerill,  John,  Blowing  Engines  by,  927. 

Colliery  Guardian : — Mr.  Mackworth  on  Ven- 
tilation of  Mines,  480. 

Conservatoire  des  Arts  et  Matters,  Annales 
du: — Hot-Air  Engines  by  Laubereau,  and 
by  Belou,  917-919 ;  Gas- Engines  by  Lenoir, 
920;  by  Hugon,  921;  by  Otto  &  Langen, 
923- 

Cooper,  J.  H.,  on  Very  Wide  Belts,  747,  749. 

Cornet,  on  the  Work  of  a  Labourer  in  France, 
720. 

Cornut,  E.,  on  Mill-Shafting,  766  ;  on  Machin- 
ery of  Flax-Mills,  957 ;  on  Flow  of  Air  in 
Pipes,  896. 

Cotterill,  J.  H.,  on  Work  of  Compression  of 
Air,  903. 

Cowper,  E.  A.,  Compound  Engine  by,  889. 

Craddock,  Thomas,  on  Cooling  through 
Plates,  461. 

Crighton  &  Co.,  on  Drying  Grain,  496. 

Crookewitt,  on  Specific  Gravities  of  Alloys, 
200. 

Crossley,  F.  W.,  on  Otto  &  Langen's  Gas- 
Engines,  923. 

Cubitt,  Mr.,  on  Strength  of  Cast-Iron  Beams, 
649. 

D 

Daglish,    G.  H.,  on   Resistance  of  Colliery 

Winding  Engines,  956. 
Dalton,  Dr.,  on  "  Spontaneous  "  Evaporation 

of  Water,  491. 


Daniel,  W.,  on  Ventilation  of  Mines,  925. 
Danvers,  F.  C.,  on  Coal  Economy,  410. 
Darcy,  on  Cooling  Hot  Water  in  Pipes,  469. 
D'Aubuisson,  on   Flow  of  Compressed  Air, 

896  ;  on  Hydraulic  Rams,  949. 
Davey,  Paxman,  &  Co.,  Work  of  Steam  in 

Portable  Engine  by,  883. 
Davies,   Thomas,    on   Strength  of  Rivetted 

Joists,  658. 
Davison,    R.,    on    Resistance    of   Shafting, 

766;  Duty  of  Pumps  by,  944;  on  Resist- 
ance of  Grain  Machinery,  961. 
Day,  Summers,  &  Co.,  Work  of  Steam   in 

Marine  Engines  by,  882. 
Debauve,  on  Resistance  on  Common  Roads, 

961. 
Delabeche  &  Playfair,  on  British  and  Foreign 

Coals,  206,  413,  416,  770. 
Despretz,  on  Conducting  Powers  of  Bodies, 

33*' 
Deville,    Sainte-Claire,    on    Composition    of 

Petroleum  and  other  Oils,  456,  457. 
Dewrance,  John,  on  the  Heating  Surface  of  a 

Locomotive,  803. 
Donkin,  Bryan,  &  Co.,  Work  of  Steam  in 

Stationary  Engines  by,  882. 
Downing,  on  Flow  of  Water  in  Pipes,  933, 

934- 

Dunlop,  on  Strength  of  Cast  Iron,  565. 
Dupuit,  on  Resistance  on  Common  Roads, 

961. 

Durie,  James,  on  Rope-Gearing,  753. 
Duvoir,  Rene,  Drying  House  by,  495. 


Eastons  &  Anderson,  on  Portable  Steam 
Engines,  801 ;  on  Rider's  Hot-Air  Engine, 
917 ;  on  Resistance  of  Waggons,  962. 

Elder,  John,  &  Co.,  on  the  Strength  of  Boilers, 
638,  693 ;  Work  of  Steam  in  Marine  Engine 
by,  882. 

Emery,  on  American  Marine  Engines,  884. 

Engineer,  The: — Crighton  &  Co.  on  Drying 
Grain,  496  ;  Mr.  W.  S.  Hall  on  the  Strength 
of  Rivetted  Joints,  641 ;  Messrs.  Woods  & 
Dewrance  on  Locomotive  Boilers,  803  ;  Mr. 
C.  L.  Hett  on  Hydraulic  Rams,  949. 

Engineering: — on  Heating  Water  by  Steam, 
464 ;  on  Cooling  Wort,  470,  471 ;  Mr.  B. 
G.  Nichol  on  Surface  Condensation,  476 ; 
Mr.  G.  Graham  Smith  on  Strength  of 
Timber,  544;  Factor  of  Safety  for  Wrought 
Iron,  by  Roebling,  625  ;  Mr.  W.  S.  Hall 
on  the  Strength  of  Rivetted  Joints,  641 ; 
Mr.  John  Mason  on  Strength  of  Untanned 
Leather  Belts,  680;  Mr.  Phillips  on  Strength 
of  Flat  Plates,  686;  Mr.  Bury  on  the  Strength 
of  Flat  Stayed  Surfaces,  686 ;  Messrs.  John 
Elder  &  Co.  on  the  Strength  of  Boilers, 
638,  693  ;  Mr.  J.  Durie  on  Rope  Gearing, 


AUTHORITIES  CONSULTED  OR  QUOTED. 


XXlll 


753;  Dr.  Hartig  on  Resistance  of  Tools,    j 

951 ;  Resistance  of  Waggons,  by  Messrs. 

Eastons  &  Anderson,  962. 
English  Mechanic: — Mr.  W.  A.  Bradford  on 

Otto  &  Langen's  Gas-Engine,  924. 
Evrard,  A.,  on  the  Work  of  Animals,  720. 


Fagersta  Steel  Works,  Strength  of  Steel  made 
at,  604,  618,  619,  620,  621. 

Fairbairn,  Sir  William,  on  Hot-Blast  Iron, 
556 ;  on  the  Strength  of  Cast  Iron,  557 ;  on 
the  Strength  of  Wrought  Iron,  567-569; 
of  Rivetted  Joints,  633 ;  of  Screwed  Stay- 
Bolts  and  Flat  Stayed  Plates,  685  ;  on  the 
Proportions  of  Spur  Wheels,  729,  734,  737  ; 
on  the  Load  on  Journals,  766,  767;  on 
Water  Wheels,  938. 

Fairbairn  &  Tate,  on  the  Expansion  of  Steam, 

383- 
Fairweather,  James  C. ,  on  Resistance  of  Air, 

897. 
Faraday,  Dr.,  on  the  Liquefaction  of  Gases, 

372. 
Favre  &  Silbermann,  on  the  Heating  Powers 

of  Combustibles,  404. 

Field,  Joshua,  on  the  Work  of  Labourers,  719. 
Fincham,  on  Strength  of  Timber,  542,  543, 

549- 

Flagg,  J.  F.,  on  Steam- vacuum  Pumps,  969. 

Fletcher,  L.  E.,  on  the  Strength  of  a  Boiler, 
638,  693  ;  his  Reports,  696  ;  his  Report  on 
Boiler  and  Smoke  Prevention  Trials,  771- 
784. 

Fowke,  Captain,  on  Colonial  Woods,  209. 

Fowler,  G.,  on  Resistance  of  Waggons  in 
Coal  Pits,  956. 

Fowler,  John,  Strength  of  Steel  Rails  de- 
signed by,  666,  670. 

Fowler,  J.,  &  Co.,  Compressed-air  Machinery 
by,  916. 

Fox,  Head  &  Co.,  on  Condensation  of  Steam 
in  a  Boiler,  475. 

Francis,  J.  B.,  on  a  Swain  Turbine,  943. 

Franklin  Institute,  Journal  of: — the  Shear- 
ing Resistance  of  Bar  Iron,  by  Chief 
Engineer  W.  H.  Shock,  588 ;  Mr.  R.  H. 
Buel  on  Belts,  742 ;  Mr.  H.  R.  Towne  on 
Belts,  742,  745;  Mr.  J.  H.  Cooper  on 
Belts,  747 ;  Mr.  S.  Webber  on  Mill  Shaft- 
ing, 763,  764;  Mr.  Emery  on  American 
Marine  Engines,  884;  Mr.  Briggs  on 
Blowing  Engines,  927;  Mr.  J.  B.  Francis 
on  a  Swain  Turbine,  943 ;  Mr.  E.  D. 
Leavitt's  Pumping  Engines,  948. 


Gammelbo  &  Co.,  Hammered  Bars  made  by, 
Strength  of,  581. 


Gaudillot,  on  Heating  Apparatus,  486. 
Gay-Lussac,  on  Cold  by  Evaporation,  376. 
Glynn,  Mr.,  on  Strength  of  Ropes,  673 ;  on 

the  Work  of  a  Labourer,  718. 
Gooch,  Sir  Daniel,  on  Consumption  of  Water 

by  the  "Great  Britain"  Locomotive,  884. 
Gordon,  L.  D.  B.,  on  Strength  of  Columns, 

645- 
Graham,  John,  on  Heating  Water,  461 ;  on 

Heating  Surface,  802. 
Grant,  on  Strength  of  Cements,  &c.,  630. 
Greaves,  on  Pumping  Engines,  948. 
Grouvelle,  on  Condensation  of  Steam  in  Pipes, 

474;  on  Heating  Factories,  486,  487. 


H 


Hackney,  W.,  on  Anthracitic  Coke,  432. 

Haines,  R.,  on  Indian  Coals,  423. 

Hall,  W.  S.,  on  the  Strength  of  Rivetted 
Joints,  641. 

Harcourt,  Vernon,  on  Analysis  of  Coal-Gas, 
458. 

Harmegnies,  Dumont,  &  Co. ,  on  French  Wire 
Ropes,  677. 

Hartig,  Dr.,  on  Driving  Belts,  743;  on  Re- 
sistance of  Tools,  951 ;  on  Resistance  of 
Machinery  of  Woollen  Mills,  959. 

Havrez,  P.,  on  Heating  Water  by  Steam, 
464,  468 ;  on  Heating  Surface  of  Loco- 
motives, 803. 

Hawksley,  Thomas,  on  Flow  of  Air  through 
Pipes,  894 ;  on  Velocity  of  Air  in  Up-cast 
Shaft,  897 ;  on  Flow  of  Water  in  Pipes, 
933  ;  on  Thickness  of  Water  Pipes,  935. 

Hett,  C.  L.,  on  Hydraulic  Rams,  949. 

Hick,  John,  M.P.,  on  Friction  of  Leather 
Collars,  950. 

Hirn,  on  Work  of  Expanded  Steam  in  Sta- 
tionary Engines,  886. 

Hodgkinson,  on  the  Strength  of  Cast  Iron, 
553-555-  558,  559-  563.  564:  of  Columns, 
643,  646 ;  of  Cast-Iron  Flanged  Beams, 
647-650. 

Holtzapffel,  his  Wire-Gauges,  131,  132,  134. 

Hood,  on  Warming  and  Ventilation,  477-485. 

Hopkinson,  on  the  Performance  of  a  Corliss 
Engine,  88 1. 

Hunt,  R.,  on  Combustion  of  Coal,  770. 

Hutton,  Dr.,  Law  of  Resistance  of  Air  by, 

i. 

Institute  of  Naval  Architects,  Transactions 
of  the: — Strength  of  Rivet  Joints  of  Steel 
Plates,  642. 

Institution  of  Civil  Engineers,  Proceedings 
of: — Mr.  Wm.  Anderson  on  Heating  Water 
by  Steam,  465;  M.  Burnat  on  Condensation 
of  Steam  in  Pipes,  472;  Dr.  Pole  on  Spon- 
taneous Evaporation,  493 ;  Regenerative 
Hot-Blast  Stoves,  556 ;  Mr.  Bramwell  on 


XXIV 


AUTHORITIES   CONSULTED   OR   QUOTED. 


Strength  of  Cast  Iron,  556;  Mr.  Grant  on  | 
the  Strength  of  Cements,  &c.,  630;  Mr.  J. 
Cochrane  on  the  Strength  of  Punched  Bar 
Iron,  633 ;  Mr.  R.  Price  Williams  on 
Strength  of  Rails,  662 ;  Mr.  J.  T.  Smith 
on  the  Strength  of  Bessemer  Steel  Rails, 
664;  Mr.  R.  Davison  on  Resistance  of 
Shafting,  766 ;  Evaporative  Performance  of 
Steam  Boilers  in  France,  796  ;  Composition 
of  Coals  and  Lignites,  797;  M.  Paul  Havrez 
on  Heating  Surface  of  Locomotives,  803 ; 
Mr.  Emery  on  American  Marine  Engines, 
884;  Mr.  Hawksley  on  Flow  of  Air  through 
Pipes,  894 ;  and  on  Velocity  of  Air  in  Up- 
cast Shaft,  897;  M.  Piccard  on  the  Work 
of  Compressed  Air,  911;  Mr.  J.  B.  Francis' 
trial  of  a  Swain  Turbine,  943 ;  Mr.  R. 
Davison  on  Duty  of  Pumps,  944 ;  Hon.  R. 

C.  Parsons  on  Centrifugal  Pumps,  947 ;  Mr. 
Henry  Robinson  on  Armstrong's  Hydraulic 
Machines,  950. 

Institution  of  Engineers  and  Ship-Builders  in 
Scotland,  Transactions  of  the: — on  Strength 
of  Helical  Springs,  672 ;  Report  on  Safety 
Valves,  893;  Mr.  J.  Brownlee's  Experi- 
ments on  Flow  of  Water,  931. 

Institution  of  Mechanical  Engineers,  Pro- 
ceedings of: — Mr.  C.  Little  on  the  Shearing 
and  Punching  Strength  of  Wrought  Iron, 
587 ;  Mr.  Vickers  on  the  Strength  of  Steel, 
621;  Mr.  W.  R.  Browne's  paper  on  Rivetted 
Joints,  637;  Mr.  Robertson  on  Grooved 
Frictional  Gearing,  741 ;  Mr.  H.  M.  Mor- 
rison on  Hirn's  Rope  Transmitter,  755; 
Mr.  Ramsbottom  on  Cotton-Rope  Trans- 
mitter, 755;  Mr.  Westmacott  and  Mr.  B. 
Walker  on  Resistance  of  Shafting,  766 ;  Mr. 

D.  K.  Clark  on  the  Expansive  Working  of 
Steam  in  Locomotives,  879,  880;   Data  of 
the  Practical  Performance  of  Steam,  880; 
Mr.  F.  J.  Bramwell  on  Economy  of  Fuel 
in  Steam  Navigation,  889 ;  Compressed-Air 
Machinery  by  Messrs.  John  Fowler  &  Co., 
916 ;  Wenham's  Hot-Air  Engine,  919 ;  Mr. 
F.  W.  Crossley  on  Otto  and  Langen's  Gas- 
Engine,   923 ;    Mr.  Buckle  on  Fans,  924 ; 
Mr.J.S.E.  Swindell  on  Ventilation  of  Mines, 
925;  Mr.  W.  Daniel  on  Ventilation  of  Mines, 
925 ;  Mr.  A.  C.  Hill  on  Blowing  Engines, 
927 ;  Mr.  J.  F.  Bateman's  Experiments  on 
Flow  of  Water,  930  ;  Mr.  David  Thomson 
on    Pumping   Engines,    948 ;     Mr.  G.   H. 
Daglish  on  Winding  Engines,  956 ;  Mr.  G. 
Fowler  on  Resistance  of  Waggons  in  Coal 
Pits,  956 ;  Mr.  Westmacott  on  Corn- Ware- 
housing Machinery,  961. 

Iron  and  Steel  Institute,  Journal  of  the: — 
Mr.  J.  Lothian  Bell  on  the  Cleveland  Blast 
Furnaces,  498. 

Isherwood,  Trials  of  Evaporative  Performance 
of  a  Marine  Boiler,  795. 


J 
James,  Captain,  on  the  Strength  of  Cast  Iron, 

555- 

Jardine,  Mr.,  on  the  Strength  of  Lead  Pipes, 

696. 
Johnson,  Professor  W.  R.,  on  American  Coals, 

418,  770,  79!-79S- 
Joule,  Dr.,  Mechanical  Equivalent  of  Heat, 

K 

Kane,  Sir  Robert,  on  Peat,  453. 

Kennedy,  Colonel  J.  P.,  on  Weight  and 
Volume  of  Goods  carried  on  Railways,  213. 

Kirkaldy,  David,  on  Compressive  Strength 
of  Timber,  546,  547,  647 ;  on  the  Tensile 
Strength  of  Wrought  Iron  and  Steel,  571- 
578  ;  of  Swedish  Hammered  Bars,  581,  590; 
of  Krupp  and  of  Yorkshire  Iron  Plates, 
583-586  ;  of  Borsig's  Iron  Plates,  586 ;  Ten- 
sile Strength  of  Bar  Steel,  593,  594 ;  of  He- 
matite Steel,  594 ;  of  Krupp  Steel,  595 ;  of 
Steel  Bars,  for  the  Steel  Committee,  597- 
600;  of  Fagersta  Steel,  604-611 ;  of  Siemens- 
Steel  Plates  and  Tyres,  612-614;  on  Shear- 
ing Strength  of  Steel,  617 ;  on  Strength  of 
Phosphor-Bronze,  628,  629;  of  Wires,  629; 
of  Rolled  Wrought-iron  Joists,  654;  of  Rails, 
662,  663,  666-668;  of  Ropes,  674;  of  Belt- 
ing, 680;  of  Plates  of  a  Marine  Boiler,  694. 

Krest,  on  the  Slip  of  Belts,  742. 

Krupp,  Herr,  Strength  of  Wrought-iron  Plates 
made  by,  583 ;  of  his  Cast  Steel,  595,  618- 
621. 

L 

Landore  Siemens-Steel  Company,  Strength  of 
Steel  Plates  and  Tyres  made  by,  612-614. 

Laslett,  Thomas,  on  the  Strength  of  Timber, 
538-542,  546,  548,  550,  647. 

Leavitt,  E.  D.,  Pumping  Engines  by,  948. 

Legrand,  on  Boiling  Points,  370. 

Leigh,  Evan,  on  Belting,  746. 

Leloutre,  on  M.  Hirn's  Experiments  on  Work 
of  Steam,  886. 

Leplay,  on  Moisture  in  Wood,  439;  on 
Drying  Wood,  496. 

Literary  and  Philosophical  Society  of  Man- 
chester, Memoirs  of : —  Dr.  Dalton  on 
"Spontaneous"  Evaporation,  491 1  Mr. 
John  Graham  on  Heating  Surface,  802. 

Little,  C.,  on  the  Shearing  and  Punching 
Strength  of  Wrought  Iron,  587. 

Lloyd,  Thomas,  on  the  Slrength  of  Bar  Iron, 

569.  570. 

London  Association  of  Foremen  Engineers, 
Proceedings  of: — Mr.  David  Thomson  on 
Expansive  Work  of  Steam,  822. 

Longridge,  J.  A.,  on  Combustion  and  Evap- 
orative Power  of  Coals,  770,  785. 

Longsdon,  Mr.,  on  Strength  of  Krupp  Steel, 
595- 


AUTHORITIES  CONSULTED  OR  QUOTED. 


XXV 


M 


MacColl,  on  the  Strength  of  Rivetted  Joints, 

641. 
Mackintosh,  Charles,  Weight  of  Belt-Pulleys 

by,  752. 
Mackworth,   H.,   on  Ventilation   of  Mines, 

479- 
Maclure,  H.  H.,  on  Strength  of  Timber,  542, 

543-  549- 
Macneil,  Sir  John,  on  Resistance  on  Common 

Roads,  964. 
Mahan,  Lieutenant  F.  A.,  on  Outward-Flow 

Turbines,  941. 
Mallard,  on  Compressed- Air  Machines,  902 ; 

on  Compressed  Air,  907,  912. 
Mallet,  R.,  Strength  of  Buckled  Iron  Plates 

by,  660. 
Marshall,   Sons,   &  Co.,  Work  of  Steam  in 

Portable  Engine  by,  883. 
Mason,  John,  Strength  of  Untanned  Leather 

Belts  by,  680. 

M'Donnell,  A.,  on  Composition  of  Peat,  454. 
Menelaus,  on  Portable  Steam  Engines,  801. 
Miller,  T.  W.,  Trials  of  Coals  by,  790. 
Miller  &  Taplin,  Trials  of  Coals  by,  787. 
Montgolfier,  on  Drying  by  Forced  Currents, 

494- 

Monthly  Reports  to  the  Manchester  Steam- 
Users  Association:—^.  L.  E.  Fletcher's 
Data,  696. 

Morin,  on  Transverse  Strength  of  Timber, 
537;  on  the  Friction  of  Journals,  722  ;  and 
of  Solid  Bodies,  723  ;  on  Leather  Belts, 
743-745 ;  on  Breast  Wheels,  938 ;  on  a 
Fourneyron  Turbine,  940;  on  Centrifugal 
Pumps,  946. 

Morrison,  H.  M.,  on  M.  Hirn's  Rope  Trans- 
mitter, 755. 

Morton,  Francis,  &  Co.,  Weight  of  Iron 
Sheets  by,  245 ;  Strength  of  Cable  Fencing 
Stands  by,  676. 

Moser,  Strength  of  Beams  tested  for,  654. 

Muspratt,  Dr.,  Analyses  of  Coke  by,  433. 


N 


Napier,  James  R.,  on  Transmission  of  Heat, 
460 ;  on  Drying  Stuffs,  496. 

Nau,  on  Moisture  in  Charcoal,  451. 

Newall,  R.  S.,  &  Co.,  Strength  of  Hemp  and 
Wire  Ropes  by,  674. 

Nichol,  B.  G.,  on  Condensation  of  Steam  in 
Pipes  and  Tubes,  474,  476. 

Nicoll  &  Lynn,  Trials  of  Coals  by,  784. 

Norris  &  Co.,  Strength  of  Leather  Belts  by, 
680. 

North  British  Rubber  Company,  Driving  Belts 
by,  750. 

North  of  England  Mining  Institute,  Transac- 
tions of: — Rivetted  Joints,  588. 


Oldham,  Dr.,  on  Indian  Coals,  424. 
Ott,  Karl  Von,  on  Strength  of  Ropes,  674, 
679. 

P 

Parsons,  on  Strength  of  Oak  Trenails,  551. 
Parsons,  Hon.  R.  C.,  on  Centrifugal  Pumps, 

947- 

Payen,  on  Explosive  Mixture  of  Gas  and  Air, 
921. 

Pearce,  W.  A.,  on  Rope  Gearing,  754. 

Peclet,  on  Radiation  of  Heat,  329 ;  on  French 
Coals,  420 ;  on  Coke,  431 ;  on  Moisture  in 
Tan,  455 ;  on  Transmission  of  Heat,  459, 
462,  463,  468;  on  Condensing  Power  of 
Air  and  Water,  475 ;  on  Ventilation,  477 ; 
on  Heating  Apparatus,  488,  489 ;  on  Drying 
by  Air  Currents,  494 ;  on  a  Drying  House, 
495  ;  on  Cupola  Furnaces,  497. 

Penot,  on  Drying  Houses,  496. 

Penrose  &  Richards,  their  Anthracitic  Coke, 

432. 

Perkins,  Heating  Apparatus  by,  486. 

Perkins,  Jacob,  Invention  of  the  Ice-Making 
Machine  by,  373. 

Person,  on  the  Latent  Heat  of  Fusion,  367. 

Phillips,  on  Strength  of  Flat  Plates,  686. 

Piccard,  on  Work  of  Compressed  Air,  911. 

Poiree,  on  Friction  on  Rails  by,  724. 

Pole,  Dr.,  on  Spontaneous  Evaporation,  493; 
on  the  Strength  of  Steel  Wire,  617. 

Poncelet,  on  Water  Wheels,  938. 

Portefeuille  de  John  Cockerill :—  Blowing 
Engines,  927. 

Porter,  C.  T.,  on  Expansion  of  Steam,  886. 

Pouillet,  on  Luminosity  at  High  Temper- 
atures, 328. 

R 

Radford,  R.  Heber,  Weight  of  Belt-Pulleys 

by,  7SJ.  752. 
Ramsbottom,  J.,  on  Cotton- Rope  Transmitter, 

755- 

Rankine,  Dr.,  on  Expansion  of  Water,  340; 
on  the  Melting  Point  of  Ice,  364;  on 
Transmission  of  Heat,  461 ;  on  Shearing 
Strength  of  Oak  Trenails,  551;  and  of 
Cast  Iron,  561 ;  Factors  of  Safety,  625,  626  ; 
on  Stresses  in  Roofs,  715,  717 ;  on  Load  on 
Working  Surfaces,  767. 

Reading  Engine  Works  Co.,  Work  of  Steam 
in  Portable  Engine  by,  883. 

Reclus,  Specific  Gravity  of  Sea  Water  by,  126. 

Regnault,  Air  Thermometer  by,  325 ;  on  the 
Expansion  of  Air,  344 ;  on  Specific  Heat 
of  Metals,  353;  and  Gases,  359;  Boiling 
Points  of  Vapours,  371 ;  on  Steam,  378, 
379-  3^3.  384 ;  on  the  Mixture  of  Gases  and 


XXVI 


AUTHORITIES  CONSULTED   OR   QUOTED. 


Vapours,  392  ;  on  French  Coals,  420,  421 ; 

on  Lignite  and  Asphalte,  436. 
Reilly,  Calcott,  on   the  Varieties   of  Stress, 

500. 

Rennie,  on  the  Work  of  Horses,  720. 
Revue  Industrielle: — Atmospheric  Hammer 

by  M.  Chenot  Aine",  967. 
Reynolds,  Dr.,  on  Peat,  454. 
Richardson,   Dr.,   on  Coals,  412  ;   on  Coke, 

433 ;  Report  on  Evaporative  Power  of  Coals, 

785- 
Robertson,   James,    on   Grooved    Frictional 

Gearing,  741. 
Robinson,  Henry,  on  Armstrong's  Hydraulic 

Machines,  950. 
Roebling,  on  the  Strength  of  Iron  Wire,  587 ; 

and  of  Steel  Wire,  617 ;  Factor  of  Safety  for 

Iron,  625;   on  the  Strength  of  Wire  Rope 

and  Hemp  Rope,  676. 
Ross,  Owen  C.  D.,  on  Coal  Gas,  457. 
Rouget  de  Lisle,  on  Drying  Stuffs,  496. 
Royal  Society  of  Edinburgh,  Proceedings  of: — 

Mr.  Fairweather  on  Resistance  of  Air,  897. 
Royer,  on  Drying  Houses,  496;   on  Drying 

Stuffs,  496. 
Russell    &    Sons,  J.,   on    the    Strength   of 

Wrought-Iron  Tubes,  692,  693. 
Ryland  Brothers,  Warrington  Wire  Gauge  by, 

133.  247- 


Sauvage,  on  Charcoal,  447,  449,  452. 

Scheurer  -  Kestner  &  Meunier  -  Dollfus,  on 
French  and  other  Coals,  and  Lignites,  422, 
797- 

Sharp,  Henry,  on  Rivetted  Joints  of  Steel 
Plates,  642. 

Shields,  F.  W.,  on  Cast-Iron  Columns,  645. 

Shock,  Chief  Engineer  W.  H.,  on  Shearing 
Strength  of  Bar  Iron,  587. 

Siemens,  Dr.  C.  W.,  on  Isolated  Steam,  383; 
on  the  Consumption  of  Fuel  in  Metallurgical 
Furnaces,  497;  on  the  Strength  of  Hot- 
Blast  Iron,  556;  on  Hot-Air  Engines,  920; 
his  Water  Pyrometer,  967. 

Simms,  F.  W.,  on  the  Work  of  Horses,  720. 

Smeaton,  on  the  Power  of  Labourers,  718. 

Smith,  C.  Graham,  on  Strength  of  Timber, 

543-  544.  549- 

Smith,  J.  T.,  on  Punching  Resistance  of  Steel, 
617 ;  on  the  Strength  of  Rails,  664. 

Snelus,  G.  J.,  Analysis  of  Welsh  Coal  by,  413. 

Socidte"  Industrielle  de  Mulhouse: — on  Steam 
Boilers,  796. 

Socidtd  Industrielle  Minerale,  Bulletin  de 
la: — M.  Cornut  on  Compressed- Air  Machi- 
nery, 896 ;  M.  Mallard  on  Compressed- Air 
Machines,  902. 

Sod  did  des  Ingdnieurs  Civils,  Comptes  Rendus 
de  la: — Anemometer  by  M.  Arson,  892. 

Socidtd  Vaudoise  des  Ingdnieurs  et  des  Archi- 


tectes,  Bulletin  de  la: — M.  Piccard  on 
Compressed  Air,  911. 

Society  of  Arts,  Committee  of,  on  Resistance 
on  Common  Roads,  963. 

Society  of  Arts,  Journal  of: — on  Resistance 
on  Common  Roads,  963. 

Spill,  Strength  of  Belting  by,  680. 

Steel  Committee  of  Civil  Engineers,  on  the 
Strength  of  Wrought  Iron,  579,  580 ;  and  of 
Steel,  596-603, 

Stephenson,  Robert,  on  the  Strength  of  Cast 
Iron,  555,  561. 

Stoney,  on  Stress  in  a  Curved  Flange,  525 ; 
on  Sectional  Area  of  a  Continuous  Web, 
526;  on  Shearing  Strength  of  Cast  Iron, 
561 ;  his  Factors  of  Safety,  625 ;  on  the  Re- 
sistance of  Columns,  643,  645,  646 ;  on 
Stresses  in  Roofs,  715. 

Sullivan,  Dr.,  on  Peat,  207. 

Sutcliffe,  on  Condensation  of  Steam  in  the 
Cylinder,  880. 

Swindell,  J.  S.  E.,  on  Ventilation  of  Mines, 
925- 

Sylvester,  Cockle  Stove  by,  488. 


Tangye,  J.,  on  the  Compressive  Resistance  of 
Wrought  Iron,  582. 

Tasker,  Work  of  Steam  in  Portable  Engine 
by,  883. 

Telford,  Thomas,  on  the  Strength  of  Wrought 
Iron,  567 ;  and  of  Iron  Wire,  586. 

Thomas  &  Laurens,  on  Brown  Charcoal,  449; 
on  Heating  by  Steam,  463,  468. 

Thomson,  David,  on  Expansive  Action  of 
Steam,  822,  882;  on  Centrifugal  Pumps, 
946 ;  Duty  of  Pumping  Engines,  948. 

Thomson,  Professor  James,  Vortex  Wheel 
by,  943- 

Thurston,  on  the  Strength  of  Iron  Wire,  587. 

Thwaites  &  Carbutt,  on  Root's  Blower,  928. 

Towne,  H.  R.,  on  Leather  Belts,  679,  742, 
745,  748-750- 

Tredgold,  Weight  and  Volume  of  Various 
Substances  by,  213  ;  on  Cooling  Hot  Water, 
469;  on  Cooling  of  Steam  in  Pipes,  472, 
474 ;  on  the  Work  of  a  Horse,  720. 

Tresca,  on  Laubereau's  Hot-Air  Engine, 
917 ;  on  Gas-Engines,  920,  921,  923 ;  on 
Pumps,  945,  946 ;  on  Resistance  of  Tram- 
way Omnibus,  961. 

Turner,  Work  of  Steam  in  Portable  Engine 
by,  883. 

Tweddell,  R.  H.,  on  Shafting,  763. 


U 


Umber,  on  M.  Hirn's  Wire  Ropes,  754. 
Unwin,  on  Strength  of  Columns,  645. 
Ure,  Specific  Gravity  of  Alloys  by,  200. 


AUTHORITIES   CONSULTED   OR   QUOTED. 


XXVll 


V 


Vickers,  T.  E.,  on  the  Strength  of  Steel,  621, 

622. 
Violette,  on  Wood,  439,  441,  442,  445;   on 

Charcoal,  446-448,  450,  451 


w 


Wade,  Major,  on  the  Strergth  of  Cast  Iron, 

557- 

Walker,  B.,  on  Resistance  of  Shafting,  766. 
Walker,  John,  on  the  Work  of  Labourers, 

718. 
Webb,  F.  W.,  on  the  Strength  of  Steel,  614, 

621. 

Webber,  S.,  on  Mill  Shafting,  763,  764,  766. 
Westmacott,    Percy,    on   Shafting,    766 ;    on 

Corn  -  Warehousing   Machinery,    961 ;    on 

Armstrong's  Hydraulic  Machines,  950. 
Whitelaw,  James,  Water  Mill  by,  939. 


Whitworth,  Sir  Joseph,  Standard  Wire-Gauge 
by,  133,  134;  Strength  of  his  Fluid-Com- 
pressed Steel,  and  of  Iron,  614,  615 ;  on 
Resistance  of  Steel  and  Iron  to  Explosive 
Force,  622;  his  System  of  Standard  Sizes 
of  Bolts  and  Nuts,  68  r  ;  Standard  Pitches 
of  Screwed-Iron  Piping,  683. 

Wiesbacb,  Coefficients  for  Flow  of  Water, 
892. 

Williams,  R.  Price,  on  the  Transverse  Strength 
of  Rails,  662,  664. 

Williams,  Foster,  &  Co.,  Weight  of  Sheet 
Copper  by,  261. 

Wilson,  A.,  on  the  Work  of  Bullocks,  720. 

Wilson,  R.,  on  Strength  of  Perforated  Iron 
Plates,  633. 

Wilson,  Robert  (Patricroft),  on  Teeth  of 
Wheels,  732. 

Wood,  J.  &  E.,  Work  of  Steam  in  Stationary 
Engine  by,  882. 

Woods,  E.,  andj.  Dewrance,  on  the  Efficiency 
of  Heating  Surface  of  a  Locomotive,  803. 

Wright,  J.  G.,  on  Rivetted  Joints,  637. 


A  MAN  UAL 


OF 


RULES,  TABLES,  AND  DATA* 


FOR 


MECHANICAL    ENGINEERS. 


GEOMETRICAL   PROBLEMS. 


PROBLEMS  ON  STRAIGHT  LINES. 

PROBLEM  I. — To  bisect  a  straight 
line,  or  an  arc  of  a  circle,  Fig.  i. — 
From  the  ends  A,  B,  as  centres,  de- 
scribe arcs  intersecting  at  c  and  D, 
and  draw  c  D,  which  bisects  the  line, 
or  the  arc,  at  the  point  E  or  F. 

PROBLEM  II. — 72?  draw  a  perpen- 
dicular to  a  straight  line,  or  a  radial 
line  to  a  circular  arc,  Fig.  i. — Operate 


Fig.  i.— Probs.  I.  and  II. 

as  in  the  foregoing  problem.  The 
line  CD  is  perpendicular  to  A  B :  the 
line  c  D  is  also  radial  to  the  arc  A  B. 

PROBLEM  III. — To  draw  a  perpen- 
dicular to  a  straight  line,  from  a  given 
point  in  that  line,  Fig.  2. — With  any 
radius,  from  the  given  point  A,  in  the 


line  B  c,  cut  the  line  at  B  and  c ;  with 
a  longer  radius  describe  arcs  from  B 


A  ~Tc 

Fig.  2.—  Prob.  III. 


and  c,  cutting  each  other  at  D,  and 
draw  the  perpendicular  D  A. 

2d  Method,  Fig.  3.  —  From  any  cen- 
tre F,  above  BC,  describe  a  circle 
passing  through  the  given  point  A, 


Fig.  3. — Prob.  III.  ad  method. 

and  cutting  the  given  line  at  D  ;  draw 
D  F,  and  produce  it  to  cut  the  circle 
at  E;  and  draw  the  perpendicular  A  E. 


GEOMETRICAL   PROBLEMS 


$d  Method,  Fig.  4. —  From  A  de- 
scribe an  arc  E  c,  and  from  E,  with 
the  same  radius,  the  arc  A  c,  cutting 


E  A 

Fig.  4.— Prob.  III.  3d  method. 

the  other  at  c ;  through  c  draw  a  line 
E  c  D,  and  set  oft"  c  D  equal  to  c  E  ; 
and  through  D  draw  the  perpendicu- 
lar A  D. 

4//z  Method,  Fig.   5.  —  From   the 
given  point  A  set  off  a  distance  A  E 


E1  3  A 

Fig.  5.— Prob.  III.  4th  method. 

equal  to  three  parts,  by  any  scale; 
and  on  the  centres  A  and  E,  with 
radii  of  four  and  five  parts  respec- 
tively, describe  arcs  intersecting  at  c. 
Draw  the  perpendicular  A  c. 

Note. — This  method  is  most  useful 
on  very  large  scales,  where  straight 
edges  are  inapplicable.  Any  multi- 
ples of  the  numbers  3,  4,  5  may  be 
taken  with  the  same  effect,  as  6,  8, 
10,  or  9,  12,  15. 

PROBLEM  IV. — To  draw  a  perpen- 
dicular to  a  straight  line  from  any 
point  without  it,  Fig.  6. — From  the 
point  A,  with  a  sufficient  radius,  cut 
the  given  line  at  Fanda;  and  from 
these  points  describe  arcs  cutting  at 
E.  Draw  the  perpendicular  A  E. 

Note. — If  there  be  no  room  below 


the  line,  the  intersection  may  be  taken 
above  the  line;  that  is  to  say,  be- 
tween the  line  and  the  given  point. 


2d  Method,  Fig.  7. — From  any  two 
points  B,  c,  at  some  distance  apart, 


Fig.  7.— Prob.  IV.  ad  method. 

in  the  given  line,  and  with  the  radii 
B  A,  c  A,  respectively,  describe  arcs 
cutting  at  A  D.  Draw  the  perpendi- 
cular A  D. 

PROBLEM  V. — To  draw  a  straight 
line  parallel  to  a  given  line,  at  a  given 
distance  apart,  Fig.  8. — From  the  cen- 


Fig.  8.-Prob.  V. 

tres  A,  B,  in  the  given  line,  with  the 
given  distance  as  radius,  describe  arcs 
c, D;  and  draw  the  parallel  line  CD 
touching  the  arcs. 

PROBLEM  VI. — To  draw  a  parallel 
through  a  given  point,  Fig.  9.— With 
a  radius  equal  to  the  distance  of  the 


ON   STRAIGHT   LINES. 


given  point  c  from  the  given  line 
A  B,  describe  the  arc  D  from  B,  taken 


Fig.  9.— Prob.  VI. 

considerably  distant  from  c.  Draw 
the  parallel  through  c  to  touch  the 
arc  D. 

2d  Method,  Fig.  10. — From  A,  the 


A/; 


E  F 

Fig.  10. — Prob.  VI.  ad  method. 

given  point,  describe  the  arc  F  D,  cut- 
ting the  given  line  at  F;  from  F,  with 
the  same  radius,  describe  the  arc  E  A, 
and  set  off  F  D  equal  to  E  A.  Draw 
the  parallel  through  the  points  A,  D. 

Note,  Fig.  n. — When  a  series  of 
parallels  are  required  perpendicular 
to  a  base  line  A  B,  they  may  be  drawn, 
as  in  Fig.  i,  through  points  in  the 
base  line,  set  off  at  the  required  dis- 


Fig.  n.— Prob.  VI. 

tances  apart.  This  method  is  con- 
venient also  where  a  succession  of 
parallels  are  required  to  a  given  line, 


CD;  for  the  perpendicular  A  B  may  be 
drawn  to  it,  and  any  number  of  par- 
allels may  be  drawn  upon  the  per- 
pendicular. 

PROBLEM  VII. — To  divide  a  straight 
line  into  a  number  of  equal  parts,  Fig. 
12. — To  divide  the  line  AB  into,  say, 
five  parts.  From  A  and  B  draw  par- 
allels A  c,  B  D,  on  opposite  sides.  Set 
off  any  convenient  distance  four  times 


A-"' 

JK"  \ 
•A""  \      \ 
\ 


Fig.  12.— Prob.  VII. 


(one   less   than  the   given  number) 


from  A  on  AC,  and  from  B  on  B  D  ; 
join  the  first  on  AC  to  the  fourth  on 
B  D,  and  so  on.  The  lines  so  drawn 
divide  A  B  as  required. 

2d  Method,  Fig.  13. — Draw  the  line 
A  c  at  an  angle  from  A,  set  off,  say, 


2  3         4 

Fig.  13.— Prob.  VII.  sd  method. 

five  equal  parts;  draw  B  5,  and  draw 
parallels  to  it  from  the  other  points 
of  division  in  AC.  These  parallels 
divide  A  B  as  required. 

Note. — By  a  similar  process  a  line 
may  be  divided  into  a  number  of 
unequal  parts;  setting  off  divisions 
on  A  c,  proportional  by  a  scale  to  the 
required  divisions,  and  drawing  par- 
allels cutting  A  B. 

PROBLEM  VIII. —  Upon  a  straight 


GEOMETRICAL   PROBLEMS 


line  to  draw  an  angle  equal  to  a  given 
angle,  Fig.  14. — Let  A  be  the  given 
angle,  and  FG  the  line.  With  any 


radius,  from  the  points  A  and  F,  de- 
scribe arcs  D  E,  i  H,  cutting  the  sides 
of  the  angle  A,  and  the  line  F  G.  Set 


Fig.  14.— Prob.  VIII. 


off  the  arc  i  H  equal  to  D  E,  and  draw 
F  H.  The  angle  F  is  equal  to  A,  as 
required. 

To  draw  angles  of  60°  and  30°,  Fig. 
15. — From  F,  with  any  radius  F  i,  de- 
scribe an  arc  i  H  ;  and  from  i,  with 
the  same  radius,  cut  the  arc  at  H,  and 


F          x         T 

Fig.  15.— Prob.  VIII. 

draw  F  H  to  form  the  required  angle 
i  F  H.  Draw  the  perpendicular  H  K 
to  the  base  line,  to  form  the  angle  of 
30°  F  H  K. 

To  draw  an  angle  of  45°,  Fig.  16. 
— Set  off  the  distance  F  i,  draw  the 


«F  I 

Fig.  16.— Prob.  VIII. 

perpendicular  i  H  equal  to  i  F,  and 
join  H  F,  to  form  the  angle  at  F  as  re- 
quired. The  angle  at  H  is  also  45°. 
PROBLEM  IX. — To  bisect  an  angle, 
Fig.  17. — Let  ACB  be  the  angle;  on 
the  centre  c  cut  the  sides  at  A,  B.  On 
A  and  B,  as  centres,  describe  arcs 


cutting  at  D.    Draw  c  D,  dividing  the 
angle  into  two  equal  parts. 


PROBLEM  X. — To  bisect  the  inclina- 
tion of  two  lines,  of  which  the  intersec- 
tion is  inaccessible,  Fig.  18. — Upon  the 


Fig.  18.— Prob.  X. 

given  lines  c  B,  CH,  at  any  points, 
draw  perpendiculars  E  F,  G  H,  of  equal 
lengths,  and  through  F  and  G  draw 
parallels  to  the  respective  lines,  cut- 
ting at  s;  bisect  the  angle  FSG,  so 
formed,  by  the  line  s  D,  which  divides 
equally  the  inclination  of  the  given 
lines. 


ON   STRAIGHT   LINES   AND   CIRCLES. 


PROBLEMS  ON   STRAIGHT   LINES 
AND  CIRCLES. 

PROBLEM  XI. — Through  two  given 
points  to  describe  an  arc  of  a  circle  with 
a  given  radius,  Fig.  19. — On  the  points 


Fig.  19.— Prob.  XI. 

A  and  B  as  centres,  with  the  given 
radius,  describe  arcs  cutting  at  c ;  and 
from  c,  with  the  same  radius,  describe 
an  arc  A  B  as  required. 

PROBLEM  XII. — To  find  the  centre 
of  a  circle,  or  of  an  arc  of  a  circle. 
ist,  for  a  circle,  Fig.  20. — Draw  the 


PROBLEM  XIII. — To  describe  a  cir- 
cle passing  through  three  given  points, 
Fig.  21. — Let  A,  B,  c  be  the  given 
points,  and  proceed  as  in  last  pro- 


Fig.  21.— Prob.  XII.  XIII. 

blem  to  find  the  centre  o,  from  which 
the  circle  may  be  described. 

Note. — This  problem  is  variously 
useful: — in  striking  out  the  circular 
arches  of  bridges  upon  centerings, 
when  the  span  and  rise  are  given; 
describing  shallow  pans,  or  dished 


Fig.  20.— Prob.  XII. 

chord  A  B,  bisect  it  by  the  perpendi- 
cular c  D,  bounded  both  ways  by  the 
circle ;  and  bisect  c  D  for  the  centre  G. 
2d,  for  a  circle  or  an  arc,  Fig.  21. 
— Select  three  points,  A,  B,  c,  in  the 
circumference,  well  apart;  with  the 
same  radius,  describe  arcs  from  these 
three  points,  cutting  each  other;  and 
draw  the  two  lines,  D  E,  F  G,  through 
their  intersections,  according  to  Fig.  i. 
The  point  o,  where  they  cut,  is  the 
centre  of  the  circle  or  arc. 


Fig.  22.— Prob.  XIV.  ist  method. 

covers  of  vessels ;  or  finding  the  dia- 
meter of  a  fly-wheel  or  any  other 
object  of  large  diameter,  when  only 
a  part  of  the  circumference  is  ac- 
cessible. 

PROBLEM  XIV. — To  describe  a  circle 
passing  through  three  given  points  when 
the  centre  is  not  available. 

\st  Method,  Fig.  22.  —  From  the 
extreme  points  A,  B,  as  centres,  de- 
scribe arcs  AH,  EG.  Through  the 
third  point  c,  draw  A  E,  B  F,  cutting 


GEOMETRICAL   PROBLEMS 


the  arcs.  Divide  A  F  and  B  E  into 
any  number  of  equal  parts,  and  set 
off  a  series  of  equal  parts  of  the  same 
length  on  the  upper  portions  of  the 
arcs  beyond  the  points  E,  F.  Draw 
straight  lines,  B  L,  B  M,  &c.,  to  the  divi- 
sions in  A  F;  and  A  i,  A  K,  &c.,  to  the 
divisions  in  EG;  the  successive  inter- 


sections N,  o,  &c.,  of  these  lines,  are 
points  in  the  circle  required,  between 
the  given  points  A  and  c,  which  may 
be  filled  in  accordingly:  similarly  the 
remaining  part  of  the  curve  B  c  may 
be  described. 

zd  Method,  Fig.  23. — Let  A,  D,B  be 
the  given  points.    Draw  A  B,  A  D,  D  B, 


Fig.  23.— Prob.  XIV.  2d  method. 


and  ef  parallel  to  A  B.  Divide  D  A 
into  a  number  of  equal  parts  at  i,  2, 3, 
&c.,  and  from  D  describe  arcs  through 
these  points  to  meet  ef.  Divide  the 
arc  A  e  into  the  same  number  of  equal 
parts,  and  draw  straight  lines  from  D 
to  the  points  of  division.  The  inter- 
sections of  these  lines  successively 
with  the  arcs  i,  2,  3,  &c.,  are  points 
in  the  circle  which  may  be  filled  in  as 
before. 

Note. — The  second  method  is  not 
perfectly  exact,  but  is  sufficiently  near 
to  exactness  for  arcs  less  than  one- 
fourth  of  a  circle.  When  the  middle 
point  is  equally  distant  from  the  ex- 
tremes, the  vertical  c  D  is  the  rise  of 
the  arc;  and  this  problem  is  service- 
able for  setting  circular  arcs  of  large 
radius,  as  for  bridges  of  very  great 


Fig.  24.— Prob.  XV. 

span,  when  the  centre  is  unavailable; 
and  for  the  outlines  of  bridge-beams, 


and  of  beams  and  connecting-rods  of 
steam-engines,  and  the  like. 

PROBLEM  XV. — To  draw  a  tangent 
to  a  circle  from  a  given  point  in  the 
circumference,  Fig.  24. — Through  the 
given  point  A,  draw  the  radial  line 


Fig.  25.— Prob.  XV.  2d  method. 

A  c,  and  the  perpendicular  F  G  is  the 
tangent. 

2d  Method,  when  the  centre  is  not 
available,  Fig.  25. — From  A,  set  off 
equal  segments  A  B,  A  D;  join  B  D,  and 
draw  A  E  parallel  to  it  for  the  tangent. 

PROBLEM  XVI. — To  draw  tangents 
to  a  circle  from  a  point  without  it. 


Fig.  26.— Prob.  XVI.  ist  method. 

ist  Method,  Fig.  26. — Draw  AC 
from  the  given  point  A  to  the  centre 


ON   STRAIGHT   LINES   AND   CIRCLES. 


c;  bisect  it  at  D,  and  from  the  centre 
D,  describe  an  arc  through  c,  cutting 
the  circle  at  E,  F.  Then  A  E,  A  F,  are 
tangents. 

2d  Method,  Fig.  27. — From  A,  with 
the  radius  A  c,  describe  an  arc  BCD, 
and  from  c,  with  a  radius  equal  to  the 


Fig.  27.— Prob.  XVI.  2d  method. 

diameter  of  the  circle,  cut  the  arc  at 
B,  D;  join  EC,  CD,  cutting  the  circle 
at  E,  F,  and  draw  A  E,  A  F,  the  tan- 
gents. 

Note. — When  a  tangent  is  already 
drawn,  the  exact  point  of  contact 
may  be  found  by  drawing  a  perpen- 
dicular to  it  from  the  centre. 

PROBLEM  XVII. — Between  two  in- 
clined  lines  to  draw  a  series  of  circles 
touching  these  lines  and  touching  each 
other,  Fig.  28. — Bisect  the  inclination 


Fig.  28.— Prob.  XVII. 


of  the  given  lines  A  B,  c  D  by  the  line 
NO.     From  a  point  P  in  this  line, 


draw  the  perpendicular  P  B  to  the  line 
A  B,  and  on  P  describe  the  circle  B  D 
touching  the  lines  and  cutting  the 
centre  line  at  E.  From  E  draw  E  F 
perpendicular  to  the  centre  line,  cut- 
ting A  B  at  F,  and  from  F  describe  an 
arc  E  G,  cutting  A  B  at  G.  Draw  G  H 
parallel  to  B  p,  giving  H,  the  centre 
of  the  next  circle,  to  be  described 
with  the  radius  H  E,  and  so  on  for  the 
next  circle  i  N. 

Inversely,  the  largest  circle  may 
be  described  first,  and  the  smaller 
ones  in  succession. 

Note. — This  problem  is  of  frequent 
use  in  scroll  work. 

PROBLEM  XVIII. — Between  two 
inclined  lines  to  draw  a  circular  seg- 
ment to  fill  the  angle,  and  touching  the 
lines,  Fig.  29. — Bisect  the  inclination 


F  A 

Fig.  29.— Prob.  XVIII. 

of  the  lines  A  B,  D  E  by  the  line  F  c, 
and  draw  the  perpendicular  A  F  D  to 
define  the  limit  within  which  the  cir- 
cle is  to  be  drawn.  Bisect  the  angles 
A  and  D  by  lines  cutting  at  c,  and 
from  c,  with  radius  c  F,  draw  the  arc 
H  F  G  as  required. 

PROBLEM  XIX. — To  describe  a  cir- 
cular arc  joining  two  circles,  and  to 
touch  one  of  them  at  a  given  point,  Fig. 
30. — To  join  the  circles  A  B,  F  G,  by 
an  arc  touching  one  of  them  at  F, 
draw  the  radius  E  F,  and  produce  it 
both  ways;  set  off  FH  equal  to  the 
radius  AC  of  the  other  circle,  join  CH 


8 


GEOMETRICAL    PROBLEMS 


and  bisect  it  with  the  perpendicular 
L  i,  cutting  E  F  at  i.    On  the  centre  i, 


Fig.  30.— Prob.  XIX. 

with  radius  i  F,  describe  the  arc  F  A  as 
required. 


PROBLEMS   ON   CIRCLES  AND 
RECTILINEAL   FIGURES. 

PROBLEM  XX. — To  construct  a  tri- 
angle on  a  given  base,  the  sides  being 
given. 

i  st.  An  equilateral  triangle,  Fig.  31. 


Fig.  31. -Prob.  XX. 

— On  the  ends  of  the  given  base,  A,  B, 
with  A  B  as  radius,  describe  arcs  cut- 
ting at  c,  and  draw  AC,  c B. 

2d.  A  triangle  of  unequal  sides, 
Fig.  32. — On  either  end  of  the  base 
A  D,  with  the  side  B  as  radius,  describe 
an  arc ;  and  with  the  side  c  as  radius, 
on  the  other  end  of  the  base  as  a 
centre,  cut  the  arc  at  E.  Join  A  E,  D  E. 

Note. — This  construction  may  be 
used  for  finding  the  position  of  a 
point,  c  or  E,  at  given  distances  from 


the  ends  of  a  base,  not  necessarily  to 
form  a  triangle. 


Fig.  32.— Prob.  XX. 


PROBLEM  XXI.  —  To  construct  a 
square  or  a  rectangle  on  a  given  straight 
line. 

ist.  A  square,   Fig.  33.— On  the 


Fig.  33.— Prob.  XXI. 

ends  A,  B,  as  centres,  with  the  line  A  B 
as  radius,  describe  arcs  cutting  at  c; 
on  c,  describe  arcs  cutting  the  others 
at  D  E  ;  and  on  D  and  E,  cut  these  at 
F G.  Draw  A  F,  EG,  and  join  the  in- 
tersections H,  i. 

2d.  A  rectangle,  Fig.  34. — On  the 
base  E  F,  draw  the  perpendiculars  E  H, 


Fig.  34.— Prob.  XXI. 

F  G,  equal  to  the  height  of  the  rect- 
angle, and  join  G  H. 

When  the  centre  lines,  A  B,  CD, 
Fig-  35,  of  a  square  or  a  rectangle  are 
given,  cutting  at  E. — Set  off  E  F,  EG, 


ON   CIRCLES   AND   RECTILINEAL   FIGURES. 


the  half  lengths  of  the  figure,  and  E  H, 
E  T,  the  half  heights.  On  the  centres 
H,  T,  with  a  radius  of  half  the  length, 


M 


Fig.  35.— Prob.  XXI. 

describe  arcs;  and,  on  the  centres  F, 
G,  with  a  radius  of  half  the  height,  cut 
these  arcs  at  K,  L,  M,  N.  Join  these 
intersections. 

PROBLEM  XXII. — To  construct  a 
parallelogram,  of  which  the  sides  and 
me  of  the  angles  are  given,  Fig.  36. — 


Fig.  36.— Prob.  XXII. 

Draw  the  side  D  E  equal  to  the  given 
length  A,  and  set  off  the  other  side 
D  F  equal  to  the  other  length  B,  form- 
ing the  given  angle  c.  From  E,  with 
D  F  as  radius,  describe  an  arc,  and 
from  F,  with  the  radius  D  E,  cut  the 
arc  at  G.  Draw  F  G,  EG. 

Or,  the  remaining  sides  may  be 
drawn  as  parallels  to  D  E,  D  F. 

The  formation  of  the  angle  D  is 
readily  done  as  indicated,  by  taking 
the  straight  length  of  the  arc  H  i  and 
c  i  as  radius,  and  finding  the  inter- 
section L. 

PROBLEM  XXIII. — To  describe  a 
circle  about  a  triangle,  Fig.  37. — Bisect 
two  sides  A  B,  A  c  of  the  triangle  at 
E,  F,  and  from  these  points  draw  per- 
pendiculars cutting  at  K.  On  the 


centre  K,  with  the  radius  K  A,  draw 
the  circle  ABC. 


Fig.  37.— Prob.  XXIII. 

PROBLEM  XXIV.  —  To  inscribe  a 
circle  in  a  triangle,  Fig.  38. — Bisect 
two  of  the  angles  A,  c,  of  the  triangle 
by  lines  cutting  at  D;  from  D  draw  a 
perpendicular  D  E  to  any  side,  and 
with  D  E  as  radius  describe  a  circle. 

When  the  triangle  is  equilateral, 
the  centre  of  the  circle  may  be  found 
by  bisecting  two  of  the  sides,  and 


Fig.  38.— Prob.  XXIV. 

drawing  perpendiculars  as  in  the  pre- 
vious problem.  Or,  draw  a  perpen- 
dicular from  one  of  the  angles  to  the 
opposite  side,  and  from  the  side  set 
off  one-third  of  the  perpendicular. 


Fig.  39.— Prob.  XXV. 

PROBLEM  XXV.  —  To  describe  a 
circle  about  a  square,  and  to  inscribe 
a  square  in  a  circle,  Fig.  39. 


10 


GEOMETRICAL   PROBLEMS 


i  st.  To  describe  the  circle.  Draw 
the  diagonals  A B,  CD  of  the  square, 
cutting  at  E;  on  the  centre  E,  with  the 
radius  E  A,  describe  the  circle. 

2d.  To  inscribe  the  square. — Draw 
the  two  diameters  A  B,  CD  at  right 
angles,  and  join  the  points  A,  B,  c,  D 
to  form  the  square. 

Note. — In  the  same  way  a  circle 
may  be  described  about  a  rectangle. 

PROBLEM  XXVI.  —  To  inscribe  a 
circle  in  a  square,  and  to  describe  a 
square  about  a  circle.  Fig.  40. 

i  st.  To  inscribe  the  circle. — Draw 


Fig.  40.— Prob.  XXVI. 

the  diagonals  A  B,  c  D  of  the  square, 
cutting  at  E;  draw  the  perpendicular 
E  F  to  one  side,  and  with  the  radius 
E  F  describe  the  circle. 

2(1.  To  describe  the  square. — Draw 
two  diameters  A  B,  c  D  at  right  angles, 
and  produce  them;  bisect  the  angle 
D  E  B  at  the  centre  by  the  diameter 
F  G,  and  through  F  and  G  draw  per- 


Fig.  41.— Prob.  XXVII. 

pendiculars  AC,  ED,  and  join  the 
points  A  D  and  B  c,  where  they  cut  the 
diagonals,  to  complete  the  square. 


PROBLEM  XXVII. — To  inscribe  a 
pentagon  in  a  circle,  Fig.  41. — Draw 
two  diameters  A  c,  B  D  at  right  angles, 
cutting  at  o;  bisect  A  o  at  E,  and  from 
E,  with  radius  E  B,  cut  A  c  at  F  ;  from  B, 
with  radius  B  F,  cut  the  circumference 
at  G,  H,  and  with  the  same  radius  step 
round  the  circle  to  i  and  K;  join  the 
points  so  found  to  form  the  pentagon. 

PROBLEM  XXVIII. — To  construct 
a  hexagon  upon  a  given  straight  line, 
Fig.  42. — From  A  and  B,  the  ends  of 
the  given  line,  describe  arcs  cutting 
at  g\  from  g,  with  the  radius  g  A,  de- 
F 


Fig.  42.— Prob.  XXVIII. 

scribe  a  circle;  with  the  same  radius 
set  off  the  arcs  A  G,  G  F,  and  B  D,  D  E. 
Join  the  points  so  found  to  form  the 
hexagon. 

PROBLEM  XXIX.  —  To  inscribe  a 
hexagon  in  a  circle,  Fig.  43. — Draw  a 
diameter  ACB;  from  A  and  B  as  centres, 
with  the  radius  of  the  circle  A  c,  cut 
the  circumference  at  D,  E,  F,  G;  and 
draw  AD,  D  E,  &c.  to  form  the  hexagon. 


Fig.  43.— Prob.  XXIX. 

The  points  D,  E,  &c.,  may  also  be 
found  by  stepping  the  radius  six 
times  round  the  circle. 


ON  CIRCLES   AND   RECTILINEAL  FIGURES. 


II 


PROBLEM  XXX. — To  describe  a  hex- 
agon about  a  circle,  Fig.  44. — Draw  a 


Fig.  44.— Prob.  XXX. 

diameter  ADB,  and  with  the  radius 
A  D,  on  the  centre  A,  cut  the  circum- 
ference at  c;  join  AC,  and  bisect  it 
with  the  radius  D  E:  through  E  draw 
the  parallel  F  G  cutting  the  diameter 
at  F,  and  with  the  radius  D  F  describe 
the  circle  F  H.  Within  this  circle  de- 
scribe a  hexagon  by  the  preceding 
problem;  it  touches  the  given  circle. 

PROBLEM  XXXI. — To  describe  an 
octagon  on  a  given  straight  line,  Fig.  45. 


H 


k  A.  B  7 

Fig.  45.— Prob.  XXXI. 

— Produce  the  given  line  AB  both 
ways,  and  draw  perpendiculars  A  E, 
B  F;  bisect  the  external  angles  A  and 
B,  by  the  lines  A  H,  B  c,  which  make 
equal  to  A  B.  Draw  c  D  and  H  G  par- 
allel to  A  E,  and  equal  to  A  B  ;  from  the 
centres  G,  D,  with  the  radius  A  B,  cut 
the  perpendiculars  at  E.  F,  and  draw 
E  F  to  complete  the  octagon. 

PROBLEM  XXXII. — To  convert  a 
square  into  an  octagon,  Fig.  46. — Draw 
the  diagonals  of  the  square  cutting  at 


e;  from  the  corners  A,  B,  c,  D,  with  A  e 
as  radius,  describe  arcs  cutting  the 


Fig.  46.— Prob.  XXXII. 

sides  at  g,  h,  &c.;  and  join  the  points 
so  found  to  form  the  octagon. 

PROBLEM   XXXIII.  —  To  inscribe 
an  octagon  in  a  circle,  Fig.  47. — Draw 


Fig.  47.—  Prob.  XXXIII. 

two  diameters  AC,  B  D  at  right  angles ; 
bisect  the  arcs  AB,  BC,  &c.,  at  e,f, 
&c.,  and  join  A  e,  <?B,  &c.,  to  form  the 
octagon. 

PROBLEM    XXXIV.  —  To  describe 
an  octagon  about  a  circle,  Fig.  48. — 

k     c  ^ 


Fig.  48. -Prob.  XXXIV. 

Describe  a  square  about  the  given 
circle  A  B,  draw  perpendiculars  h  k, 


12 


GEOMETRICAL   PROBLEMS 


&c.,  to  the  diagonals,  touching  the 
circle,  to  form  the  octagon. 

Or,  the  points  h,  k,  &c.,  may  be 
found  by  cutting  the  sides  from  the 
corners  of  the  square,  as  in  the  second 
last  problem. 

PROBLEM  XXXV. — To  describe  a 
polygon  of  any  number  of  sides  iipon  a 
given  straight  line,  Fig.  49. — Produce 


Fig.  49.-Prob.  XXXV. 

the  given  line  A  B,  and  on  A,  with 
the  radius  A  B,  describe  a  semicircle, 
divide  the  semi-circumference  into  as 
many  equal  parts  as  there  are  to  be 
sides  in  the  polygon;  say,  in  this  ex- 
ample, five  sides.  Draw  lines  from 
A  through  the  divisional  points  D,  b, 
and  c,  omitting  one  point  a;  and  on 
the  centres  B,  D,  with  the  radius  A  B, 
cut  A  b  at  E  and  A  c  at  F.  Draw 
D  E,  E  F,  F  B  to  complete  the  polygon, 

PROBLEM  XXXVI. — To  inscribe  a 
circle  within  a  polygon,  Figs.  50,  51. — 
When  the  polygon  has  an  even  num- 
ber of  sides,  Fig.  50,  bisect  two  op- 


Fig.  50.— Prob.  XXXVI.  XXXVII. 

posite  sides  at  A  and  B,  draw  A  B,  and 
bisect  it  at  c  by  a  diagonal  D  E;  and 
with  the  radius  c  A  describe  the  circle. 


When  the  number  of  sides  is  odd, 
Fig.  51,  bisect  two  of  the  sides  at  A 


Fig.  51.— Prob.  XXXVI.  XXXVII. 

and  B,  and  draw  lines  A  E,  B  D  to  the 
opposite  angles,  intersecting  at  c; 
from  c,  with  the  radius  c  A,  describe 
the  circle. 

PROBLEM  XXXVII.— To  describe  a 
circle  without  a  polygon,  Figs.  50, 5 1. — 
Find  the  centre  c  as  before,  and  with 
the  radius  c  D  describe  the  circle. 

The  foregoing  selection  of  prob- 
lems on  regular  figures  are  the  most 
useful  in  mechanical  practice  on  that 
subject.  Several  other  regular  figures 
may  be  constructed  from  them  by 
bisection  of  the  arcs  of  the  circum- 
scribing circles.  In  this  way  a  de- 
cagon, or  ten-sided  polygon,  may  be 
formed  from  the  pentagon,  as  shown 
by  the  bisection  of  the  arc  B  H  at  k 
in  Fig.  41.  Inversely,  an  equilateral 
triangle  may  be  inscribed  by  joining 
the  alternate  points  of  division  found 
for  a  hexagon. 

PROBLEM  XXXVIII.— To  inscribe 
a  polygon  of  any  number  of  sides 
within  a  circle,  Fig.  52. — Draw  the 
diameter  A  B,  and  through  the  centre 
E  draw  the  perpendicular  E  c,  cutting 
the  circle  at  F.  Divide  E  F  into  four 
equal  parts,  and  set  off  three  parts 
equal  to  those  from  F  to  c.  Divide 
the  diameter  A  B  into  as  many  equal 
parts  as  the  polygon  is  to  have  sides; 
and  from  c  draw  c  D  through  the 
second  point  of  division,  cutting  the 
circle  at  D.  Then  A  D  is  equal  to  one 


ON   THE  ELLIPSE. 


side  of  the  polygon,  and  by  stepping 
round    the   circumference   with   the 


Fig.  52.— Prob.  XXXVIII. 

length  A  D,  the  polygon  may  be  com- 
pleted. 

The  constructions  for  inscribing 
regular  polygons  in  circles  are  suit- 
able also  for  dividing  the  circumfer- 
ence of  a  circle  into  a  number  of 
equal  parts.  To  supply  a  means  of 
dividing  the  circumference  into  any 
number  of  parts,  including  cases  not 
provided  for  in  the  foregoing  prob- 
lems, the  annexed  table  of  angles 
relating  to  polygons,  expressed  in 
degrees,  will  be  found  of  general 
utility.  In  this  table  the  angle  at 

TABLE  OF  POLYGONAL  ANGLES. 


Number 
of  Sides. 

Angle 
at  Centre. 

Number 
of  Sides. 

Angle 
at  Centre. 

No. 

Degrees. 

No. 

Degrees. 

3 

120 

12 

30 

4 

5 

90 
72 

13 
14 

lit 

6 

60 

15 

24 

7 

5  ^ 

1  6 

22^ 

8 

45 

'I 

21^ 

9 

40 

18 

2O 

10 

36 

19 

19 

ii 

M 

20 

18 

the  centre  is  found  by  dividing  360°, 
the  number  of  degrees  in  a  circle,  by 
the  number  of  sides  in  the  polygon; 
and  by  setting  off  round  the  centre 
of  the  circle  a  succession  of  angles 
by  means  of  the  protractor,  equal  to 
the  angle  in  the  table  due  to  a  given 


number  of  sides,  the  radii  so  drawn 
will  divide  the  circumference  into  the 
same  number  of  parts.  The  triangles 
thus  formed  are  termed  the  elemen- 
tary triangles  of  the  polygon. 

PROBLEM  XXXIX. — To  inscribe 
any  regular  polygon  in  a  given  circle; 
or  to  divide  the  circumference  into  a 
given  number  of  equal  parts ,  by  means 
of  the  angle  at  the  centre.  Fig.  53. — 


Fig.  53.— Prob.  XXXIX. 

Suppose  the  circle  is  to  contain  a 
hexagon,  or  is  to  be  divided  at  the 
circumference  into  six  equal  parts. 
Find  the  angle  at  the  centre  for  a 
hexagon,  or  60°;  draw  any  radius  B  c, 
and  set  off,  by  a  protractor  or  other- 
wise, the  angle  at  the  centre  CBD 
equal  to  60°;  then  the  interval  CD  is 
one  side  of  the  figure,  or  segment  of 
the  circumference;  and  the  remaining 
points  of  division  may  be  found  either 
by  stepping  along  the  circumference 
with  the  distance  c  D  in  the  dividers, 
or  by  setting  off  the  remaining  five 
angles,  of  60°  each,  round  the  centre. 


PROBLEMS  ON  THE  ELLIPSE. 

An  ellipse  is  an  oval  figure,  like  a 
circle  in  perspective.  The  line  A  B, 
Fig.  54,  that  divides  it  equally  in  the 
direction  of  its  greatest  dimension, 
is  the  transverse  axis;  and  the  per- 
pendicular c  D,  through  the  centre, 
is  the  conjugate  axis.  Two  points, 
F,  G,  in  the  transverse  axis,  are  the 


GEOMETRICAL   PROBLEMS 


foci  of  the  curve,  each  being  called  a 
focus;  being  so  placed  that  the  sum 
of  their  distances  from  either  end  of 
the  conjugate  axis,  c  or  D,  is  equal 


Fig.  54.-Prob.  XL. 

to  the  transverse  axis.  In  general,  the 
sum  of  their  distances  from  any  other 
point  in  the  curve  is  equal  to  the 
transverse  axis,  A  line  drawn  at 
right  angles  to  either  axis,  and  termi- 
nated by  the  curve,  is  a  double  ordi- 
nate, and  each  half  of  it  is  an  ordinate. 
The  segments  of  an  axis  between  an 
ordinate  and  its  vertices  are  called 
abscisses.  The  double  ordinate  drawn 
through  a  focus  is  called  the  para- 
meter of  the  axis. 

The  squares  of  any  two  ordinates 
to  the  transverse  axis,  are  to  each 
other  as  the  rectangles  of  their  respec- 
tive abscisses. 

PROBLEM  XL. — To  describe  an  el- 
lipse when  the  length  and  breadth  are 
%iven,  Fig.  54. — On  the  centre  c,  with 
A  E  as  radius,  cut  the  axis  A  B  at  F 
and  G,  the  foci;  fix  a  couple  of  pins 
into  the  axis  at  F  and  G,  and  loop  on 
a  thread  or  cord  upon  them  equal  in 
length  to  the  axis  A  B,  so  as  when 
stretched  to  reach  to  the  extremity  c 
of  the  conjugate  axis,  as  shown  in  dot- 
lining.  Place  a  pencil  or  drawpoint 
inside  the  cord,  as  at  H,  and  guiding 
the  pencil  in  this  way,  keeping  the 
cord  equally  in  tension,  carry  the 
pencil  round  the  pins  F,  G,  and  so 
describe  the  ellipse. 

Note. — This  method  is  employed 


in  setting  off  elliptical  garden-plots, 
walks,  &c. 

2d  Method,  Fig.  55. — Along  the 
straight  edge  of  a  slip  of  stiff  paper, 
mark  off  a  distance  a  c  equal  to  A  c, 


Fig-  55-— Prob.  XL.  2d  method. 

half  the  transverse  axis;  and  from 
the  same  point  a  distance  a  b  equal 
to  c  D,  half  the  conjugate  axis.  Place 
the  slip  so  as  to  bring  the  point  b  on 
the  line  A  B  of  the  transverse  axis,  and 
the  point  c  on  the  line  DE;  and  set 
off  on  the  drawing  the  position  of  the 
point  a.  Shifting  the  slip,  so  that 
the  point  b  travels  on  the  transverse 
axis,  and  the  point  c  on  the  conjugate 
axis,  any  number  of  points  in  the 
curve  may  be  found,  through  which 
the  curve  may  be  traced. 

$d  Method^  Fig.   56. — The  action 
of  the  preceding  method  may  be  em- 


Fig.  56.— Prob.  XL.  sd  method. 

bodied  so  as  to  afford  the  means  of 
describing  a  large  curve  continuously, 
by  means  of  a  bar  mk,  with  steel 
points  m,  /,  k,  rivetted  into  brass  slides 
adjusted  to  the  length  of  the  semi- 
axes,  and  fixed  with  setscrews.  A  rec- 
tangular cross  E  G,  with  guiding  slots, 
is  placed  coinciding  with  the  two 


ON   THE   ELLIPSE. 


axes  of  the  ellipse,  A  c  and  B  H  ;  by 
sliding  the  points  k,  /,  in  the  slots,  and 
carrying  round  the  point  m,  the  curve 
may  be  continuously  described.  A 
pen  or  pencil  may  be  fixed  at  m. 

qth  Method,  Fig.   57. — Bisect  the 
transverse  axis  at  c,  and  through  c 


Fig.  57.— Prob.  XL.  4th  method. 

draw  the  perpendicular  D  E,  making 
c  D  and  c  E  each  equal  to  half  the 
conjugate  axis.  From  D  or  E,  with 
the  radius  A  c,  cut  the  transverse  axis 
at  F,  F',  for  the  foci.  Divide  A  c  into 
a  number  of  parts  at  the  points  i,  2, 
3,  &c.  With  the  radius  A  i,  on  F  and 
F'  as  centres,  describe  arcs ;  and  with 
the  radius  B  i,  on  the  same  centres, 
cut  these  arcs  as  shown.  Repeat  the 
operation  for  the  other  divisions  of 
the  transverse  axis.  The  series  of 
intersections  thus  made  are  points  in 
the  curve,  through  which  the  curve 
may  be  traced. 

5M  Method,  Fig.  58.— On  the  two 


Fig.  58.— Prob.  XL.  sth  method. 

axes  A  B,  D  E  as  diameters,  on  centre 
c,  describe  circles;   from  a  number 


of  points,  a,  b,  &c.,  in  the  circumfer- 
ence. A  FB,  draw  radii  cutting  the  in- 
ner circle  at  a',  b',  &c.  From  a,  b, 
&c.,  draw  perpendiculars  to  AB;  and 
from  a',  b',  &c.,  draw  parallels  to  A  B, 
cutting  the  respective  perpendiculars 
at  n,  ot  &c.  The  intersections  are 
points  in  the  curve,  through  which 
the  curve  may  be  traced. 

6th  Method,  Fig.  59.— When  the 
transverse  and   conjugate  diameters 


Fig.  59.— Prob.  XL.  6th  method. 

are  given,  A  B,  CD,  draw  the  tangent 
E  F  parallel  to  A  B.  Produce  c  D, 
and  on  the  centre  G,  with  the  radius 
of  half  A  B,  describe  a  semicircle  HDK; 
from  the  centre  G  draw,  any  number 
of  straight  lines  to  the  points  E,  r, 
&c.,  in  the  line  E  F,  cutting  the  cir- 
cumference at  /,  m,n,  &c.;  from  the 
centre  o  of  the  ellipse  draw  straight 
lines  to  the  points  E,  r,  &c.,  and 
from  the  points  /,  m,  n,  &c.,  draw 
parallels  to  G  c,  cutting  the  lines  o  E, 
o>,  &c.,  at  L,  M,  N,  &c.  These  are 
points  in  the  circumference  of  the 
ellipse,  and  the  curve  may  be  traced 
through  them.  Points  in  the  other 
half  of  the  ellipse  are  formed  by  ex- 
tending the  intersecting  lines  as  indi- 
cated in  the  figure. 

PROBLEM  XLI.  —  To  describe  an 
ellipse  approximately  by  means  of  cir- 
cular arcs. — First,  with  arcs  of  two 
radii,  Fig.  60. — Find  the  difference 


i6 


GEOMETRICAL    PROBLEMS 


of  the  two  axes,  and  set  it  off  from  the 
centre  o  to  a  and  c,  on  o  A  and  o  c ; 


Fig.  60.— Prob.  XLI. 

draw  a  c,  and  set  off  half  a  c  to  d; 
draw  d  i  parallel  to  a  c,  set  ofT  o  e 
equal  to  od,  join  ei,  and  draw  the 
parallels  em,  d  m.  From  m,  with 
radius  m  c,  describe  an  arc  through 
c  ;°and  from  i  describe  an  arc  through 
D  ;  from  d  and  e  describe  arcs  through 
A  and  B.  The  four  arcs  form  the 
ellipse  approximately. 

Note. — This  method  does  not  ap- 
ply satisfactorily  when  the  conjugate 
axis  is  less  than  two-thirds  of  the 
transverse  axis. 


o  M  equal  to  c  L,  and  on  D  describe 
an  arc  with  radius  DM;  on  A,  with 
radius  OL,  cut  this  arc  at  a.  Thus 
the  five  centres  D,  a,  b,  H,  H'  are  found, 
from  which  the  arcs  are  described  to 
form  the  ellipse. 

Note. — This  process  works  well  for 
nearly  all  proportions  of  ellipses.  It 
is  employed  in  striking  out  vaults  and 
stone  bridges. 

PROBLEM  XLII. — To  draw  a  tan- 


Fig.  61.— Prob.  XLI.  ad  method. 

Second,  with  arcs  of  three  radii, 
Fig.  6 1. — On  the  transverse  axis  AB 
draw  the  rectangle  B  G,  on  the  height 
o  c ;  to  the  diagonal  A  c  draw  the  per- 
pendicular G  H  D  ;  set  off  OK  equal 
to  o  c,  and  describe  a  semicircle  on 
A  K,  and  produce  oc  to  L;  set  off 


Fig.  62.— Prob.  XLII. 

gent  to  an  ellipse  through  a  given  point 
in  the  curve,  Fig.  62. — From  the  given 
point  T  draw  straight  lines  to  the 
foci  F,  F'J  produce  F  T  beyond  the 
curve  to  c,  and  bisect  the  exterior 
angle  c  T  F,  by  the  line  T  d,  which 
is  the  tangent. 

PROBLEM  XLIII.  —  To  draw  a 
tangent  to  an  ellipse  from  a  given 
point  without  the  curve,  Fig.  63. — 
From  the  given  point  T,  with  a 
radius  to  the  nearest  focus  F,  de- 
scribe an  arc  on  the  other  focus 
F',  with  a  radius  equal  to  the  trans- 
verse axis,  cut  the  arc  at  K  L,  and 


Fig.  63.— Prob.  XLIII. 

draw  K  F',  L  F',  cutting  the  curve  at 
M,  N.    The  lines  T  M,  T  N  are  tangents. 


ON   THE   PARABOLA. 


PROBLEMS  ON  THE    PARABOLA. 

A  parabola,  DAC,  Fig.  64,  is  a 
curve  such  that  every  point  in  the 
curve  is  equally  distant  from  the  di- 
rectrix K  L  and  the  focus  F.  The 
focus  lies  in  the  axis  A  B  drawn  from 
the  vertex  or  head  of  the  curve  A,  so 
as  to  divide  the  figure  into  two  equal 
parts.  The  vertex  A  is  equidistant 
from  the  directrix  and  the  focus,  or 
A  e=  A  F.  Any  line  parallel  to  the 
axis  is  a  diameter.  A  straight  line, 
as  E  G  or  D  c,  drawn  across  the  figure 
at  right  angles  to  the  axis  is  a  double 
ordinate,  and  either  half  of  it  is  an 
ordinate.  The  ordinate  to  the  axis 
E  F  G,  drawn  through  the  focus,  is 
called  the  parameter  of  the  axis.  A 
segment  of  the  axis,  reckoned  from  the 
vertex,  is  an  absciss  of  the  axis;  and 
it  is  an  absciss  of  the  ordinate  drawn 
from  the  base  of  the  absciss.  Thus, 
A  B  is  an  absciss  of  the  ordinate  B  c. 

Abscisses  of  a  parabola  are  as  the 
squares  of  their  ordinates. 

PROBLEM  XLIV.  —  To  describe  a 
parabola  when  an  absciss  and  its  ordi- 
nate are  given;  that  is  to  say,  when 
the  height  and  breadth  are  given, 
Fig.  64. — Bisect  the  given  ordinate 


7 


Fig.  64.— Prob.  XLIV. 

B  c  at  a ;  draw  A  a,  and  then  a  b  per- 
pendicular to  it,  meeting  the  axis  at 
b.  Set  off  A  e,  A  F,  each  equal  to  B  b; 
and  draw  K  e  L  perpendicular  to  the 
axis.  Then  K  L  is  the  directrix  and 
F  is  the  focus.  Through  F  and  any 


number  of  points,  <?,  o,  &c.,  in  the 
axis,  draw  double  ordinates,  n  o  n, 
&c. ;  and  on  the  centre  F,  with  the 
radii  Ye,oe,  &c.,  cut  the  respective 
ordinates  at  E,  G,  n,  n,  &c.  The  curve 
may  be  traced  through  these  points 
as  shown. 

zd  Method; .  by  means  of  a  square 
and  a  cord,  Fig.  65. — Place  a  straight- 


Fig.  65.— Prob.  XLIV.  ad  method. 

edge  to  the  directrix  E  N,  and  apply 
to  it  a  square  LEG.  Fasten  to  the 
end  G,  one  end  of  a  thread  or  cord 
equal  in  length  to  the  edge  E  G,  and 
attach  the  other  end  to  the  focus  F; 
slide  the  square  along  the  straight- 
edge, holding  the  cord  taut  against 
the  edge  of  the  square  by  a  draw- 
point  or  pencil  D,  by  which  the  curve 
is  described. 

$d  Method;  when  the  height  and  the 
base  are  given,  Fig.  66. — Let  A  B  be 


^^- 

7? 

t 

e 

^ 



^-~-- 

/ 

1 

? 

c 

/ 

\ 

y 

^ 

L  — 
^ 

s 

1 

* 

\ 

a. 

'     c/   ~b     <3>  _2?     a,     b     c     d  J. 

Fig.  66.—  Prob.  XLIV.  sd  method. 

the  given  axis,  and  c  D  a  double  ordi- 
nate or  base;  to  describe  a  parabola 
2 


18 


GEOMETRICAL   PROBLEMS 


of  which  the  vertex  is  at  A.  Through 
A  draw  E  F  parallel  to  c  D,  and 
through  c  and  D  draw  c  E  and  D  F 
parallel  to  the  axis.  Divide  B  c  and 
B  D  into  any  number  of  equal  parts, 
say  five,  at  #,  b,  &c.,  and  divide 
c  E  and  D  F  into  the  same  number  of 
parts.  Through  the  points  a,  b,  c,  d 
in  the  base  c  D,  on  each  side  of  the 
axis,  draw  perpendiculars,  and  through 
a,  b,  c,  d,  in  c  E  and  D  F,  draw  lines  to 
the  vertex  A,  cutting  the  perpendicu- 
lars at  e,f,  £-,  h.  These  are  points  in 
the  parabola,  and  the  curve  c  A  D  may 
be  traced  as  shown,  passing  through 
them. 


PROBLEMS  ON  THE  HYPERBOLA. 

The  vertices  A,  B,  Fig.  67,  of  oppo- 
site hyperbolas,  are  the  heads  of  the 
curves,  and  are  points  in  their  centre 
or  axial  lines.  The  transverse  axis 
A  B  is  the  distance  between  the  ver- 
tices, of  which  the  centre  c  is  the 
centre.  The  conjugate  axis  G  H  is  a 
straight  line  drawn  through  the  centre 
at  right  angles  to  the  transverse  axis. 
An  ordinate  F  K  is  a  straight  line 
drawn  from  any  point  of  the  curve 
perpendicular  to  the  axis.  The  seg- 
ments of  the  transverse  axis  A  F,  B  F, 
between  an  ordinate  F  K  and  the  ver- 
tices of  the  curves,  are  abscisses. 
The  parameter  is  the  double  ordinate 
drawn  through  the  focus.  The 
asymptotes  are  two  straight  lines,  s  s, 
R  R,  drawn  from  the  centre  through 
the  ends  of  a  tangent  ED  at  the  vertex, 
equal  and  parallel  to  the  conjugate 
axis,  and  bisected  by  the  transverse 
axis. 

The  nature  of  the  hyperbola  is  such 
that  the  difference  of  the  distances 
of  any  point  in  the  curve  from  the 
foci  is  always  the  same,  and  is  equal 
to  the  transverse  axis. 

In  a  hyperbola  the  squares  of  any 
two  ordinates  to  the  transverse  axes 
are  to  each  other  as  the  rectangles  of 
their  abscisses. 


PROBLEM  XLV.  —  To  describe  a 
hyperbola,  the  transverse  and  conjugate 
axes  being  given.  Fig.  67. — Draw  AB 


Fig.  67.— Prob.  XLV. 

equal  to  the  transverse  axis,  and  D  E 
perpendicular  to  it  and  equal  to  the 
conjugate  G  H.  On  c,  with  the  radius 
c  E,  describe  a  circle  cutting  A  B  pro- 
duced, at  F/;  these  points  are  the  foci. 
In  A  B  produced  take  any  number  of 
points  o,  o,  &c.,  with  the  radii  KO,  BO, 
and  on  centres  F,/ describe  arcs  cut- 
ting each  other  at  n,  n,  &c.  These 
are  points  in  the  curve,  through  which 
it  may  be  traced. 

2d  Method,  Fig.  67. — The  curve 
may  be  drawn  thus: — Let  the  ends 
of  two  threads/p  Q,  F  p  Q,  be  fastened 
at  the  points  /,  F,  and  be  made  to 
pass  through  a  small  bead  or  pin  p, 
and  knotted  together  at  Q.  Take 
hold  of  Q,  and  draw  the  threads  tight; 
move  the  bead  along  the  threads,  and 
the  point  P  will  describe  the  curve. 
If  the  end  of  the  long  thread  be  fixed 
at  F,  and  the  short  thread  at  f,  the 
opposite  curve  may  be  described  in 
the  same  manner. 

Or,  the  line  /Q  may  be  replaced 
by  a  straight-edge  turning  on  a  pin 
at/  and  the  cord  F  Q  joined  to  it  at  Q. 
The  curve  may  then  be  described  by 
means  of  a  point  or  pencil  in  the  same 
manner  as  for  the  parabola,  Fig.  65. 

$d  Method ;  when  the  breadth  c  D, 


ON    THE   HYPERBOLA,    CYCLOID,    EPICYCLOID. 


A  B,  and  transverse  axis  A  A.'  of 
the  curve  are  given.  Fig.  68. — Divide 


Os       O         v       &•       -B       Ct>        ~fa        O       ct,      J) 

Fig.  68.— Prob.  XLV.  3d  method. 

the  base  or  double  ordinate  c  D  into 
a  number  of  equal  parts  on  each  side 
of  the  axis  at  a,  b,  &c. ;  and  divide 
the  parallels  c  E,  D  F,  into  the  same 
number  of  equal  parts  at  a,  b,  &c. 
From  the  points  a,  b,  &c.,  in  the  base, 
draw  lines  to  A',  and  from  the  points 
a,  b,  &c.,  in  the  verticals,  draw  lines 
to  A,  cutting  the  respective  lines  from 
the  base.  Trace  the  curve  through 
the  intersections  thus  obtained. 


THE  CYCLOID  AND  EPICYCLOID. 

PROBLEM  XLVI.  —  To  describe  a 
cycloid,  Fig.  69. — When  a  wheel  or  a 
circle  D  G  c  rolls  along  a  straight  line 


A  B,  Fig.  69,  beginning  at  A  and  end- 
ing at  B,  where  it  has  just  completed 


one  revolution,  it  measures  off  a 
straight  line  A  B  exactly  equal  to  the 
circumference  of  the  circle  D  G  c, 
which  is  called  the  generating  circle, 
and  a  point  or  pencil  fixed  at  the 
point  D  in  the  circumference  traces 
out  a  curvilinear  path  A  D  B,  called  a 
cycloid.  A  B  is  the  base  and  c  D  is  the 
axis  of  the  cycloid. 

Place  the  generating  circle  in  the 
middle  of  the  cycloid,  as  in  the  figure, 
draw  a  line  E  H  parallel  to  the  base, 
cutting  the  circle  at  G;  and  the  tan- 
gent H  i  to  the  curve  at  the  point  H. 
Then  the  following  are  some  of  the 
properties  of  the  cycloid : — 

The  horizontal  line  H  G  =  arc  of  the 
circle  G  D. 

The  half-base  A  c  =  the  half-circum- 
ference CGD. 

The  arc  of  the  cycloid  D  H  =  twice 
the  chord  D  G. 

The  half-arc  of  the  cycloid  D  A  = 
twice  the  diameter  of  the  circle  D  c. 

Or,  the  whole  arc  of  the  cycloid 
A  D  B  =  four  times  the  axis  c  D. 

The  area  of  the  cycloid  A  D  B  A  = 
three  times  the  area  of  the  generating 
circle  D  c. 

The  tangent  H  i  is  parallel  to  the 
chord  G  D. 

PROBLEM  XLVII. — To  describe  an 


Fig.  7o.-Prob.  XLVII. 

exterior  epicycloid,  Fig.  70. — The  epicy- 
cloid differs  from  the  cycloid  in  this, 


20 


GEOMETRICAL   PROBLEMS 


that  it  is  generated  by  a  point  D  in 
one  circle  D  c  rolling  upon  the  cir- 
cumference of  another  circle  A  c  B, 
instead  of  on  a  flat  surface  or  line; 
the  former  being  the  generating  circle, 
and  the  latter  the  fundamental  circle. 
The  generating  circle  is  shown  in  four 
positions,  in  which  the  generating 
point  is  successively  marked  D,  D'.  D", 
D'".  A  D'"  B  is  the  epicycloid. 

PROBLEM   XLVIIL—  To  describe 


Fig.  71.— Prob.  XLVIIL 

an  interior  epicycloid,  Fig.  71. — If  the 
generating  circle  be  rolled  on  the  in- 
side of  the  fundamental  circle,  as  in 
Fig.  71,  it  forms  an  interior  epicycloid, 
or  hypocycloid,  A  D'"  B,  which  becomes 
in  this  case  nearly  a  straight  line.  The 
other  points  of  reference  in  the  figure 
correspond  to  those  in  Fig.  70. 
When  the  diameter  of  the  generating 
circle  is  equal  to  half  that  of  the  fun- 
damental circle,  the  epicycloid  be- 
comes a  straight  line,  being  in  fact  a 
diameter  of  the  larger  circle. 


THE   CATENARY. 

When  a  perfectly  flexible  string,  or 
a  chain  consisting  of  short  links,  is 
suspended  from  two  points  M,  N,  Fig. 
72,  it  is  stretched  by  its  own  weight, 
and  it  forms  a  curve  line  known  as 
the  catenary,  M  c  N.  The  point  c, 
where  the  catenary  is  horizontal,  is 
the  vertex. 

PROBLEM   XLIX. — To  describe  a 


catenary,  Fig.  72. — Draw  the  vertical 
c  G  equal  to -the  length  of  the  arc  of  the 
chain,  M  c,  on  one  side  of  the  vertex, 
and  divide  it  into  a  great  number  of 
equal  parts,at  ( i ),  ( 2),  (3),&c.  Draw  the 
horizontal  line  c  H  equal  to  the  length 
of  so  much  of  the  rope  or  chain  as 
measures  by  its  weight  the  horizontal 
tension  of  the  chain.  From  the  point 
c  as  the  vertex,  set  off  c  ( i)  on  the 
horizontal  line  equal  to  c  i"  on  the 
vertical;  and  (i)  (2)  from  the  point 
(i),  parallel  to  H  i  and  equal  to  c  (i); 
and  again  (2)  (3)  from  the  point  (2) 
parallel  to  H  2  and  equal  to  c  (i); 
and  so  on  till  the  last  segment  (6)  M 
is  drawn  parallel  to  H  G.  The  poly- 
gon c  (i)  (2)  (3)  .  .  .  M,  thus  formed, 
is  approximately  the  catenary  curve, 
which  may  be  traced  through  the 
middle  points  of  the  sides  of  the 
polygon.  A  similar  process  being 
performed  for  the  other  side  of  the 
curve,  the  catenary  is  completed. 


Fig.  72.— Prob.  XLIX. 

2(t  Method.  —  Suspend  a  finely 
linked  chain  against  a  vertical  wall. 
The  curve  may  be  traced  from  it,  on 
the  wall,  answering  the  conditions  of 
given  length  and  height,  or  of  given 
width  or  length  of  arc.  A  cord  having 
numerous  equal  weights  suspended 
from  it  at  short  and  equal  distances 
may  be  used. 


CIRCLES,    PLANE   TRIGONOMETRY. 


21 


CIRCLES. 

The  circumference  of  a  circle  is  commonly  signified  in  mathematical 
discussions  by  the  symbol  T,  which  indicates  the  length  of  the  circumfer- 
ence when  the  diameter  =  i. 

The  area  of  a  circle  is  as  the  square  of  the  diameter,  or  the  square  of  the 
circumference. 

The  ratio  of  the  diameter  to  the  circumference  is      as- 1  to  3*141593 — 

commonly  abbreviated, as  i  to  3*1416 

approximately, as  i  to  3^ 

or  as  7  to  22 

When  the  diameter  =  i,  the  area  is  equal  to 785398  + 

or,  commonly  abbreviated, 7854 

approximately, -fths. 

When  the  circumference  =  i,  the  area  is  equal  to '° 795 7 7  + 

or,  abbreviated, '0796 

approximately, Aths,  or  -08. 

In  these  ratios,  the  diameter  and  the  circumference  are  taken  lineally, 
and  the  area  superficially.  So  that  if  the  diameter  =  i  foot,  the  circum- 
ference is  equal  to  '3*1416  feet,  and  the  area  is  equal  to  "7854  square  foot. 

Note. — If  the  first  three  odd  figures,  i,  3,  5,  be  each  put  down  twice,  the 
first  three  of  these  will  be  to  the  last  three,  that  is  113  is  to  355,  as  the 
diameter  to  the  circumference. 


PLANE   TRIGONOMETRY. 


The  circumference  of  a  circle  is  supposed  to  be  divided  into  360  degrees 
or  divisions,  and  as  the  total  angularity  about  the  centre  is  equal  to  four  right 
angles,  each  right  angle  contains  90  degrees,  or  90°,  and  half  a  right  angle 


Fig-  73- — Definitions  in  Plane  Trigonometry. 

contains  45°.  Each  degree  is  divided  into  60  minutes,  or  60';  and,  for  the 
sake  of  still  further  minuteness  of  measurement,  each  minute  is  divided  into 
60  seconds,  or  60".  In  a  whole  circle  there  are,  therefore,  360  x  60  x  60  = 


22  GEOMETRICAL   PROBLEMS. 

1,296,000  seconds.  The  annexed  diagram,  Fig.  73,  exemplifies  the  rela- 
tive positions  of  the  sine,  cosine,  versed  sine,  tangent,  co-tangent,  secant, 
and  co-secant  of  an  angle.  It  may  be  stated,  generally,  that  the  correlated 
quantities,  namely,  the  cosine,  co-tangent,  and  co-secant  of  an  angle,  are 
the  sine,  tangent,  and  secant,  respectively,  of  the  complement  of  the  given 
angle,  the  complement  being  the  difference  between  the  given  angle  and  a 
right  angle.  The  supplement  of  an  angle  is  the  amount  by  which  it  is 
less  than  two  right  angles. 

When  the  sines  and  cosines  of  angles  have  been  calculated  (by  means  of 
formulas  which  it  is  not  necessary  here  to  particularize),  the  tangents,  co-tan- 
gents, secants,  and  co-secants  are  deduced  from  them  according  to  the 
following  relations : — 

rad.  x  sin.  rad.2  rad.'-'  rad.2 

tan.  =  -          ;  cotan.  =  -   —  j        sec.  =  —   — ;  cosec.  = . 

cos.  tan.  cos.  sin. 

For  these  the  values  will  be  amplified  in  tabular  form. 

A  triangle  consists  of  three  sides  and  three  angles.  When  any  three 
of  these  are  given,  including  a  side,  the  other  three  may  be  found  by  cal- 
culation : — 

CASE  i. —  When  a  side  and  its  opposite  angle  are  two  of  the  given  parts. 
RULE  i.   To  find  a  side,  work  the  following  proportion: — 
as  the  sine  of  the  angle  opposite  the  given  side 
is  to  the  sine  of  the  angle  opposite  the  required  side, 
so  is  the  given  side 
to  the  required  side. 
RULE  2.   To  find  an  angle: — 

as  the  side  opposite  to  the  given  angle 
is  to  the  side  opposite  to  the  required  angle, 
so  is  the  sine  of  the  given  angle 
to  the  sine  of  the  required  angle. 

RULE  3.  In  a  right-angled  triangle,  when  the  angles  and  one  side  next  the 
right  angle  are  given,  to  find  the  other  side: — 
as  radius 

is  to  the  tangent  of  the  angle  adjacent  to  the  given  side, 
so  is  this  side 
to  the  other  side. 

CASE  2. —  When  two  sides  and  the  included  angle  are  given. 
RULE  4.   To  find  the  other  side: — 

as  the  sum  of  the  two  given  sides 
is  to  their  difference, 

so  is  the  tangent  of  half  the  sum  of  their  opposite  angles 
to  the  tangent  of  half  their  difference — 

add  this  half  difference  to  the  half  sum,  to  find  the  greater  angle;  and 
subtract  the  half  difference  from  the  half  sum,  to  find  the  less  angle.  The 
other  side  may  then  be  found  by  Rule  i. 

RULE  5.  When  the  sides  of  a  right-angled  triangle  are  given,  to  find  the 
angles: — 


MENSURATION   OF   SURFACES.  23 

as  one  side 
is  to  the  other  side, 
so  is  the  radius 

to  the  tangent  of  the  angle  adjacent  to  the  first  side. 
CASE  3. —  When  the  three  sides  are  given. 

RULE  6.  To  find  an  angle.  Subtract  the  sum  of  the  logarithms  of  the 
sides  which  contain  the  required  angle,  from  20;  to  the  remainder  add  the 
logarithm  of  half  the  sum  of  the  three  sides,  and  that  of  the  difference 
between  this  half  sum  and  the  side  opposite  to  the  required  angle.  Half 
the  sum  of  these  three  logarithms  will  be  the  logarithmic  cosine  of  half  the 
required  angle.  The  other  angles  may  be  found  by  Rule  i. 

RULE  7.  Subtract  the  sum  of  the  logarithms  of  the  two  sides  which  con- 
tain the  required  angle,  from  20,  and  to  the  remainder  add  the  logarithms 
of  the  differences  between  these  two  sides  and  half  the  sum  of  the  three 
sides.  Half  the  result  will  be  the  logarithmic  sine  of  half  the  required 
angle. 

Note. — In  all  ordinary  cases  either  of  these  rules  gives  sufficiently  accur- 
ate results.  It  is  recommended  that  Rule  6  should  be  used  when  the 
required  angle  exceeds  90°;  and  Rule  7  when  it  is  less  than  90°. 


MENSURATION  OF  SURFACES. 

To  find  the  area  of  a  parallelogram.  Multiply  the  length  by  the  height,  or 
perpendicular  breadth. 

Or,  multiply  the  product  of  two  contiguous  sides  by  the  natural  sine 
of  the  included  angle. 

To  find  the  area  of  a  triangle.  Multiply  the  base  by  the  perpendicular 
height,  and  take  half  the  product. 

Or,  multiply  half  the  product  of  two  contiguous  sides  by  the  natural 
sine  of  the  included  angle. 

To  find  the  area  of  a  trapezoid.  Multiply  half  the  sum  of  the  parallel 
sides  by  the  perpendicular  distance  between  them. 

To  find  the  area  of  a  quadrilateral  inscribed  in  a  circle.  From  half  the 
sum  of  the  four  sides  subtract  each  side  severally;  multiply  the  four  re- 
mainders together;  the  square  root  of  the  product  is  the  area. 

To  find  the  area  of  any  quadrilateral  figure.  Divide  the  quadrilateral  into 
two  triangles;  the  sum  of  the  areas  of  the  triangles  is  the  area. 

Or,  multiply  half  the  product  of  the  two  diagonals  by  the  natural  sine  of 
the  angle  at  their  intersection. 

Note. — As  the  diagonals  of  a  square  and  a  rhombus  intersect  at  right  angles 
(the  natural  sine  of  which  is  i),  half  the  product  of  their  diagonals  is  the 
area. 

To  find  the  area  of  any  polygon.  Divide  the  polygon  into  triangles  and 
trapezoids  by  drawing  diagonals ;  find  the  areas  of  these  as  above  shown, 
for  the  area. 

To  find  the  area  of  a  regular  polygon.  Multiply  half  the  perimeter  of  the 
polygon  by  the  perpendicular  drawn  from  the  centre  to  one  of  the  sides. 

Note. — To  find  the  perpendicular  when  the  side  is  given — 


24 


GEOMETRICAL   PROBLEMS. 


as  radius 

to  tangent  of  half-angle  at  perimeter  (see  table  No.  i), 
so  is  half  length  of  side 
to  perpendicular. 

Or,  multiply  the  square  of  a  side  of  any  regular  polygon  by  the  corres- 
ponding area  in  the  following  table: — 

TABLE  No.   i. — ANGLES  AND  AREAS  OF  REGULAR  POLYGONS. 


NAME. 

Number 
of 
Sides. 

One  half 
Angle  at  the 
Perimeter. 

Area. 
(Side=i) 

Perpendi- 
cular. 
(Side=i) 

Equilateral  triangle 

?0° 

O'A.'Z'ZO 

0-2887 

Square,  . 

3 

4' 

ow 

4c° 

*  ^TOOW 
I  'OOOO 

O'  '(OOO 

Pentagon,  

$ 

54° 

I'72O^ 

0*6882 

Hexagon,  

6 

60° 

2x081 

o'S66o 

Heptagon,  

7 

64°? 

*  Jy^JJ 
•2*6770 

i  '0787 

Octagon,  

8 

67°A 

4-8284 

wo"o 
I  '2O71 

Nonagon 

Q 

70° 

6'i8i8 

I  *7  1  "2  1 

Decagon, 

y 
IO 

72° 

7'6oA2 

1  61  61 
T'^88 

Undecagon 

1  1 

7V3  7 

vjy^^s 

Q'76^6 

1    JO00 

I  '7028 

Dodecagon, 

I  2 

/  O    1  1 

7c° 

y  o^D^ 
1  1  '1962 

I  '8660 

/  J 

To  find  the  circumference  of  a  circle.     Multiply  the  diameter  by  3 '141 6. 

Or,  multiply  the  area  by  12  '5664;  the  square  root  of  the  product  is  the 
circumference. 

To  find  the  diameter  of  a  circle.     Divide  the  circumference  by  3 '141 6. 

Or,  multiply  the  circumference  by  '3183. 

Or,  divide  the  area  by  7854;  the  square  root  of  the  quotient  is  the 
diameter. 

To  find  the  area  of  a  circle.     Multiply  the  square  of  the  diameter  by  7854. 

Or,  multiply  the  circumference  by  one-fourth  of  the  diameter. 

Or,  multiply  the  square  of  the  circumference  by  '07958. 

To  find  the  length  of  an  arc  of  a  circle.  Multiply  the  number  of  degrees 
in  the  arc  by  the  radius,  and  by  '01745. 

Or,  the  length  may  be  found  nearly,  by  subtracting  the  chord  of  the  whole 
arc  from  eight  times  the  chord  of  half  the  arc,  and  taking  one-third  of  the 
remainder. 

To  find  the  area  of  a  sector  of  a  circle.  Multiply  half  the  length  of  the 
arc  of  the  sector  by  the  radius. 

Or,  multiply  the  number  of  degrees  in  the  arc  by  the  square  of  the  radius, 
and  by  '008727. 

To  find  the  area  of  a  segment  of  a  circle.  Find  the  area  of  the  sector 
which  has  the  same  arc  as  the  segment;  also  the  area  of  the  triangle 
formed  by  the  radial  sides  of  the  sector  and  the  chord  of  the  arc;  the 
difference  or  the  sum  of  these  areas  will  be  the  area  of  the  segment,  ac- 
cording as  it  is  less  or  greater  than  a  semicircle. 

To  find  the  area  of  a  ring  included  between  the  circumferences  of  two  con- 


MENSURATION   OF   SURFACES. 


centric  circles.     Multiply  the  sum  of  the  diameters  by  their  difference,  and 
by  7854- 

To  find  the  area  of  a  cycloid.     Multiply  the  area  of  the  generating  circle 

by  3- 

To  find  the  length  of  an  arc  of  a  parabola,  cut  off  by  a  double  ordinate 
to  the  axis.  To  the  square  of  the  ordinate  add  four-fifths  of  the  square  of 
the  absciss ;  twice  the  square  root  of  the  sum  is  the  length  nearly. 

Note. — This  rule  is  an  approximation  which  applies  to  those  cases  only 
in  which  the  absciss  does  not  exceed  half  the  ordinate. 

To  find  the  area  of^  a  parabola.  Multiply  the  base  by  the  height;  two- 
thirds  of  the  product  is  the  area. 

To  find  the  circumference  of  an  ellipse.  Multiply  the  square  root  of  half  the 
sum  of  the  squares  of  the  two  axes  by  3*1416. 

To  find  the  area  of  an  ellipse.    Multiply  the  product  of  the  two  axes  by 

7854- 

Note. — The  area  of  an  ellipse  is  equal  to  the  area  of  a  circle  of  which 
the  diameter  is  a  mean  proportional  between  the  two  axes. 

To  find  the  area  of  an  elliptic  segment,  the  base  of  which  is  parallel  to 
either  axis  of  the  ellipse.  Divide  the  height  of  the  segment  by  the  axis  of 
which  it  is  a  part,  and  find  the  area  of  a  circular  segment,  by  table  No. 
VII. ,  of  which  the  height  is  equal  to  this  quotient;  multiply  the  area  thus 
found  by  the  two  axes  of  the  ellipse  successively;  the  product  is  the  area. 

To  find  the  length  of  an  arc  of  a  hyperbola,  beginning  at  the  vertex.  To 
19  times  the  transverse  axis  add  21  times  the  parameter  to  this  axis,  and 
multiply  the  sum  by  the  quotient  of  the  absciss  divided  by  the  transverse. 
2d.  To  9  times  the  transverse  add  2 1  times  the  parameter,  and  multiply  the 
sum  by  the  quotient  of  the  absciss  divided  by  the  transverse.  3d.  To 
each  of  these  products  add  1 5  times  the  parameter,  and  then 

as  the  latter  sum 

is  to  the  former  sum, 

so  is  the  ordinate 

to  the  length  of  the  arc,  nearly. 

To  find  the  area  of  a  hyperbola.  To  the  product  of  the  transverse  and 
absciss  add  five-sevenths  of  the  square  of  the  absciss,  and  multiply  the  square 
root  of  the  sum  by  21;  to  this  product  add  4  times  the  square  root  of 
the  product  of  the  transverse  and  absciss;  multiply  the  sum  by  4  times  the 
product  of  the  conjugate  and  absciss,  and  divide  by  75  times  the  transverse. 
The  quotient  is  the  area  nearly. 

To  find  the  area  of  any  curvilineal  figure, 
bounded  at  the  ends  by  parallel  straight  lines, 
Fig.  74.  Divide  the  length  of  the  figure 
a  b  into  any  even  number  of  equal  parts, 
and  draw  ordinates  c,  d,  e,  &c.,  through  the 
points  of  division,  to  touch  the  boundary 
lines.  Add  together  the  first  and  last 
ordinates  (c  and  k),  and  call  the  sum  A; 
add  together  the  even  ordinates  (that  is,  Fis-  74--For  Area  of  Curvilinear  Figure. 
d,f<>  h)j\  and  call  the  sum  B;  add  together 

the  odd  ordinates,  except  the  first  and  last  (e,g,  i},  and  call  the  sum  c.    Let 
D  be  the  common  distance  of  the  ordinates,  then 


26 


GEOMETRICAL   PROBLEMS. 


(A  +  4  B  +  2  c) 


x  D  =  area  of  figure. 


This  is  known  as  Simpson's  Rule. 

zd  Method,  Fig.  74. — Having  divided  the  figure  into  an  even  or  an  odd 
number  of  equal  parts,  add  together  the  first  and  last  ordinates,  making 
the  sum  A;  and  add  together  all  the  intermediate  ordinates,  making  the 
sum  B.  Let  L  =  the  length  of  the  figure,  and  n  =  the  number  of  divisions, 
then 


A  +  2  B 

2  n 


x  L  =  area  of  figure. 


That  is  to  say,  twice  the  sum  of  the  intermediate  ordinates,  plus  the  first 
and  last  ordinates,  divided  by  twice  the  number  of  divisions,  and  multi- 
plied by  the  length,  is  equal  to  the  area  of  the  figure. 

This  method  is  that  commonly  used;  it  is  sufficiently  near  to  exactness 
for  most  purposes. 

$d  Method,  Fig.  74. — Having  divided  the  figure  as  above,  measure  by  a 
scale  the  mean  depth  of  each  division,  at  the  middle  of  the  division;  add 
together  the  depths  of  all  the  divisions,  and  divide  the  sum  by  the  number 
of  divisions,  for  the  average  depth;  multiply  the  average  depth  by  the  length, 
which  gives  the  area. 

For  the  sake  of  obtaining  a  more  nearly  exact  result,  the  figure  may  be 
divided  into  two  half-parts,  c,  k,  Fig.  75,  one  at  each  end,  and  a  number 
of  whole  equal  parts,  d,e,f,g,h,i,j,  intermediately.  Then  the  ordinates 
separating  these  parts,  excluding  the  extreme  ordinates,  may  be  measured 


rT 

x*-* 

.  —  •  — 

7 

c   * 

«. 

f 

^ 

*. 

,; 

7  t 

Fig.  75-                  For  Area 

Fig.  76. 


direct,  and  the  sum  of  the  measurements  divided  by  the  number  of  them, 
and  multiplied  by  the  length,  for  the  area. 

Note. — In  dealing  with  figures  of  excessively  irregular  outline,  as  in  Fig. 
76,  representing  an  indicator-diagram  from  a  steam-engine,  mean  lines,  ab, 
cd,  may  be  substituted  for  the  actual  lines,  being  so  traced  as  to  intersect 
the  undulations,  so  that  the  total  area  of  the  spaces  cut  off  may  be  com- 
pensated by  that  of  the  extra  spaces  inclosed. 

Note  2. — The  figures  have  been  supposed  to  be  bounded  at  the  ends  by 
parallel  planes.  But  they  may  be  terminated  by  curves  or  angles,  as  in 
Fig.  76,  at  b,  when  the  extreme  ordinates  become  nothing. 


MENSURATION   OF   SOLIDS.  2/ 


MENSURATION   OF   SOLIDS. 

To  find  the  surface  of  a  prism  or  a  cylinder.  The  perimeter  of  the  end 
multiplied  by  the  height  gives  the  upright  surface;  add  twice  the  area  of 
an  end. 

To  find  the  cubic  contents  of  a  prism  or  a  cylinder.  Multiply  the  area  of 
the  base  by  the  height. 

To  find  the  surface  of  a  pyramid  or  a  cone.  Multiply  the  perimeter  of  the 
base  by  half  the  slant  height,  and  add  the  area  of  the  base. 

To  find  the  cubic  contents  of  a  pyramid  or  a  cone.  Multiply  the  area  of 
the  base  by  one-third  of  the  perpendicular  height. 

To  find  the  surface  of  a  frustum  of  a  pyramid  or  a  cone. — Multiply  the 
sum  of  the  perimeters  of  the  ends  by  half  the  slant  height,  and  add  the 
areas  of  the  ends. 

To  find  the  cubic  contents  of  a  frustum  of  a  pyramid,  or  a  cone. — Add 
together  the  areas  of  the  two  ends,  and  the  mean  proportional  between  them 
(that  is,  the  square  root  of  their  product),  and  multiply  the  sum  by  one-third 
of  the  perpendicular  height. 

Or,  when  the  ends  are  circles,  add  together  the  square  of  each  diameter, 
and  the  product  of  the  diameters,  and  multiply  the  sum  by  7854,  and  by 
one-third  of  the  height. 

To  find  the  cubic  contents  of  a  wedge. — To  twice  the  length  of  the  base 
add  the  length  of  the  edge;  multiply  the  sum  by  the  breadth  of  the  base, 
and  by  one-sixth  of  the  height. 

To  find  the  cubic  contents  of  a  prismoid  (a  solid  of  which  the  two  ends  are  un- 
egual  but  parallel  plane  figures  of  the  same  number  of  sides}. — To  the  sum 
of  the  areas  of  the  two  ends,  add  four  times  the  area  of  a  section  parallel  to 
and  equally  distant  from  both  ends;  and  multiply  the  sum  by  one-sixth  of 
the  length. 

Note. — This  rule  gives  the  true  content  of  all  frustums,  and  of  all  solids 
of  which  the  parallel  sections  are  similar  figures;  and  is  a  good  approxima- 
tion for  other  kinds  of  areas  and  solidities. 

To  find  the  surface  of  a  sphere. — Multiply  the  square  of  the  diameter  by 
3-1416. 

Note. — The  surface  of  a  sphere  is  equal  to  4  times  the  area  of  one  of  its 
great  circles. 

2.  The  surface  of  a  sphere  is  equal  to  the  convex  surface  of  its  circum- 
scribing cylinder. 

3.  The  surfaces  of  spheres  are  to  one  another  as  the  squares  of  their 
diameters. 

To  find  the  curve  surf  ace  of  any  segment  or  zone  of  a  sphere. — Multiply  the 
diameter  of  the  sphere  by  the  height  of  the  zone  or  segment,  and  by  3 '141 6. 

Note. — The  curve  surfaces  of  segments  or  zones  of  the  same  sphere  are 
to  one  another  as  their  heights. 

To  find  the  cubic  contents  of  a  sphere. — Multiply  the  cube  of  the  diameter 
by  -5236. 

Or,  multiply  the  surface  by  one-sixth  of  the  diameter. 


28  GEOMETRICAL   PROBLEMS. 

Note. — The  contents  of  a  sphere  are  two-thirds  of  the  contents  of  its 
circumscribing  cylinder. 

2.  The  contents  of  spheres  are  'to  one  another  as  the  cubes  of  their 
diameters. 

To  find  the  cubic  contents  of  a  segment  of  a  sphere. — From  3  times  the 
diameter  of  the  sphere  subtract  twice  the  height  of  the  segment;  multiply 
the  difference  by  the  square  of  the  height,  and  by  '5236. 

Or,  to  3  times  the  square  of  the  radius  of  the  base  of  the  segment,  add 
the  square  of  its  height;  and  multiply  the  sum  by  the  height,  and  by  '5236. 

To  find  the  cubic  contents  of  a  frustum  or  zone  of  a  sphere. — To  the  sum 
of  the  squares  of  the  radii  of  the  ends  add  ^3  of  the  square  of  the  height ; 
multiply  the  sum  by  the  height,  and  by  1-5708. 

To  find  the  cubic  contents  of  a  spheroid. — Multiply  the  square  of  the  re- 
volving axis  by  the  fixed  axis  and  by  '5236. 

Note. — The  contents  of  a  spheroid  are  two-thirds  of  the  contents  of  its 
circumscribing  cylinder. 

2.  If  the  fixed  and  revolving  axes  of  an  oblate  spheroid  be  equal  to  the 
revolving  and  fixed  axes  of  an  oblong  spheroid  respectively,  the  contents  of 
the  oblate  are  to  those  of  the  oblong  spheroid  as  the  greater  to  the  less  axis. 

To  find  the  cubic  contents  of  a  segment  of  a  spheroid. — ist.  When  the  base 
is  parallel  to  the  revolving  axis.  Multiply  the  difference  between  thrice  the 
fixed  axis  and  double  the  height  of  the  segment,  by  the  square  of  the  height, 
and  the  product  by  "5236.  Then, 

as  the  square  of  the  fixed  axis 

is  to  the  square  of  the  revolving  axis, 

so  is  the  last  product 

to  the  content  of  the  segment. 

2d.  When  the  base  is  perpendicular  to  the  revolving  axis.     Multiply  the 
difference  between  thrice  the  revolving  axis  and  double  the  height  of  the 
segment,  by  the  square  of  the  height,  and  the  product  by  '5236.     Then, 
as  the  revolving  axis 
is  to  the  fixed  axis, 
so  is  the  last  product 
to  the  content  of  the  segment. 

To  find  the  solidity  of  the  middle  frustum  of  a  spheroid. — ist.  When  the 
ends  are  circular,  or  parallel  to  the  revolving  axis.  To  twice  the  square  of 
the  middle  diameter,  add  the  square  of  the  diameter  of  one  end;  multiply 
the  sum  by  the  length  of  the  frustum,  and  the  product  by  "2618  for  the 
content. 

2d.  When  the  ends  are  elliptical,  or  perpendicular  to  the  revolving  axis. 
To  twice  the  product  of  the  transverse  and  conjugate  diameters  of  the  middle 
section,  add  the  product  of  the  transverse  and  conjugate  diameters  of  one 
end;  multiply  the  sum  by  the  length  of  the  frustum,  and  by  '2618  for  the 
content. 

To  find  the  cubic  contents  of  a  parabolic  conoid. — Multiply  the  area  of  the 
base  by  half  the  height. 

Or,  multiply  the  square  of  the  diameter  of  the  base  by  the  height,  and 
by  -3927. 

To  find  the  cubic  contents  of  a  frustum  of  a  parabolic  conoid. — Multiply 
half  the  sum  of  the  areas  of  the  two  ends  by  the  height  of  the  frustum. 


MENSURATION   OF   SOLIDS. 


29 


Or,  multiply  the  sum  of  the  squares  of  the  diameters  of  the  two  ends  by 
the  height,  and  by  '3927. 

To  find  the  cubic  contents  of  a  parabolic  spindle, — Multiply  the  square  of  the 
middle  diameter  by  the  length,  and  by  '41888. 

To  find  the  cubic  contents  of  the  middle  frustum  of  a  parabolic  spindle. — 
Add  together  8  times  the  square  of  the  largest  diameter,  3  times  the  square 
of  the  diameter  at  the  ends,  and  4  times  the  product  of  the  diameters; 
multiply  the  sum  by  the  length  of  the  frustum,  and  by  '05236. 

To  find  the  surface  and  the  cubic  contents  of  any  of  the  five  regular  solids,  Figs. 


Fig.  77. 


Fig.  78 


Fig.  80. 


Fig.  81. 


77)  78>  79?  8o>  8 1. — For  the  surface,  multiply  the  tabular  area  below,  by  the 
square  of  the  edge  of  the  solid. 

For  the  contents,  multiply  the  tabular  contents  below,  by  the  cube  of  the 
given  edge. 

Note. — A  regular  solid  is  bounded  by  similar  and  regular  plane  figures. 
There  are  five  regular  solids,  shown  by  Figs.  77  to  81,  namely: — 

The  tetrahedron,  bounded  by  four  equilateral  triangles. 

The  hexahedron,  or  cube,  bounded  by  six  squares. 

The  octahedron,  bounded  by  eight  equilateral  triangles. 

The  dodecaAedronJxyvs&ed.  by  twelve  pentagons. 

The  icosahedron,  bounded  by  twenty  equilateral  triangles. 

Regular  solids  may  be  circumscribed  by  spheres;  and  spheres  may  be 
inscribed  in  regular  solids. 

SURFACES  AND  CUBIC  CONTENTS  OF  REGULAR  SOLIDS. 


Number 
of  sides. 

Name. 

AREA. 
Edge  =  i. 

CONTENTS. 
Edge  =  i. 

4 
6 
8 

Tetrahedron  
Hexahedron  
Octahedron  

17320 
6'oooo 
3*464.1 

0*1178 

I  *OOOO 

0*4.714 

12 

Dodecahedron  

20*64=18 

7*6631 

2O 

Icosahedron.  . 

8*6603 

2*1817 

To  find  the  cubic  contents  of  an  irregular  solid. — Suppose  it  divided  into 
parts,  resembling  prisms  or  other  bodies  measurable  by  preceding  rules;  find 
the  content  of  each  part;  the  sum  of  the  contents  is  the  cubic  contents  of 
the  solid. 

Note. — The  content  of  a  small  part  is  found  nearly  by  multiplying  half 
the  sum  of  the  areas  of  each  end  by  the  perpendicular  distance  between 
them. 


30  GEOMETRICAL   PROBLEMS. 

Or,  the  contents  of  small  irregular  solids  may  sometimes  be  found  by  im- 
mersing them  under  water  in  a  prismatic  or  cylindrical  vessel,  and  observing 
the  amount  by  which  the  level  of  the  water  descends  when  the  solid  is 
withdrawn.  The  sectional  area  of  the  vessel  being  multiplied  by  the  descent 
of  the  level,  gives  the  cubic  contents. 

Or,  when  the  solid  is  very  large,  and  a  great  degree  of  accuracy  is  not 
requisite,  measure  its  length,  breadth,  and  depth  in  several  different  places, 
and  take  the  mean  of  the  measurement  for  each  dimension,  and  multiply 
the  three  means  together. 

Or,  when  the  surface  of  the  solid  is  very  extensive,  it  is  better  to  divide 
it  into  triangles,  to  find  the  area  of  each  triangle,  and  to  multiply  it  by  the 
mean  depth  of  the  triangle  for  the  contents  of  each  triangular  portion ;  the 
contents  of  the  triangular  sections  are  to  be  added  together. 

The  mean  depth  of  a  triangular  section  is  obtained  by  measuring  the 
depth  at  each  angle,  adding  together  the  three  measurements,  and  taking 
one-third  of  the  sum. 


MENSURATION   OF   HEIGHTS  AND   DISTANCES. 

To  find  the  height  of  an  accessible  object. — Measure  the  distance  from  the 
base  of  the  object  to  any  convenient  station  on  the  same  horizontal  plane; 
and  at  this  station  take  the  angle  of  altitude.     Then 
as  radius 

to  tangent  of  the  angle  of  altitude, 
so  is  the  horizontal  distance 

to  the  height  of  the  object  above  the  horizontal  plane  passing 
through  the  eye  of  the  observer.  Add  the  height  of  the  eye,  and  the  sum 
is  the  height  of  the  object. 

Note. — The  station  should  be  chosen  so  that  the  angle  of  altitude  should 
be  as  near  to  45°  as  practicable;  because  the  nearer  to  45°,  the  less  is  the 
error  in  altitude  arising  from  error  of  observation. 

When  the  angle  of  elevation  is  45°,  the  height  above  the  plane  of  the 
eye  is  equal  to  the  distance.  When  it  is  26°  34',  the  height  is  half  the  dis- 
tance. 

To  find  approximately  the  height  of  an  accessible  object. — There  are  four 
methods  based  on  the  principle  of  similar  triangles, 
i st  By  a  geometrical  square,  Fig.  82. — This  is  a 
square,  a  b,  with  two  sights  on  one  of  its  sides,  a  n, 
a  plumb-line  hung  from  one  extremity,  n,  of  that 
side,  and  each  of  the  two  sides  opposite  to  that 
extremity,  mb,ma,  divided  into  100  equal  parts; 
the  division  beginning  at  the  remote  ends,  so  that 
the  looth  divisions  meet  at  the  corner  m.  Let  re 
be  the  object,  and  the  sights  be  directed  to  the 
summit  e,  at  the  known  distance  ad.  When  the 
'~Hee£hTati°n  °f  a  plummet  cuts  the  side  b  m  at,  say,  c,  then  by  similar 
triangles,  n  b  :b  c :\  a  d :  d  e.  Or,  if  the  plumb-line 

cuts  the  side  a  m,  then  the  part  of  a  m  cut  off  is  to  a  n  : :  a  d  :  de.     Adding 
to  de  the  height  of  the  eye  rd,  the  sum  is  the  height  of  the  object,  re. 


MENSURATION    OF    HEIGHTS   AND   DISTANCES. 


2d.  By  shadows.  Fig.  83. — When  the  sun  shines,  fix  a  pole  be  in  the 
ground,  vertically,  and  measure  its  shadow  ab.    Measure  also  the  shadow  de 


Fig.  83. 


Mensuration  of  a  Height. 


Fig.  84. 


of  the  object  e  m;  then,  by  similar  triangles,  a  b  :  b  c  : :  de :  e  m,  the  height  of 
the  object. 

3d.  By  reflection,  Fig.  84. — Place  a  basin  of  water,  or  any  horizontal 
reflecting  surface,  at  a,  level  with  the  base  of  the  object  de,  and  retire  from 
it  till  the  eye  at  c  sees  the  top  of  the  object  e, 
in  the  centre  of  the  basin  at  a.  Then,  by  similar 
triangles,  ab\  be  \  \ad\de. 

4th.  By  two  poles,  Fig.  85. — Fix  two  poles  a  m, 
en,  of  unequal  lengths,  parallel  to  the  object  er, 
so  that  the  eye  of  the  observer  at  a,  the  top  of 
the  shorter  pole,  may  see  c,  the  top  of  the  longer 
pole,  in  a  line  with  e,  the  summit  of  the  object  re. 
By  similar  triangles,  a  b  :  b  c  :  :  a  d :  de;  and  adding 
r  d,  the  height  of  the  eye,  to  de,  the  sum  re  is  the 
height  of  the  object. 

To  find  the  distance  of  the  visible  horizon. — To 
half  the  logarithm  of  the  height  of  the  eye,  add  3-8105;  the  sum  is  the 
logarithm  of  the  distance  in  feet,  nearly. 

To  find  the  distance  of  an  object  by  the  motion  of  sound. — Multiply  the 
number  of  seconds  that  elapse  between  the  flash  or  other  sign  of  the  gene- 
ration of  the  sound  and  the  arrival  of  the  sound  to  the  ear,  by  1120.  The 
product  is  the  distance  in  feet. 

Note. — When  a  sound  generated  near  the  ear  returns  as  an  echo,  half  the 
interval  of  time  is  to  be  taken,  to  find  the  distance  of  the  reflecting  surface. 


Fig.  85.  Mensuration  of  a 
Height. 


32  MATHEMATICAL   TABLES. 

MATHEMATICAL   TABLES. 

TABLE  No.  I. — OF  LOGARITHMS  OF  NUMBERS  FROM  i  TO  10,000. 

Logarithms  consist  of  integers  and  decimals;  but,  for  the  sake  of  com- 
pactness, the  integers  have  been  omitted  in  the  table,  except  in  the  short 
preliminary  section  containing  the  complete  logarithms  of  numbers  from  i  to 
100.  The  table  No.  I.  contains  the  decimal  parts,  to  six  places,  of  the  loga- 
rithms of  numbers  from  i  to  10,000.  The  integer,  or  index,  or  character- 
istic of  a  logarithm,  standing  on  the  left-hand  side  of  the  decimal  point,  is 
a  number  less  by  i  than  the  number  of  figures  or  places  in  the  integer 
of  the  number.  If  a  number  contains  both  integers  and  decimals,  the 
index  is  regulated  according  to  the  integers.  If  it  contain  only  decimals, 
the  index  is  equal  to  the  number  of  cyphers  next  the  decimal  point,  plus  i; 
moreover,  the  index  is  negative,  and  is  so  distinguished  by  the  sign  minus, 
- ,  written  over  it. 

For  example,  to  illustrate  the  adjustment  of  the  integer  of  the  logarithm 
to  the  composition  of  the  number : — 

Number.  Logarithm. 

4743 3-676053 

474-3 2.676053 

47-43 1-676053 

4-743 0.676053 

-4743 1-676053 

•04743 ^.676053 

•004743 3-676053 

Still  more  for  the  sake  of  compactness,  the  first  two  figures  of  the  loga- 
rithms are  given  only  at  the  beginning  of  each  line  of  logarithms,  to  save 
repetition,  only  the  remaining  four  decimal  places  being  given  for  each 
logarithm.  In  seeking  for  a  logarithm,  the  eye  readily  takes  in  the  prefixed 
two  digits  at  the  commencement  of  each  line. 

Rules. — To  find  the  logarithm  of  a  number  containing  one  or  two  digits, 
look  for  the  number  in  the  preliminary  tablet  in  one  of  the  columns 
marked  No.,  and  find  the  logarithm  next  it.  Or,  look  in  the  body  of 
the  table  for  the  given  number  in  the  columns  marked  N,  with  one  or 
two  cyphers  following  it;  the  decimal  part  of  the  logarithm  is  in  the 
column  next  to  it.  For  example,  the  decimal  part  of  the  logarithm  of  3 
is  found,  in  the  column  next  to  the  number  300,  to  be  .477121,  and  as 
there  is  but  one  digit,  the  logarithm  is  completed  with  a  cypher,  thus, 
0.477121.  The  same  logarithm  stands  for  30,  except  that,  when  completed, 
it  becomes  1.477121.  Again,  take  the  number  37;  look  for  370  in  column 
N,  and  the  decimal  part  of  the  logarithm  is  found,  in  the  column  next  it, 
to  be  .568202,  which,  being  completed,  becomes  1.568202.  If  the  number 
be  .37,  the  logarithm  becomes  1.568202. 

To  find  the  logarithm  of  a  number  consisting  of  three  digits,  look  for  the 


EXPLANATION   AND   USES   OF   THE   TABLES.  33 

number  in  column  N,  and  find  the  logarithm  in  the  column  next  it,  as 
already  exemplified,  for  which  the  index  is  to  be  settled  and  prefixed  as 
before. 

If  the  number  consist  of  four  digits,  look  for  the  first  three  in  column  N, 
and  the  fourth  in  the  horizontal  line  at  the  head  or  at  the  foot  of  the  table. 
The  decimal  part  of  the  logarithm  is  found  opposite  the  three  first  digits 
and  under  or  over  the  fourth.  Take  the  number  5432;  opposite  543  in 
column  N,  and  in  the  column  headed  2,  is  the  logarithm  .734960,  to  which 
3  is  to  be"  prefixed,  making  3.734960.  If  the  number  be  5.432,  the 
complete  logarithm  is  0.734960. 

If  the  number  consist  of  five  or  more  digits,  find  the  logarithm  for  the 
first  four  as  above;  multiply  the  difference,  in  column  D,  by  the  remaining 
digits,  and  divide  by  10  if  there  be  only  one  digit  more,  by  100  if  there 
be  two  more,  and  so  on;  add  the  quotient  to  the  logarithm  for  the  first  four. 
The  sum  is  the  decimal  part  of  the  required  logarithm,  to  which  the  index 
is  to  be  prefixed.  For  example,  take  3.1416.  The  logarithm  of  3141  is 
.497068,  decimal  part;  and  the  difference,  138  x  6  -=-  10  =  83,  is  to  be  added, 
thus  — 

0.497068 


making  the  complete  logarithm,  ..................  0.497151 

To  find  the  number  corresponding  to  a  given  logarithm,  look  'fce       W 
logarithm  without  the  index.     If  it  be  found  exactly  or  within  two  or  &re$V/x< 
units  of  the  right-hand  digit,  then  the  first  three  figures  of  the  indicated 
number  will  be  found  in  the  number  column,  in  a  line  with  the  logarithm, 
and  the  fourth  figure  at  the  top  or  the  foot  of  the  column  containing  the 
logarithm.    Annex  the  fourth  figure  to  the  first  three,  and  place  the  decimal 
point  in  its  proper  position,  on  the  principles  already  explained. 

If  the  given  logarithm  differs  by  more  than  two  or  three  units  from  the 
nearest  in  the  table,  find  the  number  for  the  next  less  tabulated  logarithm, 
which  will  give  the  four  first  digits  of  the  required  number.  To  find  the 
fifth  and  sixth  digits,  subtract  the  tabulated  logarithm  from  the  given  loga- 
rithm, add  two  cyphers,  and  divide  by  the  difference  found  in  column  D 
opposite  the  logarithm.  Annex  the  quotient  to  the  four  digits  already 
found,  and  place  the  decimal  point.  For  example,  to  find  the  number 
represented  by  the  logarithm  2.564732:  — 

2-564732  given  logarithm. 
Log  ...................  367.0=  .........  2.564666  nearest  less. 


D  118)6600  (56  nearly. 
590 


700 
708 


Showing  that  the  required  number  is  367.056. 


To  multiply  together  two  or  more  numbers,  add  together  the  logarithms 

3 


34  MATHEMATICAL   TABLES. 

of  the  numbers,  and  the  sum  is  the  logarithm  of  the  product.     Thus,  to 
multiply  365  by  3.146: — 

Log    365 =  2.562293 

Log    3.146 =0.497759 

3.060052 
Log    1148 3-059942 


29  D   380)11000  (29  nearly. 

760 


1148.29 

3400 
3420 


Showing  that  the  product  is 1148.29. 

To  divide  one  number  by  another,  subtract  the  logarithm  of  the  divisor 
from  that  of  the  dividend,  and  the  remainder  is  the  logarithm  of  the 
quotient. 

To  find  any  power  of  a  given  number,  multiply  the  logarithm  of  the  num- 
ber by  the  exponent  of  the  power.  The  product  is  the  logarithm  of  the 
power. 

To  find  any  root  of  a  given  number,  divide  the  logarithm  of  the  number 
by  the  index  of  the  root.  The  quotient  is  the  logarithm  of  the  root. 

To  find  the  reciprocal  of  a  number,  subtract  the  decimal  part  of  the 
logarithm  of  the  number  from  o.oooooo;  add  i  to  the  index  of  the  loga- 
rithm, and  change  the  sign  of  the  index.  This  completes  the  logarithm  of 
the  reciprocal.  For  example,  to  find  the  reciprocal  of  230:— 

o.oooooo 
Log    230  = 2.361728 


3.638272  =  log  0.004348  (reciprocal). 
Inversely,  to  find  the  reciprocal  of  the  decimal  .004348:— 

o.oooooo 
Log    .004348  = 3^638272 


2.361728  =  ^   230  (reciprocal). 

Note. — It  will  be  found  in  practice,  for  the  most  part,  unnecessary  to 
note  the  indices  of  logarithms,  as  the  decimal  parts  are  in  most  cases  suffi- 
ciently indicative  of  the  numbers  without  the  indices.  The  exact  calcula- 
tion of  differences  may  also  in  most  cases  be  dispensed  with — rough  mental 
approximations  being  sufficiently  near  for  the  purpose — particularly  when 
the  numbers  contain  decimals.  The  indices  are,  however,  indispensable  in 
the  calculation  of  the  roots  of  numbers. 


EXPLANATION   AND    USES   OF   THE   TABLES.  35 

TABLE  No.  II. — OF  HYPERBOLIC  LOGARITHMS  OF  NUMBERS. 

In  this  table,  the  numbers  range  from  i.oi  to  30,  advancing  by  .01,  up 
to  the  whole  number  10;  and  thence  by  larger  intervals  up  to  30.  The 
hyperbolic  logarithms  of  numbers,  or  Neperian  logarithms,  as  they  are 
sometimes  called,  are  calculated  by  multiplying  the  common  logarithms  of 
the  given  numbers,  in  table  No.  I.,  by  the  constant  multiplier,  2.302585. 
The  hyperbolic  logarithms  of  numbers  intermediate  between  those  which 
are  given  in  the  table,  may  be  readily  obtained  by  interpolating  proportional 
•differences. 

TABLE  No.  III. — OF  CIRCUMFERENCES,  CIRCULAR  AREAS,  SQUARES  AND 
CUBES;  AND  OF  SQUARE  ROOTS  AND  CUBE  ROOTS. 

It  has  been  shown  how  to  calculate  the  powers  and  roots  of  numbers  by 
means  of  logarithms.  The  table  No.  III.  will  be  useful  for  reference.  It 
contains  the  powers  and  roots  of  numbers  consecutively  from  i  to  1000. 
The  circumferences  and  areas  of  circles,  due  to  the  numbers  contained  in 
the  first  columns,  considered  as  diameters,  are  also  given. 

By  a  suitable  adjustment  of  decimal  points  the  circumferences,  areas, 
squares  and  cubes,  may  be  determined  from  the  contents  of  the  table  for 
diameters  ten  or  a  hundred  times  as  much  as,  or  less  than,  the  values  given 
in  the  first  column. 

For  example,  if  the  number  378  in  the  first  column,  page  73,  be  taken  as 
37.8,  the  corresponding  circumference,  area,  square  and  cube  are  as  follows: 

Original.  Decimalized, 

Number 378  37.8 

Circumference 1,187.52  118.752 

Circular  area 112,221.09  1122.2109 

Square 142,884  1,428.84 

Cube 54,010,152  54,010.152 

TABLE  No.   IV. — OF  CIRCUMFERENCES  AND  AREAS  OF  CIRCLES  WITH 
SIDES  OF  EQUAL  SQUARES. 

The  table  No.  IV.  gives  the  circumferences  and  areas  of  circles  from 
Jg-  inch  to  120  inches  in  diameter,  advancing  by  sixteenths  of  an  inch  up 
to  6  inches  diameter;  thence  by  eighths  of  an  inch  to  50  inches  diameter; 
thence  by  quarters  of  an  inch  to  100  inches  diameter;  and  thence  by  half- 
inches  to  1 20  inches  diameter. 

Whilst  the  diameters  are  here  expressed  as  inches,  they  may  be  taken  as 
feet,  or  as  measures  of  any  other  denomination. 

The  column  of  sides  of  equal  squares,  contains  the  sides  of  squares  having 
the  same  area  as  the  circles  in  the  same  lines  of  the  table  respectively. 

TABLES  Nos.  V.  AND  VI. — OF  LENGTHS  OF  CIRCULAR  ARCS. 

The  lengths  of  circular  arcs  are  given  proportionally  to  that  of  the  radius, 
and  to  that  of  the  chord,  in  the  tables  Nos.  V.  and  VI.  In  the  first  of  these 
tables,  the  radius  is  taken  -  i,  and  the  number  of  degrees  in  the  arc  are 
given  in  the  first  column.  The  length  of  the  arc  as  compared  with  the 
radius  is  given  decimally  in  the  second  column. 


36  MATHEMATICAL   TABLES. 

In  the  second  table,  the  chord  is  taken  =  i,  and  the  rise  or  height  of  the 
arc,  expressed  decimally  as  compared  with  the  chord,  is  given  in  the  first 
column.  The  length  of  the  arc  relatively  to  the  chord  is  given  in  the 
second  column. 

To  use  the  first  table,  No.  V.,  find  the  proportional  length  of  the  arc 
corresponding  to  the  degrees  in  the  arc,  and  multiply  it  by  the  actual  length 
of  the  radius;  the  product  is  the  actual  length  of  the  arc. 

To  use  the  second  table,  No.  VI.,  divide  the  height  of  the  arc  by  the  chord 
for  the  proportional  height  of  the  arc,  which  find  in  the  first  column  of  the 
table;  the  proportional  length  of  the  arc  corresponding  to  it  being  multi- 
plied by  the  actual  length  of  the  chord,  gives  the  actual  length  of  the  arc. 

Note. — The  length  of  an  arc  of  a  circle  may  be  found  nearly  thus: — 
Subtract  the  chord  of  the  whole  arc  from  8  times  the  chord  of  half  the  arc. 
A  third  of  the  remainder  is  the  length  nearly. 

TABLE  No.  VII. — OF  AREAS  OF  CIRCULAR  SEGMENTS. 

The  areas  of  circular  segments  are  given  in  Table  No.  VII.,  in  proportional 
superficial  measure,  the  diameter  of  the  circle  of  which  the  segment  forms 
a  portion  being  =  i.  The  height  of  the  segment,  expressed  decimally  in 
proportion  to  the  diameter,  is  given  in  the  first  column,  and  the  relative 
area  in  the  second  column. 

To  use  the  table,  divide  the  height  by  the  diameter,  find  the  quotient  in 
the  table,  and  multiply  the  corresponding  area  by  the  square  of  the  actual 
length  of  the  diameter;  the  product  will  be  the  actual  area. 

TABLE  No.  VIII. — SINES,   COSINES,  TANGENTS,  COTANGENTS,  SECANTS, 
AND  COSECANTS  OF  ANGLES  FROM  o°  TO  90°. 

This  table,  No.  VIII.,  is  constructed  for  angles  of  from  o°  to  90°,  advancing 
by  10',  or  one-sixth  of  a  degree.  The  length  of  the  radius  is  equal  to  i, 
and  forms  the  basis  for  the  relative  lengths  given  in  the  table,  and  which 
are  given  to  six  places  of  decimals.  Each  entry  in  the  table  has  a  duplicate 
significance,  being  the  sine,  tangent,  or  secant  of  one  angle,  and  at  the 
same  time  the  cosine,  cotangent,  or  cosecant  of  its  complement.  For  this 
reason,  and  for  the  sake  of  compactness,  the  headings  of  the  columns  are 
reversed  at  the  foot ;  so  that  the  upper  headings  are  correct  for  the  angles 
named  in  the  left  hand  margin  of  the  table,  and  the  lower  headings  for 
those  named  in  the  right  hand  margin. 

To  find  the  sine,  or  other  element \  to  odd  minutes ;  divide  the  difference 
between  the  sines,  &c.,  of  the  two  angles  greater  and  less  than  the  given 
angle,  in  the  same  proportion  that  the  given  angle  divides  the  difference  of 
the  two  angles,  and  add  one  of  the  parts  to  the  sine  next  it. 

By  an  inverse  process  the  angle  may  be  found  for  any  given  sine,  &c., 
not  found  in  the  table. 

TABLE  No.  IX. — OF  LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  CO- 
TANGENTS OF  ANGLES  FROM  o°  TO  90°. 

This  table,  No.  IX.,  is  constructed  similarly  to  the  table  of  natural  sines,. 
&c.,  preceding.  To  avoid  the  use  of  logarithms  with  negative  indices,  the 
radius  is  assumed,  instead  of  being  equal  to  i,  to  be  equal  to  io10,  or 


EXPLANATION   AND   USES   OF   THE   TABLES.  37 

10,000,000,000;  consequently  the  logarithm  of  the  radius  =  10  log  10=  10. 
Whence,  if,  to  log  sine  of  any  angle,  when  calculated  for  a  radius  =  i,  there 
be  added  10,  the  sum  will  be  the  log  sine  of  that  angle  for  a  radius  =  io10. 
For  example,  to  find  the  logarithmic  sine  of  the  angle  15°  50'. 

Nat.  sine  15°  50'=  "272840;  its  log  =    1*435908 

add  =  io 


Logarithmic  sine  of  15°  50'=   9*435908 

When  the  logarithmic  sines  and  cosines  have  been  found  in  this  manner, 
the  logarithmic  tangents,  cotangents,  secants,  and  cosecants  are  found  from 
those  by  addition  or  subtraction,  according  to  the  correlations  of  the 
trigonometrical  elements  already  given,  and  here  repeated  in  logarithmic 
form: — 

Log  tan =  io  +  log  sin.  —  log.  cosin. 

Log  cotan =  20  -  log  tan. 

Log  sec =  20  -  log  cosin. 

Log  cosec =  20  -  log  sin. 

To  find  the  logarithmic  sine,  tangent,  &*c.,  of  any  angle. — When  the  number 
of  degrees  is  less  than  45°,  find  the  degrees  and  minutes  in  the  left  hand 
column  headed  angle,  and  under  the  heading  sine,  or  tangent,  &c.,  as 
required,  the  logarithm  is  found  in  a  line  with  the  angle. 

When  the  number  of  degrees  is  above  45°,  and  less  than  90°,  find  the 
degrees  and  minutes  in  the  right  hand  column  headed  angle,  and  in  the 
same  line,  above  the  title  at  the  foot  of  the  page,  sine  or  tangent,  &c.,  find 
the  logarithm  in  a  line  with  the  angle. 

When  the  number  of  degrees  is  between  90°  and  180°,  take  their  supple- 
ment to  1 80°;  when  between  180°  and  270°,  diminish  them  by  180°;  and 
when  between  270°  and  360°,  take  their  complement  to  360°,  and  find  the 
logarithm  of  the  remainder  as  before. 

If  the  exact  number  of  minutes  is  not  found  in  the  table,  the  logarithm 
of  the  nearest  tabular  angle  is  to  be  taken  and  increased  or  diminished  as 
the  case  may  be,  by  the  due  proportion  of  the  difference  of  the  logarithms 
of  the  angles  greater  and  less  than  the  given  angle. 

TABLE  No.  X. — RHUMBS,  OR  POINTS  OF  THE  COMPASS. 

The  Mariner's  Compass  is  a  circular  card  suspended  horizontally,  having 
a  thin  bar  of  steel  magnetized, — the  needle, — for  one  of  its  diameters;  the 
circumference  of  the  card  being  divided  into  32  equal  parts,  or  points,  and 
each  point  subdivided  into  quarters.  A  point  of  the  compass  is,  therefore, 
equal  to  (360° ---32  =  )  11°  15'. 

TABLE  No.  XI. — OF  RECIPROCALS  OF  NUMBERS. 

The  table  No.  XI.  contains  the  reciprocals  of  numbers  from  i  to  1000. 
It  has  already  been  shown  how  to  find  the  reciprocal  of  a  number  by  means 
of  logarithms. 


MATHEMATICAL   TABLES. 


TABLE   No.  I.— LOGARITHMS    OF   NUMBERS 

FROM    I    TO    IO,OOO. 


No. 

Log. 

No. 

Log. 

No. 

Log. 

No. 

Log. 

1 

2 

3 

4 
5 

0.000000 

0.301030 

0.477121 

0.602060 
0.698970 

26 
27 
28 
29 
30 

•414973 
•  43^64 
.447158 
.462398 
.477121 

51 

S2 
53 
54 
55 

1.707570 
1.716003 
.724276 
.732394 
•740363 

76 
77 
78 
79 
80 

.880814 
.886491 
.892095 
.897627 
.903090 

6 
9 

10 

0.778151 
0.845098 
0.903090 

0.954243 

I.OOOOOO 

31 

32 
33 
34 
35 

.491362 
•505150 
.518514 
•531479 
.544068 

56 

57 
58 

59 
60 

.748188 

:  763428 
.770852 
.778151 

81 

82 
83 
84 
85 

.908485 
.913814 
.919078 
.924279 
.929419 

11 

12 
13 
H 
15 

.041393 

.079181 

•113943 

.146128 
.176091 

36 

? 

39 
40 

•556303 
.  568202 

•579784 
.591065 
.602060 

61 

62 
63 
64 
65 

.785330 
.792392 

.799341 
.806180 
.812913 

86 

87 

88 

89 
90 

•934498 
•939519 
•944483 
•949390 
•954243 

16 

17 

18 

19 

20 

.204120 

.230449 

.255273 
.278754 

.301030 

41 

42 
43 
44 
45 

.612784 
.623249 
.633468 

.643453 
.653213 

66 
67 
68 
69 
70 

.819544 
.826075 
.832509 
.838849 
.845098 

91 

92 
93 
94 
95 

.959041 
.963788 
•968483 
.973128 
.977724 

21 

22 

23 
24 

25 

.322219 

•342423 
.361728 

.380211 

1.397940 

46 

47 
48 

49 

5° 

.662758 
.672098 
1.681241 
1.690196 
1.698970 

71 

72 
73 
74 

75 

.851258 

•857332 
.863323 
.869232 
.875061 

96 
97 
98 
99 

100 

.982271 
.986772 
.991226 
1.995635 

2.OOOOOO 

N 

01234 

56789 

D 

100 

101 
102 
102 
I03 
104 
I04 

oo-  oooo  0434  0868  1301  1734 
oo-  4321  4751  5181  5609  6038 
oo-  8600  9026  9451  9876  
01-  0300 
oi-  2837  3259  3680  4100  4521 
OI~  7°33  745  r  7868  8284  8700 

02-  

2l66   2598   3029   3461   3891 
6466   6894   7321   7748   8174 

0724   1147   1570   1993   2415 

4940  5360  5779  6197  6616 
9116  9532  9947  
0361  0775 

432 
428 

425 
424 
420 

417 
416 

105 

106 
107 
107 
108 
109 
109 

O2-  1189  1603  2Ol6  2428  2841 
02-  5306  5715  6125  6533  6942 
02-  9384  9789  
03-  0195  °600  1004 
03-  3424  3826  4227  4628  5029 
03-  7426  7825  8223  8620  9017 
04— 

3252  3664  4075  4486  4896 
7350  7757  8164  8571  8978 

1408  1812  2216  2619  3021 
5430  5830  6230  6629  7028 
94H  9811  
0207  0602  0998 

412 
408 

405 
404 
400 
398 

3Q7 

N 

01234 

56789 

D 

LOGARITHMS   OF   NUMBERS. 


39 


N 

o    i    2    34 

56789 

D 

110 

in 

1  12 

04-  1393  1787  2182  2576  2969 
04-  5323  57H  6105  6495  6885 

O4  Q2l8  0606  QQQ3 

3362  3755  4148  4540  4932 
7275  7664  8053  8442  8830 

393 

389 
388 

112 

"3 
114 
114. 

05-  0380  0766 

05-  3078  3463  3846  4230  4613 
05-  6905  7286  7666  8046  8426 
06  

1153  1538  1924  2309  2694 
4996  5378  576o  6142  6524 
8805  9185  9563  9942  
0320 

386 
383 
383 
379 

115 

116 
117 

06-  0698  1075  1452  1829  2206 

06-  4458  4832  5206  5580  5953 
06-  8186  8557  8927  9298  9668 

2582  2958  3333  3709  4083 
6326  6699  7071  7443  7815 

376 

373 
380 

117 
118 
119 

07-  
07-  1882  2250  2617  2985  3352 
°7-  5547  5912  6276  6640  7004 

0038  0407  0776  1145  1514 
3718  4085  4451  4816  5182 
7368  7731  8094  8457  8819 

370 
366 

363 

120 

O7  Ql8l  Q^43  QQO4 

362 

1  20 

08—  O266  0626 

0087  1347  1707  2067  2426 

O"-" 

360 

121 

122 

08-  2785  3144  3503  3861  4219 
08-  6360  6716  7071  7426  7781 
08  QQCX 

4576  4934  5291  5647  6004 
8136  8490  8845  9198  9552 

357 
355 

7CC 

I23 
124 

09-  0258  0611  0963  1315 
09-  3422  3772  4122  4471  4820 

1667  2018  2370  2721  3071 
5169  5518  5866  6215  6562 

353 
349 

125 

127 
128 
128 

09-  6910  7257  7604  7951  8298 

IO-  

io-  0371  0715  1059  1403  1747 
io-  3804  4146  4487  4828  5169 
io-  7210  7549  7888  8227  8565 

8644  8990  9335  9681  
0026 
2091  2434  2777  3119  3462 
5510  5851  6191  6531  6871 
8903  9241  9579  9916  
0253 

348 
346 
343 
34i 

338 

337 

129 

ii-  0590  0926  1263  1599  1934 

2270  2605  2940  3275  3609 

335 

130 

ii-  3943  4277  4611  4944  5278 
ii-  7271  7603  7934  8265  8595 

12- 

5611  5943  6276  6608  6940 
8926  9256  9586  9915  
0245 

OCOCO 

o  Co  Co 

0  HHCO 

132 
133 
134 

12-  0574  0903  1231  1560  1888 
12-  3852  4178  4504  4830  5156 

12-  7105  7429  7753  8076  8399 

2216  2544  2871  3198  3525 
5481  5806  6131  6456  6781 
8722  9045  9368  9690  

OOI2 

328 
325 
323 
323 

o<t 

135 

136 
137 
138 

13-  0334  0655  0977  1298  1619 
J3-  3539  3858  4177  4496  4814 
13-  6721  7037  7354  7671  7987 
13—  9879 

1939   2260   2580   2900   3219 

5:33  545  l  5769  6086  6403 
8303  8618  8934  9249  9564 

321 
3^ 

138 
139 

14-  0194  0508  0822  1136 
X4-  3015  3327  3639  395  i  4263 

1450  1763  2076  2389  2702 
4574  4885  5196  5507  5818 

3" 

140 

141 

14-  6128  6438  6748  7058  7367 

14-  92IQ  <K27  083? 

7676  7985  8294  8603  8911 

3°9 
308 

I4.I 

IS—  OI42  O44Q 

0756  1063  1370  1676  1982 

307 

142 
143 

14.4. 

15-  2288  2594  29OO  32O5  35IO 

I5~  5336  5640  5943  6246  6549 
is-  8362  8664  806  s  0266  <x67 

3815  4120  4424  4728  5032 
6852  7154  7457  7759  8061 
9868 

305 
303 

144 

1  6- 

0168  0469  0769  1068 

301 

145 

146 
147 

16-  1368  1667  1967  2266  2564 
16-  4353  4650  4947  5244  5541 
16-  7317  7613  7908  8203  8497 

2863  3161  3460  3758  4055 
5838  6134  6430  6726  7022 
8792  9086  9380  9674  9968 

299 
297 
295 

N 

01234 

56789 

D 

MATHEMATICAL   TABLES. 


N 

01234 

56789 

D 

148 
149 

17-  0262  0555  0848  1141  1434 
17-  3186  3478  3769  4060  4351 

1726  2019  2311  2603  2895 
4641  4932  5222  5512  5802 

293 
291 

150 

I  Cl 

17-  6091  6381  6670  6959  7248 

17  8Q77  Q264  9552  9839 

7536  7825  8113  8401  8689 

289 
287 

IDI 
151 
152 
r53 
i54 
J54 

18-  0126 
18-  1844  2129  2415  2700  2985 
18-  4691  4975  5259  5542  5825 
18-  7521  7803  8084  8366  8647 

19-  

0413  0699  0986  1272  1558 

3270  3555  3839  4123  4407 
6108  6391  6674  6956  7239 
8928  9209  9490  9771  
0051 

287 
285 
283 
281 
281 

155 

156 

157 
1^8 

iQ-  0332  0612  0892  1171  1451 
19-  3125  3403  3681  3959  4237 
ig-  5900  6176  6453  6729  7005 
IQ-  86^7  8032  0206  0481  07"^ 

1730  2010  2289  2567  2846 
4514  4792  5069  5346  5623 
7281  7556  7832  8107  8382 

279 
278 
276 

27C 

158 
159 

20-  

20-  1397  1670  1943  22l6  2488 

0029  0303  0577  0850  1124 
2761  3°33  33°5  3577  3848 

274 
272 

160 

161 

162 

20-  4120  4391  4663  4934  5204 

20-  6826  7096  7365  7634  7904 
2O-  (XIS  Q78^ 

5475  5746  6016  6286  6556 
8173  8441  8710  8979  9247 

271 
269 
268 

162 
163 
164 

21-  0051  0319  0586 
21-  2l88  2454  272O  2986  3252 

21-  4844  5109  5373  5638  5902 

0853   1  121   1388   1654   1921 

3518  3783  4049  4314  4579 
6166  6430  6694  6957  7221 

267 
266 
264 

165 

1  66 
167 
1  68 
169 
169 

21-  7484  7747  8010  8273  8536 

22-  OIOS  0370  0631  0892  1153 

22-  2716  2976  3236  3496  3755 

22-  5309  5568  5826  6084  6342 
22-  7887  8144  84CX)  8657  8913 
23-  

8798  9060  9323  9585  9846 
1414  1675  1936  2196  2456 
4015  4274  4533  4792  5051 
6600  6858  7115  7372  7630 
9170  9426  9682  9938  

OIQ3 

262 
26l 

259 

258 

257 
2^6 

170 

171 
172 
173 

173 

23-  0449  0704  0960  1215  1470 

23-  2996  3250  3504  3757  4011 
23-  5528  578i  6033  6285  6537 
23-  8046  8297  8548  8799  9049 
24— 

1724  1979  2234  2488  2742 

4264  4517  4770  5023  5276 

6789  7041  7292  7544  7795 
9299  955°  98oo  
0050  0300 

255 
253 
252 
251 

2<O 

174 

24-  0549  0799  1048  1297  1546 

1795  2044  2293  2541  2790 

249 

175 

176 
177 
177 

24-  3038  3286  3534  3782  4030 
24-  55  J3  5759  6006  6252  6499 
24-  7973  8219  8464  8709  8954 
25-  

4277  4525  4772  5019  5266 
6745  699i  7237  7482  7728 
9198  9443  9687  9932  
0176 

248 
246 
246 
24  S 

I78 
179 

25-  0420  0664  0908  1151  1395 
25-  2853  3096  3338  3580  3822 

1638  1881  2125  2368  2610 
4064  4306  4548  4790  5031 

243 
242 

180 

181 
182 

183 
184 

25-  5273  55H  5755  5996  6237 
25-  7679  79i8  8158  8398  8637 
26-  0071  0310  0548  0787  1025 
26-  2451  2688  2925  3162  3399 
26-  4818  5054  5290  5525  5761 

6477  6718  6958  7198  7439 
8877  9116  9355  9594  9833 
1263  1501  1739  1976  2214 
3636  3873  4109  4346  4582 
5996  6232  6467  6702  6937 

241 

239 
238 

237 
235 

185 

186 

26-  7172  7406  7641  7875  8110 
26-  <xi  3  0746  0080 

8344  8578  8812  9046  9279 

234 

274 

186 

27—  ...  0213  0446 

0679  0912  1144  1377  1609 

233 

187 
188 

27-  1842  2074  2306  2538  2770 
27-  4158  4389  4620  4850  5081 

3001  3233  3464  3696  3927 
5311  5542  5772  6002  6232 

232 
230 

N 

01234 

56789 

D 

LOGARITHMS  OF   NUMBERS. 


N 

01234 

56789 

D 

189 

27-  6462  6692  6921  7151  7380 

7609  7838  8067  8296  8525 

229 

190 

IQO 

27-  8754  8982  9211  9439  9667 
28- 

9895  
..  ..  0123  0351  0578  0806 

228 
228 

191 

192 

193 
194 

28-  1033  1261  1488  1715  1942 
28-  3301  3527  3753  3979  4205 
28-  5557  5782  6o°7  6232  6456 
28-  7802  8026  8249  8473  8696 

2169  2396  2622  2849  3°75 
4431  4656  4882  5107  5332 
6681  6905  7130  7354  7578 
8920  9143  9366  9589  9812 

227 
226 
225 

223 

195 

196 
197 
198 
199 
199 

29-  0035  0257  0480  0702  0925 
29-  2256  2478  2699  2920  3141 
29-  4466  4687  4907  5127  5347 
29-  6665  6884  7104  7323  7542 
29-  8853  9071  9289  9507  9725 
30-  

1147  1369  1591  1813  2034 
3363  3584  3804  4025  4246 
5567  5787  6007  6226  6446 
7761  7979  8198  8416  8635 

9943  
0161  0378  0595  0813 

222 
221 
220 
219 

218 
218 

200 

201 
2O2 
203 

2Od. 

30-  1030  1247  1464  1681  1898 
30-  3196  3412  3628  3844  4059 
30-  535i  5566  5781  5996  6211 
30-  7496  7710  7924  8137  8351 
30—  0630  084.3 

2114  2331  2547  2764  2980 
4275  4491  47o6  4921  5136 
6425  6639  6854  7068  7282 
8564  8778  8991  9204  9417 

217 

216 

215 
213 

217 

204 

31-  0056  0268  0481 

0693  0906  1118  1330  1542 

212 

205 

206 
207 
208 
209 

31-  1754  1966  2177  2389  2600 
31-  3867  4078  4289  4499  4710 
31-  5970  6180  6390  6599  6809 
31-  8063  8272  8481  8689  8898 
32-  0146  0354  0562  0769  0977 

2812  3023  3234  3445  3656 
4920  5130  5340  5551  5760 
7018  7227  7436  7646  7854 
9106  9314  9522  9730  9938 

1184   1391   1598   1805   2012 

211 
210 

209 
208 
207 

210 

211 
212 
2I3 
21^ 

32-  2219  2426  2633  2839  3046 
32-  4282  4488  4694  4899  5105 
32-  6336  6541  6745  6950  7155 
32-  8380  8583  8787  8991  9194 
33—  ... 

3252   3458   3665   3871   4077 
5310   5516   5721   5926   6131 

7359  7563  7767  7972  8176 
9398  9601  9805  
...   0008  02  1  1 

206 
205 
204 
204 
2O^ 

214 

33-  0414  0617  0819  1022  1225 

1427  1630  1832  2034  2236 

202 

215 

216 
217 
218 
218 

33-  2438  2640  2842  3044  3246 
33-  4454  4655  4856  5057  5257 
33-  6460  6660  6860  7060  7260 
33-  8456  8656  8855  9054  9253 
34-  ... 

3447  3649  3850  4051  4253 
5458  5658  5859  6059  6260 
7459  7659  7858  8058  8257 
9451  9650  9849  
0047  0246 

202 
2O  I 
200 
2OO 
199 

219 

34-  0444  0642  0841  1039  1237 

1435  1632  1830  2028  2225 

I98 

220 

221 
222 

223 

22^ 

34-  2423  2620  2817  3014  3212 
34-  4392  4589  4785  498i  5!78 
34-  6353  6549  6744  6939  7135 
34-  8305  8500  8694  8889  9083 
35-  

3409  3606  3802  3999  4196 
5374  5570  5766  5962  6157 
7330  7525  7720  7915  8110 
9278  9472  9666  9860  
0054 

197 
196 

195 
194 
194 

224 

35-  0248  0442  0636  0829  1023 

1216  1410  1603  1796  1989 

193 

225 

226 
227 

228 

22Q 

35-  2183  2375  2568  2761  2954 
35-  4108  4301  4493  4685  4876 
35-  6026  6217  6408  6599  6790 
35-  7935  8125  8316  8506  8696 
35  9835 

3H7  3339  3532  3724  39i6 
5068  5260  5452  5643  5834 
6981  7172  7363  7554  7744 
8886  9076  9266  9456  9646 

193 
192 
191 
190 
189 

229 

36-  0025  0215  0404  0593 

0783  0972  1161  1350  1539 

189 

N 

01234 

56789 

D 

MATHEMATICAL   TABLES. 


N 

01234 

56789 

D 

230 

231 

232 
233 

234 

36-  1728  1917  2105  2294  2482 
36-  3612  3800  3988  4176  4363 
36-  5488  5675  5862  6049  6236 
36-  7356  7542  7729  7915  8101 

36-  Q2l6  Q4OI  9^87  9772  99^8 

2671  2859  3048  3236  3424 
45  5  *  4739  4926  5113  5301 
6423  6610  6796  6983  7169 
8287  8473  8659  8845  9030 

iSS 
1  88 
187 
186 
1  86 

234 

235 
236 

237 
238 

239 

230 

37-  
37-  1068  1253  1437  1622  1806 
37-  2912  3096  3280  3464  3647 
37-  4748  4932  5115  5298  5481 
37-  6577  6759  6942  7124  7306 
37-  8398  8580  8761  8943  9124 
38-  .  . 

0143  0328  0513  0698  0883 
1991  2175  2360  2544  2728 
3831  4015  4198  4382  4565 
5664  5846  6029  6212  6394 
7488  7670  7852  8034  8216 
9306  9487  9668  9849  
0030 

185 

184 
184 

183 
182 
182 
181 

240 

241 
242 
243 
244 

38-  021  I  0392  0573  0754  0934 
38-  2017  2197  2377  2557  2737 

38-  3815  3995  4174  4353  4533 
38-  5606  5785  5964  6142  6321 
38-  7390  7568  7746  7923  8101 

1115  1296  1476  1656  1837 
2917  3097  3277  3456  3636 
4712  4891  5070  5249  5428 
6499  6677  6856  7034  7212 
8279  8456  8634  8811  8989 

181 

180 
179 
178 
178 

245 

38—  9l66  9343  QS2O  0608  OS?1? 

177 

245 
246 

247 
248 

249 

39-  
39-  0935  II12  I288  1464  1641 
39-  2697  2873  3048  3224  3400 
39-  4452  4627  4802  4977  5152 
39-  6199  6374  6548  6722  6896 

0051  '0228  0405  0582  0759 
1817  1993  2169  2345  2521 
3575  3751  3926  4101  4277 
5326  5501  5676  5850  6025 
7071  7245  7419  7592  7766 

177 
176 
176 

175 

174 

250 

251 

39-  7940  8114  8287  8461  8634 
39-  9674  9847 

8808  8981  9154  9328  9501 

173 
173 

20 

4O-  OO2O  0192  0365 

0538  0711  0883  1056  1228 

173 

252 
253 
254 

40-  1401  1573  1745  1917  2089 
40-  3121  3292  3464  3635  3807 
40-  4834  5005  5176  5346  5517 

2261  2433  2605  2777  2949 
3978  4149  4320  4492  4663 
5688  5858  6029  6199  6370 

172 
171 
171 

255 

256 

257 
257 
258 
259 

40-  6540  6710  688l  7051  7221 
40-  8240  8410  8579  8749  8918 

40-9933  
41-  0102  0271  0440  0609 
41-  1620  1788  1956  2124  2293 
41-  3300  3467  3635  3803  3970 

7391  7561  7731  7901  8070 
9087  9257  9426  9595  9764 

0777  0946  1114  1283  1451 
2461  2629  2796  2964  3132 
4137  43°5  4472  4639  4806 

170 
169 
169 

I67 

260 

261 

262 
263 

4i-  4973  5Ho  53°7  5474  564i 
41-  6641  6807  6973  7139  7306 
41-  8301  8467  8633  8798  8964 
41—  9956 

5808  5974  6141  6308  6474 
7472  7638  7804  7970  8135 
9129  9295  9460  9625  9791 

I67 
166 

'65 
i6<; 

263 
264 

42-  0121  0286  0451  O6l6 
42-  1604  1768  1933  2097  226l 

0781  0945  i  no  1275  1439 
2426  2590  2754  2918  3082 

'65 
164 

265 

266 
267 
268 
269 

42-  3246  3410  3574  3737  3901 
42-  4882  5045  5208  5371  5534 
42-  6511  6674  6836  6999  7161 
42-  8135  8297  8459  8621  8783 

42—  Q7^2  QQId 

4065  4228  4392  4555  4718 
5697  5860  6023  6186  6349 
7324  7486  7648  7811  7973 
8944  9106  9268  9429  9591 

164 
163 
162 
162 
162 

;:" 
269 

43-  0075  0236  0398 

°559  °72o  0881  1042  1203 

161 

270 

271 

43-  1364  1525  1685  1846  2007 
43-  2969  3130  3290  3450  3610 

2167  2328  2488  2649  2809 
3770  3930  4090  4249  4409 

161 
160 

N 

01234 

56789 

D 

LOGARITHMS   OF   NUMBERS. 


43 


N 

01234 

56789 

D 

272 

273 
274 

43-  4569  4729  .4888  .5048  5207 
43-  6163  6322  6481  6640  6799 
43-  7751  7909  8067  8226  8384 

5367  5526  5685  5844  6004 
6957  7116  7275  7433  7592 
8542  8701  8859  9017  9175 

159 

159 

T58 

275 

27G, 

43-  9333  9491  9M  9806  9964 
44— 

OI22   O27Q   O437   OSQ4   O7C.2 

158 

K8 

276 

277 
278 
279 

44-  0909  1066  1224  1381  1538 
44-  2480  2637  2793  2950  3106 
44-  4045  4201  4357  4513  4669 
44-  5604  5760  5915  6071  6226 

1695   1852   2009   2l66   2323 
3263   3419   3576   3732   3889 

4825  4981  5137  5293  5449 
6382  6537  6692  6848  7003 

157 
'57 
156 
i55 

280 

281 

281 

44-  7158  7313  7468  7623  7778 
44-  8706  8861  9015  9170  9324 
45—  • 

7933  8088  8242  8397  8552 
9478  9633  9787  994i  

OOQC. 

155 

154 

1^4. 

282 
283 
284 

45-  0249  0403  0557  0711  0865 
45-  1786  1940  2093  2247  2400 
45-  33i8  347i  3624  3777  3930 

1018  1172  1326  1479  1633 
2553  2706  2859  3012  3165 
4082  4235  4387  4540  4692 

154 
153 
153 

285 

286 

287 
288 
288 

45-  4845  4997  5150  53°2  5454 
45-  6366  6518  6670  6821  6973 
45-  7882  8033  8184  8336  8487 

45-  9392  9543  9694  9845  9995 
46-  

5606  5758  5910  6062  6214 
7125  7276  7428  7579  7731 
8633  8789  8940  9091  9242 

014.6  0206  04.47  oc,Q7  074.8 

152 
152 
151 

151 
J.CI 

289 

46-  0898  1048  1198  1348  1499 

1649  1799  1948  2098  2248 

150 

290 

291 
292 

293 
294 

46-  2398  2548  2697  2847  2997 
46-  3893  4042  4191  4340  4490 
46-  5383  5532  5680  5829  5977 
46-  6868  7016  7164  7312  7460 
46-  8347  8495  8643  8790  8938 

3146  3296  3445  3594  3744 
4639  4788  4936  5085  5234 
6126  6274  6423  6571  6719 
7608  7756  7904  8052  8200 
9085  9233  9380  9527  9675 

150 
149 

149 
148 
148 

295 

46-  0822  0060 

147 

3 

297 
298 
299 

47-  01  16  0263  0410 
47-  1292  1438  1585  1732  1878 
47-  2756  2903  3049  3195  3341 
47-  4216  4362  4508  4653  4799 
47-  5671  5816  5962  6107  6252 

°557  °7°4  0851  0998  1145 
2025  2171  2318  2464  2610 

3487  3633  3779  3925  4071 
4944  5090  5235  5381  5526 
6397  6542  6687  6832  6976 

147 
146 
146 
146 
145 

300 

301 
302 
303 
304 

47-  7121  7266  7411  7555  7700 
47-8566  8711  8855  8999  9143 
48-  0007  0151  0294  0438  0582 
48-  1443  J586  1729  1872  2016 
48-  2874  3016  3159  3302  3445 

7844  7989  8133  8278  8422 
9287  9431  9575  9719  9863 

0725   0869   IOI2   1156   1299 
2159   2302   2445   2588   2731 
3587   3730   3872   4015   4157 

145 

144 
144 
H3 
143 

305 

306 

307 
308 

1OQ 

48-  4300  4442  4585  4727  4869 
48-  5721  5863  6005  6147  6289 
48-  7138  7280  7421  7563  7704 
48-  8551  8692  8833  8974  9114 

4.8  Q(X8 

5011  5153  5295  5437  5579 
6430  6572  6714  6855  6997 
7845  7986  8127  8269  8410 
9255  9396  9537  9677  9818 

142 
142 
141 
141 
140 

ouv 
3°9 

49-  0099  0239  0380  0520 

O66l   080I   0941   I08l   1222 

140 

310 

3" 

312 

313 
3H 

49-  1362  1502  1642  1782  1922 
49-  2760  2900  3040  3179  3319 
49-4155  4294  4433  4572  4711 
49-  5544  5683  5822  5960  6099 
49-  6930  7068  7206  7344  7483 

2062   2201   2341   2481   2621 

3458  3597  3737  3876  4015 
4850  4989  5128  5267  5406 
6238  6376  6515  6653  6791 
7621  7759  7897  8035  8173 

140 
139 
139 
139 

138 

N 

01234 

56789 

D 

44 


MATHEMATICAL   TABLES. 


N 

01234 

56789 

D 

315 

49-  8311  8448  8586  8724  8862 

AQ  Q687  0824.  QQ62 

8999  9137  ,  9275  9412  955° 

138 

•*  , 

ijO  OO99  0236 

0374  0511  0648  0785  0922 

137 

OJ  CO  CO  O 

vo  oo-^  < 

50-  1059  1196  1333  1470  1607 
50-  2427  2564  2700  2837  2973 

50-  3791  3927  4063  4199  4335 

1744  1880  2017  2154  2291 
3109  3246  3382  3518  3655 
4471  4607  4743  4878  5014 

'37 

136 

136 

320 

321 
322 

50-  5150  5286  5421  5557  5693 
50-  6505  6640  6776  6911  7046 
50-  7856  7991  8126  8260  8395 

c.O  Q2O3  Q337  Q4.7I  0606  Q74.O 

5828  5964  6099  6234  6370 
7181  7316  7451  7586  7721 
8530  8664  8799  8934  9068 

9874. 

136 
135 
135 

I  34. 

323 

CJ  

OOOQ   OI4.3   O277   O4.I  I 

324 

51-  0545  0679  0813  0947  1081 

1215  1349  1482  1616  1750 

134 

325 

326 
327 

329 

51-  1883  2017  2151  2284  2418 
51-  3218  3351  3484  3617  3750 
51-  4548  4681  4813  4946  5079 
51-  5874  6006  6139  6271  6403 
51-  7196  7328  7460  7592  7724 

2551   2684   28l8   2951   3084 
3883   4016   4149  4282  4415 

5211  5344  5476  5609  5741 
6535  6668  6800  6932  7064 
7855  7987  8119  8251  8382 

CO  CO  CO  N  M 

CO  CO  CO  CO  CO 

330 

51-  8514  8646  8777  8909  9040 

c.1  0828  QQ^Q 

9i7i  9303  9434  9566  9697 

131 

331 

tj2  0090  O22I  0353 

0484  0615  0745  0876  1007 

131 

JO1 
332 

333 
334 

52-  1138  1269  1400  1530  1661 

52-  2444  2575  2705  2835  2966 
52-  3746  3876  4006  4136  4266 

1792  1922  2053  2183  2314 
3096  3226  3356  3486  3616 
4396  4526  4656  4785  4915 

130 
130 

335 

336 
337 
338 
338 

52-  5045  5174  53°4  5434  5563 
52-  6339  6469  6598  6727  6856 
52-  7630  7759  7888  8016  8145 
52-  8917  9045  9174  9302  9430 

CO  

5693  5822  5951  6081  6210 
6985  7114  7243  7372  7501 
8274  8402  8531  8660  8788 

9559  9687  9815  9943  
0072 

129 
129 
129 
128 
128 

339 

53-  O2OO  0328  0456  0584  O7I2 

0840  0968  1096  1223  1351 

128 

340 

342 
343 
344 

53-  1479  1607  1734  1862  1990 
53-  2754  2882  3009  3136  3264 
53-  4026  4153  4280  4407  4534 
53-  5294  5421  5547  5674  58oo 
53-  6558  6685  6811  6937  7063 

2117  2245  2372  2500  2627 
339i  35  18  3645  3772  3899 
4661  4787  4914  5041  5167 
5927  6053  6180  6306  6432 
7189  73J5  744i  7567  7693 

128 
127 
127 
126 
126 

345 

346 

53-  7819  7945  8071  8197  8322 

53—  9O76  92O2  9327  Q4^2  CK78 

8448  8574  8699  8825  8951 

Q7O3  0820  QQ<?4 

126 
126 

346 
347 
348 
349 

54-  
54-  0329  0455  0580  0705  0830 
54-  1579  1704  1829  1953  2078 
54-  2825  2950  3074  3199  3323 

0079  0204 
0955   I080   1205   1330   1454 
2203   2327   2452   2576  27OI 

3447  3571  3696  3820  3944 

125 
125 
125 
124 

350 

351 

352 
353 
354 
354 

54-  4068  4192  4316  4440  4564 
54-  53°7  543i  5555  5678  5802 
54-  6543  6666  6789  6913  7036 
54-  7775  7898  8021  8144  8267 
54-  9003  9126  9249  9371  9494 
55-  

4688  4812  4936  5060  5183 
5925  6049  6172  6296  6419 
7159  7282  7405  7529  7652 
8389  8512  8635  8758  8881 
9616  9739  9861  9984  
0106 

I24 
124 
I23 
I23 
123 
123 

355 

356 
357 

55-  0228  0351  0473  0595  0717 
55-  1450  1572  1694  1816  1938 
55-  2668  2790  2911  3033  3155 

0840  0962  1084  I2o6  1328 
2060  2181  2303  2425  2547 
3276  3398  3519  3640  3762 

122 
122 
121 

N 

01234 

56789 

D 

LOGARITHMS   OF   NUMBERS. 


45 


N 

01234 

56789 

D 

358 
359 

55-  3883  4004  4126  4247  4368 
55-  5094  5215  5336  5457  5578 

4489  4610  4731  4852  4973 
5699  5820  5940  6061  6182 

121 

121 

360 

361 
362 
161 

55-  6303  6423  6544  6664  6785 
55-  75°7  7627  7748  7868  7988 
55-  8709  8829  8948  9068  9188 

ee  QQO7 

6905  7026  7146  7267  7387 
8108  8228  8349  8469  8589 
9308  9428  9548  9667  9787 

1  2O 
120 
120 
1  2O 

363 
364 

56-  ....#.  0026  0146  0265  0385 

56-  noi  1221  1340  1459  1578 

0504  0624  0743  0863  0982 
1698  1817  1936  2055  2174 

II9 
119 

365 

366 

367 
368 
369 

56-  2293  2412  2531  2650  2769 

56-  3481  3600  3718  3837  3955 
56-  4666  4784  4903  5021  5139 
56-  5848  5966  6084  6202  6320 
56-  7026  7144  7262  7379  7497 

2887  3006  3125  3244  3362 
4074  4192  4311  4429  4548 
5257  5376  5494  5612  5730 
6437  6555  6673  6791  6909 
7614  7732  7849  7967  8084 

II9 
119 

118 
118 
118 

370 

37i 
•271 

56-  8202  8319  8436  8554  8671 
56-  9374  9491  9608  9725  9842 
57~ 

8788  8905  9023  9140  9257 
9959  ......  

OO76   OIQ^   O3OQ   O4.2O 

117 
117 
117 

372 
373 

374 

57-  0543  0660  0776  0893  1010 
57-  1709  1825  1942  2058  2174 
57-  2872  2988  3104  3220  3336 

1126  1243  1359  1476  1592 
2291  2407  2523  2639  2755 
3452  3568  3684  3800  3915 

"7 

116 

116 

375 

376 
377 
378 
379 

57-  4031  4147  4263  4379  4494 

57-  5lg8  5303  5419  5534  565° 
57-  6341  6457  6572  6687  6802 
57-  7492  7607  7722  7836  7951 
57-  8639  8754  8868  8983  9097 

4610  4726  4841  4957  5072 
5765  5880  5996  6111  6226 
6917  7032  7147  7262  7377 
8066  8181  8295  8410  8525 
9212  9326  9441  9555  9669 

116 
"5 
"5 
"5 
114 

380 

57-  9784  9898  

114 

380 
38i 
382 
383 
384 

58-  OOI2  OI26  O24I 
58-  0925  1039  1153  1267  1381 
58-  2063  2177  2291  2404  2518 

58-  3'99  3312  3426  3539  3652 
58-  4331  4444  4557  4670  4783 

0355  0469  0583  0697  0811 
1495  l6°8  1722  1836  1950 
2631  2745  2858  2972  3085 
3765  3879  3992  4105  4218 
4896  5009  5122  5235  5348 

114 

114 
114 
H3 
H3 

385 

386 
387 
388 
389 
389 

58-  5461  5574  5686  5799  5912 
58-  6587  6700  6812  6925  7037 
58-  7711  7823  7935  8047  8160 
58-  8832  8944  9056  9167  9279 

58-  995°  
59-  0061  0173  0284  0396 

6024  6137  6250  6362  6475 
7149  7262  7374  7486  7599 
8272  8384  8496  8608  8720 
939  i  9503  96i5  9726  9838 

0507  0619  0730  0842  0953 

"3 

112 
112 

112 
112 
112 

390 

39i 

392 
393 
394 

59-  1065  1176  1287  1399  1510 
59-  2177  2288  2399  2510  2621 
59-  3286  3397  3508  3618  3729 

59-  4393  45°3  4614  4724  4834 
59-  5496  5606  5717  5827  5937 

1621  1732  1843  1955  2066 
2732  2843  2954  3064  3175 
3840  3950  4061  4171  4282 
4945  5055  5J65  5276  5386 
6047  6157  6267  6377  6487 

III 
III 
III 
HO 
HO 

395 

396 
397 
398 

59-  6597  6707  6817  6927  7037 
59-  7695  7805  7914  8024  8134 
59-  8791  8900  9009  9119  9228 
59-  9883  9992  .... 

7146  7256  7366  7476  7586 
8243  8353  8462  8572  8681 
9337  9446  9556  9665  9774 

HO 
HO 
109 
IOQ 

398 
399 

60-  oioi  0210  0319 
60-  0973  1082  1191  1299  1408 

0428  0537  0646  0755  0864 
1517  1625  1734  1843  1951 

109 
109 

400 

60-  2060  2169  2277  2386  2494 

2603  2711  2819  2928  3036 

108 

N 

01234 

56789 

D 

46 


MATHEMATICAL  TABLES. 


N 

01234 

56789 

D 

401 

402 

403 

404 

60-  3144  3253  3361  3469  3577 
60-  4226  4334  4442  4550  4658 
60-  5305  5413  552i  5628  5736 
60-  6381  6489  6596  6704  6811 

3686  3794  3902  4010  4118 
4766  4874  4982  5089  5197 
5844  5951  6059  6166  6274 
6919  7026  7133  7241  7348 

1  08 
108 
1  08 
107 

405 

406 
407 
407 
408 
409 

60-  7455  7562  7669  7777"  7884 
60-  8526  8633  8740  8847  8954 
60-  9594  9701  9808  9914  

6l-  0021 

61-  0660  0767  0873  0979  1086 
61-  1723  1829  1936  2042  2148 

7991  8098  8205  8312  8419 
9061  9167  9274  9381  9488 

0128  0234  0341  0447  °554 
1192  1298  1405  1511  1617 
2254  2360  2466  2572  2678 

S-S-S-S-SS 

410 

411 

412 

414 

61-  2784  2890  2996  3102  3207 
6  i-  3842  3947  4053  4159  4264 
61-  4897  5003  5108  5213  5319 
61-  5950  6055  6160  6265  6370 
61-  7000  7105  7210  7315  7420 

3313  3419  3525  3630  3736 
4370  4475  4581  4686  4792 
5424  5529  5634  5740  5845 
6476  6581  6686  6790  6895 
7525  7629  7734  7839  7943 

106 
1  06 
105 

105 

415 

416 
4.16 

61-  8048  8153  8257  8362  8466 
61-  9093  9198  9302  9406  9511 
62- 

8571  8676  8780  8884  8989 
9615  9719  9824  9928  
0032 

105 
105 

IOA 

417 
418 
419 

62-  0136  0240  0344  0448  0552 
62-  1176  1280  1384  1488  1592 
62-  2214  2318  2421  2525  2628 

0656  0760  0864  0968  1072 
1695  r799  I9°3  2007  21  10 
2732  2835  2939  3042  3146 

104 
104 
104 

420 

421 
422 

423 
424 

62-  3249  3353  3456  3559  3663 
62-  4282  4385  4488  4591  4695 
62-  5312  5415  5518  5621  5724 
62-  6340  6443  6546  6648  6751 
62-  7366  7468  7571  7673  7775 

3766  3869  3973  4076  4179 
4798  4901  5004  5107  5210 
5827  5929  6032  6135  6238 
6853  6956  7058  7161  7263 
7878  7980  8082  8185  8287 

103 
103 
103 
103 
1  02 

425 

426 
426 

62-  8389  8491  8593  8695  8797 
62-  9410  9512  9613  9715  9817 
63- 

8900  9002  9104  9206  9308 

9919  
002  i  0123  0224  0326 

1  02 

102 
I  O2 

427 
428 
429 

63-  0428  0530  0631  0733  0835 
63-  1444  1545  1647  1748  1849 
63-  2457  2559  2660  2761  2862 

0936  1038  1139  1241  1342 
1951  2052  2153  2255  2356 
2963  3064  3165  3266  3367 

102 
IOI 
IOI 

430 

432 
433 
434 

63-  3468  3569  3670  3771  3872 
63-  4477  4578  4679  4779  4880 
63-  5484  5584  5685  5785  5886 
63-  6488  6588  6688  6789  6889 
63-  7490  7590  7690  7790  7890 

3973  4074  4175  4276  4376 
4981  5081  5182  5283  5383 
5986  6087  6187  6287  6388 
6989  7089  7189  7290  7390 
7990  8090  8190  8290  8389 

IOI 
IOI 
IOO 
IOO 
IOO 

435 

4^6 

63-  8489  8589  8689  8789  8888 
63-  9486  9586  9686  9785  988=; 

8988  9088  9188  9287  9387 
9984  

IOO 
IOO 

40  . 

4.^6 

64- 

0084  0183  0283  0382 

99 

437 
438 
439 

64-  0481  0581  0680  0779  0879 
64-  1474  1573  1672  1771  1871 
64-  2465  2563  2662  2761  2860 

0978  1077  1177  1276  1375 
1970  2069  2168  2267  2366 
2959  3058  3i56  3255  3354 

99 
99 
99 

440 

441 
442 
443 
444 

64-3453  3551  3650  3749  3847 
64-  4439  4537  4636  4734  4832 
64-  5422  5521  5619  5717  5815 
64-  6404  6502  6600  6698  6796 
64-  7383  748i  7579  7676  7774 

3946  4044  4143  4242  4340 
493  i  5029  5127  5226  5324 
5913  6011  6110  6208  6306 
6894  6992  7089  7187  7285 
7872  7969  8067  8165  8262 

99 
98 
98 
98 

98 

N 

01234 

56789 

D 

LOGARITHMS   OF   NUMBERS. 


47 


N 

01234 

56789 

D 

445 

446 
446 

64-  8360  8458  8555  8653  8750 

64-  9335  9432  953°  9627  9724 
65-  ?  

8848  8945  9043  9140  9237 
9821  9919  

OOl6   OII3   O2IO 

97 
97 

Q7 

447 
448 

449 

65-  0308  0405  0502  0599  0696 
65-  1278  1375  1472  1569  1666 
65-  2246  2343  2440  2536  2633 

0793  0890  0987  1084  1181 
1762  1859  1956  2053  2150 

2730  2826  2923  3019  3116 

97 
97 
97 

450 

45i 

452 
453 
454 

65-  3213  33°9  3405  35°2  3598 
65-  4177  4273  4369  4465  4562 

65-  5^8  5235  5331  5427  5523 
65-  6098  6194  6290  6386  6482 
65-  7056  7152  7247  7343  7438 

3695  3791  3888  3984  4080 

4658  4754  4850  4946  5042 
5619  5715  5810  5906  6002 
6577  6673  6769  6864  6960 
7534  7629  7725  7820  7916 

96 

96 

96 

96 
96 

455 

456 
457 

65-  8011  8107  8202  8298  8393 
65-  8965  9060  9155  9250  9346 
65-  9916  

8488  8584  8679  8774  8870 
9441  9536  9631  9726  9821 

95 
95 

oe 

457 
458 
459 

66-  oon  0106  0201  0296 
66-  0865  0960  1055  1150  1245 
66-  1813  1907  2002  2096  2191 

0391  0486  0581  0676  0771 
1339  1434  1529  1623  1718 
2286  2380  2475  2569  2663 

95 

95 
95 

460 

461 
462 

463 

464 

66-  2758  2852  2947  3041  3135 
66-  3701  3795  3889  3983  4078 
66-  4642  4736  4830  4924  5018 
66-  5581  5675  5769  5862  5956 
66-  6518  6612  6705  6799  6892 

3230  3324  3418  3512  3607 
4172  4266  4360  4454  4548 
5112  5206  5299  5393  5487 
6050  6143  6237  6331  6424 
6986  7079  7173  7266  7360 

94 
94 
94 
94 

94 

465 

466 
467 
467 
468 
469 

66~  7453  7546  7640  7733  7826 
66-  8386  8479  8572  8665  8759 
66-  9317  9410  9503  9596  9689 
67-  
67-  0246  0339  0431  0524  0617 
67-  1173  1265  1358  1451  1543 

7920  8013  8106  8199  8293 
8852  8945  9038  9131  9224 
9782  9875  9967  
0060  0153 
0710  0802  0895  0988  1080 
1636  1728  1821  1913  2005 

93 
93 
93 
93 
93 
93 

470 

47i 
472 

473 
474 

67-  2098  2190  2283  2375  2467 
67-  3021  3113  3205  3297  3390 
67-  3942  4034  4126  4218  4310 
67-  4861  4953  5045  5137  5228 
67-  5778  5870  5962  6053  6145 

2560  2652  2744  2836  2929 
3482  3574  3666  3758  3850 
4402  4494  4586  4677  4769 
5320  5412  5503  5595  5687 
6236  6328  6419  6511  6602 

92 
92 
92 
92 
92 

475 

476 
477 
478 
478 

67-  6694  6785  6876  6968  7059 
67-  7607  7698  7789  7881  7972 
67-  8518  8609  8700  8791  8882 
67-  9428  9519  9610  9700  9791 
68—  

7151  7242  7333  7424  75!6 
8063  8154  8245  8336  8427 
8973  9064  '9155  9246  9337 
9882  9973  
•  •        0063  0154  0245 

9i 
9i 
9i 
9i 
91 

479 

68~  °336  0426  0517  0607  0698 

0789  0879  0970  1060  1151 

9i 

480 

481 
482 
483 
484 

68-  1241  1332  1422  1513  1603 
68-  2145  2235  2326  2416  2506 
68-  3047  3137  3227  3317  3407 
68-  3947  4037  4127  4217  4307 
68-  4845  4935  5025  5114  5204 

1693  1784  1874,  1964  2055 
2596  2686  2777  2867  2957 

3497  3587  3677  3767  3857 
4396  4486  4576  4666  4756 

5294  5383  5473  5563  5652 

90 
90 
90 
90 
90 

485 

486 

111 
400 

480 

68-  5742  5831  5921  6010  6100 
68-  6636  6726  6815  6904  6994 
68-  7529  7618  7707  7796  7886 
68-  8420  8509  8598  8687  8776 

68—  Q3OQ  Q^Q8  Qd.86  Q^7t»  Q66<1 

6189  6279  6368  6458  6547 
7083  7172  7261  7351  7440 
7975  8064  8153  8242  8331 
8865  8953  9042  9131  9220 

Q7^3   Q84.I   QQ^O 

89 
89 
89 
89 
89 

N 

01234 

56789 

D 

48                                        MATHEMATICAL  TABLES. 

N 

01234 

56789 

D 

489 

60- 

0019    0107 

89 

490 

491 
492 
493 
494 

69-  0196  0285  0373  0462  0550 
69-  1081  1170  1258  1347  1435 
69-  1965  2053  2142  2230  2318 
69-  2847  2935  3°23  3111  3:99 
69-  3727  3815  3903  3991  4078 

0639    0728    0816    0905    0993 
1524    1612     1700    1789    1877 
2406    2494    2583    2671    2759 

3287    3375    3463    3551     3639 
4166    4254    4342    4430    4517 

88 
88 
88 

495 

496 
497 
498 
499 

6g-  4605  4693  4781  4868  4956 
69-  5482  5569  5657  5744  5832 
69-  6356  6444  6531  6618  6706 
6g-  7229  7317  7404  7491  7578 
6g-  Sioi  8188  8275  8362  8449 

5044    5131     5219    5307    5394 
5919    6007    6094    6182    6269 
6793    6880    6968    7055    7142 
7665     7752    7839    7926    8014 
8535    8622    8709    8796    8883 

88 
87 
87 
87 
87 

500 

5oi 
502 

503 
504 

6g-  8970  9057  9144  9231  9317 

60-  08^8  QQ24 

9404    9491     9578    9664    9751 

87 
87 
87 
86 
86 
86 

70-  ooi  i  0098  0184 

70-  0704  0790  0877  0963  1050 
70-  1568  1654  1741  1827  1913 
70-  2431  2517  2603  2689  2775 

0271    0358    0444    0531    0617 

1136      1222       1309       1395       1482 
1999      2086      2172      2258      2344 
2861      2947      3033      3119      3205 

505 

506 

5°7 
508 

5°9 

70-  3291  3377  3463  3549  3635 
70-  4151  4236  4322  4408  4494 
70-  5008  5094  5179  5265  5350 
70-  5864  5949  6035  6120  6206 
70-  6718  6803  6888  6974  7059 

3721     3807    3893    3979    4065 
4579    4665    4751    4837    4922 
5436    5522    5607    5693    5778 
6291     6376    6462    6547    6632 
7144    7229    7315    7400    7485 

86 
86 

86 
85 
85 

510 

5" 
512 

512 
5i3 
5H 

70-  7570  7655  7740  7826  7911 
70-  8421  8506  8591  8676  8761 
70-  9270  9355  9440  9524  9609 

7996    8081     8166    8251     8336 
8846    8931    9015    9100    9185 

9694    9779    9863    9948     
0073 

85 
85 
85 

85 
84 

71-  0117  0202  0287  0371  0456 

71-  0963  1048  1132  1217  1301 

0540    0625    0710    0794    0879 
1385     1470     1554     1639    1723 

515 

516 

518 

71-  1807  1892  1976  2060  2144 
71-  2650  2734  2818  2902  2986 
7i-  349i  3575  3659  3742  3826 
71-  4330  4414  4497  4581  4665 
71-  5167  5251  5335  54i8  5502 

2229    2313    2397    2481    2566 
3070    3154    3238    3323    3407 
39  10    3994    4078    4l62    4246 
4749    4833    49i6    5000    5084 
5586    5669    5753    5836    5920 

84 
84 
84 
84 
84 

520 

522 
523 
524 
524 

71-  6003  6087  6170  6254  6337 
71-  6838  6921  7004  7088  7171 
71-  7671  7754  7837  7920  8003 
71-  8502  8585  8668  8751  8834 
7i-  9331  94H  9497  958o  9663 
72-  

6421    6504    6588    6671     6754 
7254    7338    7421     7504    7587 
8086    8169    8253    8336    8419 
8917    9000    9083    9165    9248 
9745    9828    9911     9994     
0077 

83 
83 
83 
83 
83 
83 

525 

526 
527 
528 
529 

72-  0159  0242  0325  0407  0490 
72-  0986  1068  1151  1233  1316 
72-  1811  1893  1975  2058  2140 
72-  2634  2716  2798  2881  2963 
72-  3456  3538  3620  3702  3784 

0573    °655    0738    0821    0903 
1398    1481     1563    1646    1728 

2222      2305      2387      2469      2552 

3045    3127    3209    3291     3374 
3866    3948    4030    4112    4194 

83 
82 
82 
82 
82 

530 

53i 
532 
533 

72-  4276  4358  4440  4522  4604 
72-  5095  5!76  5258  5340  5422 
72-  5912  5993  6075  6156  6238 
72-  6727  6809  6890  6972  7053 

4685    4767    4849    4931     5013 
5503    5585     5667    5748    5830 
6320    6401     6483    6564    6646 
7134    7216    7297    7379    7460 

82 
82 
82 

81 

N 

01234 

56789 

D 

LOGARITHMS   OF   NUMBERS. 


49 


N 

01234 

56789 

D 

534 

72-  7541  7623  7704  7785  7866 

7948  8029  8110  8191  8273 

81 

535 

536 
537 
537 
538 
539 

72-  8354  8435  8516  8597  8678 
72-  9165  9246  9327  9408  9489 

72-  9974  
73-  0055  0136  0217  0298 
73-  0782  0863  0944  1024  1105 
73-  1589  1669  1750  1830  1911 

8759  8841  8922  9003  9084 
9570  9651  9732  9813  9893 

0378  0459  0540  0621  0702 
1186  1266  1347  1428  1508 
1991  2072  2152  2233  2313 

81 
81 
81 
81 

81 

Si 

540 

54i 
542 
543 
544 

73-  2394  2474  2555  2635  2715 
73-  3197  3278  3358  3438  35*8 
73-  3999  4079  4i6o  4240  4320 
73-  4800  4880  4960  5040  5120 
73-  5599  5679  5759  5838  59'8 

2796  2876  2956  3037  3117 

3598  3679  3759  3839  3919 
4400  4480  4560  4640  4720 
5200  5279  5359  5439  5519 
5998  6078  6157  6237  6317 

80 

80 
80 
80 
80 

645 

546 

547 
548 

549 
CAQ 

73-  6397  6476  6556  6635  6715 
73-  7193  7272  7352  743i  75" 
73-  7987  8067  8146  8225  8305 
73-  8781  8860  8939  9018  9097 
73-  9572  9651  973  i  98io  9889 

74— 

6795  6874  6954  7034  7113 
7590  7670  7749  7829  7908 
8384  8463  8543  8622  8701 
9177  9256  9335  9414  9493 
9968  
0047  0126  0205  0284 

So 
79 
79 
79 
79 
70 

550 

551 

552 
553 
554 

74-  0363  0442  0521  0600  0678 
74-  1152  1230  1309  1388  1467 
74-  1939  2018  2096  2175  2254 
74-  2725  2804  2882  2961  3039 
74-  35  I0  3588  3667  3745  3823 

0757  0836  0915  0994  1073 
1546  1624  1703  1782  1860 
2332  2411  2489  2568  2647 
3118  3196  3275  3353  3431 
3902  3980  4058  4136  4215 

79 
79 
79 
78 

/8 

555 

556 
557 
558 
559 

74-  4293  4371  4449  4528  4606 
74-  5075  5153  5231  5309  5387 
74-  5855  5933  ooii  6089  6167 
74-  6634  6712  6790  6868  6945 
74-  7412  7489  7567  7645  7722 

4684  4762  4840  4919  4997 

5465  5543  5621  5699  5777 
6245  6323  6401  6479  6556 
7023  7101  7179  7256  7334 
7800'  7878  7955  8033  8  1  ID 

78 
78 
78 
78 
78 

560 

56i 

562 
562 
563 
564 

74-  8188  8266  8343  8421  8498 
74-  8963  9040  9118  9195  9272 
74-  9736  9814  9891  9968  
75-  0045 
75-  0508  0586  0663  0740  0817 
75-  1279  1356  1433  ^lo  !587 

8576  8653  8731  8808  8885 
935°  9427  95°4  9582  9659 

OI23   O2OO   O277   0354   0431 
0894   0971   1048   1125   1202 

1664  1741  1818  1895  *972 

77 
77 

% 

77 

77 

565 

566 

567 
568 
569 

75-  2048  2125  2202  2279  2356 

75-  2816  2893  2970  3047  3123 
75-  3583  366o  3736  3813  3889 
75-  4348  4425  4501  4578  4654 
75-  5"2  5189  5265  5341  5417 

2433  2509  2586  2663  2740 
3200  3277  3353  3430  3506 
3966  4042  4119  4195  4272 
4730  4807  4883  4960  5036 
5494  5570  5646  5722  5799 

77 
77 
77 
76 
76 

570 

57i 

572 
573 
574 

75-  5875  5951  6027  6103  6180 
75-  6636  6712  6788  6864  6940 
75-  7396  7472  7548  7624  7700 
75-  8155  8230  8306  8382  8458 
75-  8912  8988  9063  9139  9214 

6256  6332  6408  6484  6560 
7016  7092  7168  7244  7320 
7775  7851  7927  8003  8079 
8533  8609  8685  8761  8836 
9290  9366  9441  95  i  7  9592 

76 

76 
76 
76 
76 

575 

C7C 

75-  9668  9743  9819  9894  9970 
76- 

0045   OI2I   0196   O272   0347 

76 
75 

576 

577 

76-  0422  0498  0573  0649  0724 
76-  1176  1251  1326  1402  1477 

0799  0875  0950  1025  noi 

1552   1627   1702   1778   1853 

75 
75 

N 

01234 

56789 

D 

MATHEMATICAL   TABLES. 


N 

01234 

56789 

D 

578 
579 

76-  1928  2003  2078  2153  2228 
76-  2679  2754  2829  2904  2978 

2303  2378  2453  2529  2604 
3053  3128  3203  3278  3353 

75 

75 

580 

581 
582 
533 
584 

76-  3428  3503  3578  3653  3727 
76-  4176  4251  4326  4400  4475 
76-  4923  4998  5072  5147  5221 
76-  5669  5743  5818  5892  5966 
76-  6413  6487  6562  6636  6710 

3802  3877  3952  4027  4101 
4550  4624  4699  4774  4848 
5296  5370  5445  5520  5594 
6041  6115  6190  6264  6338 
6785  6859  6933  7007  7082 

75 
75 
75 
74 
74 

585 

586 

£ 

n88 

76-  7156  7230  7304  7379  7453 
76-  7898  7972  8046  8120  8194 
76-  8638  8712  8786  8860  8934 
76-  9377  9451  9525  9599  9^73 
77— 

7527  7601  7675  7749  7823 
8268  8342  8416  8490  8564 
9008  9082  9156  9230  9303 
9746  9820  9894  9968  
0042 

74 
74 
74 
74 
74. 

DotJ 
589 

77-  0115  0189  0263  0336  0410 

0484  0557  0631  0705  0778 

74 

590 

59i 
592 
593 
594 

77-  0852  0926  0999  1073  1146 
77-  1587  1661  1734  1808  1881 
77-  2322  2395  2468  2542  2615 
77-  3°55  3128  3201  3274  3348 
77-  37S6  3860  3933  4006  4079 

I22O   1293   1367   1440   1514 
1955   2028   2102   2175   2248 
2688   2762   2835   2908   2981 

3421  3494  3567  3640  3713 
4152  4225  4298  4371  4444 

74 
73 
73 
73 
73 

595 

596 
597 
598 

599 

77-  4517  4590  4663  4736  4809 
77-  5246  5319  5392  5465  5538 
77-  5974  6047  6120  6193  6265 
77-  6701  6774  6846  6919  6992 
77-  7427  7499  7572  7644  7717 

4882  4955  5028  5100  5173 
5610  5683  5756  5829  5902 
6338  6411  6483  6556  6629 
7064  7137  7209  7282  7354 
7789  7862  7934  8006  8079 

73 
73 
73 
73 
72 

600 

60  1 
602 

77-  8151  8224  8296  8368  8441 
77-  8874  8947  9019  9091  9163 
77-  9^96  9669  9741  9813  988=; 

8513  8585  8658  8730  8802 
9236  9308  9380  9452  9524 
991:7  

72 
72 

72 

602 

78- 

0029  oioi  0173  0245 

72 

603 
604 

78-  0317  0389  0461  0533  0605 
78-  1037  1109  1181  1253  1324 

0677  0749  0821  0893  0965 
1396  1468  1540  1612  1684 

72 
72 

605 

606 
607 
608 
609 

78-  1755  1827  1899  1971  2042 
78-  2473  2544  2616  2688  2759 
78-  3189  3260  3332  3403  3475 
78-  3904  3975  4046  4118  4189 
78-  4617  4689  4760  4831  4902 

2114  2186  2258  2329  2401 
2831  2902  2974  3046  3117 
3546  3618  3689  3761  3832 
4261  4332  4403  4475  4546 
4974  5045  5  "6  5187  5259 

72 
72 

71 
71 
71 

610 

611 
612 
613 
614 

78-  5330  5401  5472  5543  56l5 
78-  6041  6112  6183  6254  6325 
78-  6751  6822  6893  6964  7035 
78-  7460  7531  7602  7673  7744 
78-  8168  8239  8310  8381  8451 

5686  5757  5828  5899  5970 
6396  6467  6538  6609  6680 
7106  7177  7248  7319  7390 
7815  7885  7956  8027  8098 
8522  8593  8663  8734  8804 

71 
71 
71 
71 
71 

615 

616 

78-  8875  8946  9016  9087  9157 

78-  9^81  Q6^I  Q722  Q7Q2  Q86^ 

9228  9299  9369  9440  9510 

QQ33 

71 
7O 

616 

7Q- 

OOO4  0074  OI44  O2  I  ^ 

70 

617 
618 
619 

7Q-  0285  0356  0426  0496  0567 

79-  0988  1059  1129  1199  1269 
79-  1691  1761  1831  1901  1971 

0637  0707  0778  0848  0918 
1340   1410   1480   1550   1620 
2041   21  1  1   2l8l   2252   2322 

70 
70 
70 

620 

621 

622 

79-  2392  2462  2532  2602  2672 
79-  3092  3162  3231  3301  3371 
79-  3790  3860  3930  4000  4070 

2742   28l2   2882   2952   3022 
3441   3511   3581   3651   3721 

4139  4209  4279  4349  4418 

70 
70 
70 

N 

01234 

56789 

D 

LOGARITHMS   OF  NUMBERS. 


N 

01234 

56789 

D 

623 
624 

79-  4488  4558  4627  4697  4767 
79-  5l85  5254  5324  5393  5463 

4836  4906  4976  5045  5115 
5532  5602  5672  5741  5811 

70 
70 

625 

626 
627 
628 
629 

79-  5880  5949  6019  6088  6158 
79-  6574  6644  6713  6782  6852 
79-  7268  7337  7406  7475  7545 
79-  7960  8029  8098  8167  8236 
79-  8651  8720  8789  8858  8927 

6227  6297  6366  6436  6505 
6921  6990  7060  7129  7198 
7614  7683  7752  7821  7890 
8305  8374  8443  8513  8582 
8996  9065  9134  9203  9272 

69 
69 

69 
69 

69 

630 

631 

632 

633 
634 

79-  9341  9409  9478  9547  9616 
80-  0029  0098  0167  0236  0305 
80-  0717  0786  0854  0923  0992 
80-  1404  1472  1541  1609  1678 
80-  2089  2158  2226  2295  2363 

9685  9754  9823  9892  9961 
0373  0442  0511  0580  0648 
1061  1129  1198  1266  1335 

1747   1815   1884   1952   2021 

2432  2500  2568  2637  2705 

69 

*9 
69 

& 

69 

635 

636 

637 
638 

639 

80-  2774  2842  2910  2979  3047 
80-  3457  3525  3594  3662  3730 
80-  4139  4208  4276  4344  4412 
80-  4821  4889  4957  5025  5093 
80-  55oi  5569  5637  5705  5773 

3116  3184  3252  3321  3389 

3798  3867  3935  4003  4071 
4480  4548  4616  4685  4753 
5161  5229  5297  5365  5433 
5841  5908  5976  6044  6112 

68 
68 
68 
68 
68 

640 

641 

642 
643 
-644 

80-  6180  6248  6316  6384  6451 
80-  6858  6926  6994  7061  7129 
80-  7535  7603  7670  7738  7806 
80-  8211  8279  8346  8414  8481 
80-  8886  8953  9021  9088  9156 

6519  6587  6655  6723  6790 
7197  7264  7332  7400  7467 
7873  7941  8008  8076  8143 
8549  8616  8684  8751  8818 
9223  9290  9358  9425  9492 

68 
68 
68 
67 
67 

645 

645 
646 
647 
648 
649 

80-  9560  9627  9694  9762  9829 
81-  
81-  0233  0300  0367  0434  0501 
81-  0904  0971  1039  1106  1173 
81-  1575  1642  1709  1776  1843 
81-  2245  2312  2379  2445  2512 

9896  9964  
0031  0098  0165 
0569  0636  0703  0770  0837 
1240  1307  1374  1441  1508 

1910   1977   2044   2III   2178 
2579   2646   2713   2780   2847 

67 
67 
67 
67 
67 
67 

650 

65i 

652 

653 
654 

81-  2913  2980  3047  3114  3181   3247  3314  3381  3448  3514 
81-  3581  3648  3714  3781  3848  3914  3981  4048  4114  4181 
81-  4248  4314  4381  4447  4514  4581  4647  4714  4780  4847 
81^.4913  4980  5046  5113  5179  5246  5312  5378  5445  5511 
8i-5578.5644  57"  5777  5843  59io  5976  6042  6109  6175 

67 
67 
67 
66 
66 

655 

656 

657 
658 

659 

81-  6241  6308  6374  6440  6506 
8  i-  6904  6970  7036  7102  7169 
81-  7565  7631  7698  7764  7830 
81-  8226  8292  8358  8424  8490 
8  i-  8885  8951  9017  9083  9149 

6573   6639   6705   6771   6838 

7235  7301  7367  7433  7499 
7896  7962  8028  8094  8160 
8556  8622  8688  8754  8820 
9215  9281  9346  9412  9478 

66 
66 
66 
66 
66 

660 

660 
66  1 
662 
663 
664 

81-  9544  9610  9676  9741  9807 
82-  
82-  020  i  0267  0333  0399  0464 

82-  0858   0924   0989   1055   I  120 

82-  1514  1579  1645  I7io  1775 

82-  2168   2233   2299   2364   2430 

9873  9939  
0004  0070  0136 
053°  °595  °66i  0727  0792 
1186  1251  1317  1382  1448 
1841  1906  1972  2037  2103 
2495  2560  2626  2691  2756 

66 
66 
66 
66 
65 
65 

665 

666 
667 
668 

82-  2822   2887   2952   3018   3083 

82-  3474  3539  3605  3670  3735 
82-  4126  4191  4256  4321  4386 
82-  4776  4841.  4906  4971  5036 

3148  3213  3279  3344  3409 
3800  3865  3930  3996  4061 
4451  4516  4581  4646  4711 
5101  5166  5231  5296  5361 

65 

55 

65 

65 

N      01234 

56789 

D 

MATHEMATICAL   TABLES. 


N 

01234 

56789 

D 

669 

82-  5426  5491  5556  5621  5686 

5751  58i5  588o  5945  6010 

65 

670 

671 
672 
673 
674 

82-  6075  6140  6204  6269  6334 
82-  6723  6787  6852  6917  6981 
82-  7369  7434  7499  7563  7628 
82-  8015  8080  8144  8209  8273 
82-  8660  8724  8789  8853  8918 

6399  6464  6528  6593  6658 
7046  7111  7175  7240  7305 
7692  7757  7821  7886  7951 
8338  8402  8467  8531  8595 
8982  9046  9111  9175  9239 

65 
65 

$ 

64 

64 

675 

676 

82-  9304  9368  9432  9497  9561 

82  QQ4.7 

9625  9690  9754  9818  9882 

64 

64. 

676 

677 
678 
679 

83-  oon  0075  0139  0204 

83-  0589  0653  0717  0781  0845 
83-  1230  1294  1358  1422  1486 

83-  1870  1934  1998  2062  2126 

0268  0332  0396  0460  0525 
0909  0973  IO37  IIQ2  IJ66 
1550  1614  1678  1742  1806 
2189  2253  2317  2381  2445 

64 

64 
64 

64 

680 

681 
682 
683 
684 

83-  2509  2573  2637  2700  2764 

83-  3H7  3211  3275  3338  3402 
83-  3784  3848  3912  3975  4039 
83-  4421  4484  4548  4611  4675 
83-  5056  5120  5183  5247  5310 

2828  2892  2956  3020  3083 
3466  3530  3593  3657  3721 
4103  4166  4230  4294  4357 
4739  4802  4866  4929  4993 
5373  5437  55°°  5564  5627 

64 
64 
64 
64 

63 

685 

686 
687 
688 
689 

83-  5691  5754  5817  588i  5944 
83-  6324  6387  6451  6514  6577 
83-  6957  7020  7083  7146  7210 
83-  7588  7652  7715  7778  7841 
83-  8219  8282  8345  8408  8471 

6007  6071  6134  6197  6261 
6641  6704  6767  6830  6894 
7273  7336  7399  7462  7525 
7904  7967  8030  8093  8156 
8534  8597  8660  8723  8786 

63 
63 
63 
63 
63 

690 

691 
6q  i 

83-  8849  8912  8975  9038  9101 

83-  9478  954i  9604  9667  9729 
84- 

9164  9227  9289  9352  9415 
9792  9855  99i8  9981  
0043 

63 
63 
63 

wyi 
692 
693 
694 

84-  0106  0169  0232  0294  0357 
84-  0733  0796  0859  0921  0984 
84-  1359  1422  1485  1547  1610 

0420  0482  0545  0608  0671 
1046  1109  1172  1234  1297 
1672  1735  1797  1860  1922 

63 
63 
63 

695 

696 

697 
698 

699 

84-  1985  2O47  21  IO  2172  2235 

84-  2609  2672  2734  2796  2859 
84-  3233  3295  3357  3420  3482 
84-  3855  3918  3980  4042  4104 
84-  4477  4539  4601  4664  4726 

2297  2360  2422  2484  2547 
2921  2983  3046  3108  3170 
3544  3606  3669  3731  3793 
4166  4229  4291  4353  4415 
4788  4850  4912  4974  5036 

62 
62 
62 
62 
62 

700 

701 

702 

703 
704 

84-  5098  5160  5222  5284  5346 
84-  5718  5780  5842  5904  5966 
84-  6337  6399  6461  6523  6585 
84-  6955  7017  7079  7141  7202 
84-  7573  7634  7696  7758  7819 

5408  5470  5532  5594  5656 
6028  6090  6151  6213  6275 
6646  6708  6770  6832  6894 
7264  7326  7388  7449  7511 
7881  7943  8004  8066  8128 

62 
62 
62 
62 
62 

705 

706 
707 
708 
709 

84-  81-89  8251  8312  8374  8435 
84-  8805  8866  8928  8989  9051 
84-  9419  9481  9542  9604  9665 
85-  o°33  0095  0156  0217  0279 
85-  0646  0707  0769  0830  0891 

8497  8559  8620  8682  8743 
9112  9174  9235  9297  9358 
9726  9788  9849  9911  9972 
0340  0401  0462  0524  0585 
0952  1014  1075  JI36  JI97 

62 
61 
61 
61 
61 

710 

711 
712 
7i3 

7i4 

85-  1258  1320  1381  1442  1503 
85-  1870  1931  1992  2053  2114 
85-  2480  2541  2602  2663  2724 
85-  3090  313°  3211  3272  3333 
85-  3698  3759  3820  3881  3941 

1564  1625  1686  1747  1809 
2175  2236  2297  2358  2419 
2785  2846  2907  2968  3029 

3394  3455  35*6  3577  3^37 
4002  4063  4124  4185  4245 

61 
61 
61 
61 
61 

N 

01234 

56789 

D 

LOGARITHMS   OF   NUMBERS. 


53 


N 

01234 

56789 

D 

715 

716 
717 
718 
719 

85-  4306  4367  4428  4488  4549 
85-  4913  4974  5°34  5°95  5^6 
85-  5519  558o  5640  57oi  576i 
85-  6124  6185  6245  6306  6366 
85-  6729  6789  6850  6910  6970 

4610  4670  4731  4792  4852 
5216  5277  5337  5398  5459 
5822  5882  5943  6003  6064 
6427  6487  6548  6608  6668 
7031  7091  7152  7212  7272 

61 
61 
61 
60 
60 

720 

721 
722 

723 

724. 

85-  7332  7393  7453  75^  7574 
85-  7935  7995  8056  8116  8176 
85-  8537  8597  8657  8718  8778 
85-  9138  9J98  9258  93i8  9379 

85—  Q7^Q  Q7QQ  Q8sQ  QQl8  QQ78 

7634  7694  7755  7815  7875 
8236  8297  8357  8417  8477 
8838  8898  8958  9018  9078 
9439  9499  9559  9619  9679 

60 

60 
60 
60 
60 

724 

86-  

0038  0098  0158  0218  0278 

60 

725 

726 
727 
728 
729 

86-  0338  0398  0458  0518  0578 
86-  0937  0996  1056  1116  1176 
86-  1534  1594  1654  1714  1773 
86-  2131  2191  2251  2310  2370 
86-  2728  2787  2847  2906  2966 

0637  0697  0757  0817  0877 
1236  1295  1355  HI5  H75 

1833   1893   !952   2012   2072 
2430   2489   2549   2608   2668 
3025   3085   3144   3204   3263 

60 

60 
60 
60 
60 

730 

73i 
732 
733 
734 

86-  3323  3382  3442  3501  3561 
86-  3917  3977  4036  4096  4155 
86-  4511  4570  4630  4689  4748 
86-  5104  5163  5222  5282  5341 
86-  5696  5755  5814  5874  5933 

3620  3680  3739  3799  3858 
4214  4274  4333  4392  4452 
4808  4867  4926  4985  5045 
5400  5459  5519  5578  5637 
5992  6051  6110  6169  6228 

59 
59 
59 
59 
59 

735 

736 
737 
738 
739 

86-  6287  6346  6405  6465  6524 
86-  6878  6937  6996  7055  7114 
86-  7467  7526  7585  7644  7703 
86-  8056  8115  8174  8233  8292 
86-  8644  8703  8762  8821  8870 

6583  6642  6701  6760  6819 
7173  7232  7291  7350  7409 
7762  7821  7880  7939  7998 
8350  8409  8468  8527  8586 
8938  8997  9056  9114  9173 

59 
59 
59 
59 
59 

740 

74i 

741 

86-  9232  9290  9349  9408  9466 
86-  9818  9877  9935  9994  
87-  °°53 

9525  9584  9642  9701  9760 
on  i  0170  0228  0287  0345 

59 
59 

en 

742 
743 
744 

87-  0404  0462  0521  0579  0638 
87-  0989  1047  1106  1164  1223 
87-  1573  l63i  1690  1748  1806 

0696  0755  °8i3  0872  0930 
1281  1339  1398  1456  1515 
1865  1923  1981  2040  2098 

| 

58 
58 

745 

746 

747 
748 

749 

87-  2156  2215  2273  2331  2389 
87-  2739  2797  2855  2913  2972 

87-  332i  3379  3437  3495  3553 
87-  3902  3960  4018  4076  4134 
87-  4482  4540  4598  4656  4714 

2448  2506  2564  2622  2681 
3030  3088  3146  3204  3262 
3611  3669  3727  3785  3844 
4192  4250  4308  4366  4424 
4772  4830  4888  4945  5003 

58 
58 
58 
58 
58 

750 

75i 
752 

753 
754 

87-  5061  5119  5177  5235  5293 
87-  5640  5698  5756  5813  5871 
87-  6218  6276  6333  6391  6449 
87-  6795  6853  6910  6968  7026 
87-  737i  7429  7487  7544  7602 

5351  5409  5466  5524  5582 
5929  5987  6045  6102  6160 
6507  6564  6622  6680  6737 
7083  7141  7199  7256  7314 
7659  7717  7774  7832  7889 

58 
58 
58 
58 
58 

755 

756 
757 
753 
758 
759 

87-  7947  8004  8062  8119  8177 
87-  8522  8579  8637  8694  8752 
87-  9096  9153  9211  9268  9325 
87-  9669  9726  9784  9841  9898 
88-  
88-  0242  0299  0356  0413  0471 

8234  8292  8349  8407  8464 
8809  8866  8924  8981  9039 
9383  9440  9497  9555  9612 

9956  .- 
0013  0070  0127  0185 
0528  0585  0642  0699  0756 

57 
57 
57 
57 
57 
57 

N 

01234 

56789 

D 

54 


MATHEMATICAL   TABLES. 


N 

01234 

56789 

D 

760 

761 

762 

763 
764 

88-  0814  0871  0928  0985  1042 
88-  1385  1442  1499  1556  1613 
88-  1955  2012  2069  2126  2183 
88-  2525  2581  2638  2695  2752 
88-  3093  3150  3207  3264  3321 

1099  1156  1213  1271  1328 
1670  1727  1784  1841  1898 
2240  2297  2354  2411  2468 
2809  2866  2923  2980  3037 
3377  3434  3491  3548  3605 

57 
57 
57 
57 
57 

765 

766 

767 
768 
769 

88-  3661  3718  3775  3832  3888 
88-  4229  4285  4342  4399  4455 
88-  4795  4852  4909  4965  5022 
88-  5361  5418  5474  5531  5587 
88-  5926  5983  6039  6096  6152 

3945  4002  4059  4115  4172 
4512  4569  4625  4682  4739 
5078  5*35  5192  5248  5305 
5644  5700  5757  5813  5870 
6209  6265  6321  6378  6434 

57 
57 
57 

P 

770 
771 

772 
773 
774 

88-  6491  6547  6604  6660  6716 
88-  7054  7111  7167  7223  7280 
88-  7617  7674  7730  7786  7842 
88-  8179  8236  8292  8348  8404 
88-  8741  8797  8853  8909  8965 

6773  6829  6885  6942  6998 

7336  7392  7449  7505  756i 
7898  7955  8011  8067  8123 
8460  8516  8573  8629  8685 
9021  9077  9134  9190  9246 

ir>\r)\D\r)\j-> 

775 

776 
776 

777 
778 

779 

88-  9302  9358  9414  9470  9526 
88-  9862  9918  9974  
89-  0030  0086 
89-  0421  0477  0533  0589  0645 
89-  0980  1035  1091  1147  1203 
89-  1537  1593  1649  1705  I76o 

9582  9638  9694  9750  9806 

0141  0197  0253  0309  0365 
0700  0756  0812  0868  0924 
1259  1314  1370  1426  1482 
1816  1872  1928  1983  2039 

56 
56 

5? 
56 

56 
56 

780 

781 
782 

783 
784 

89-  2095  2150  2206  2262  2317 
89-  2651  2707  2762  2818  2873 
89-  3207  3262  3318  3373  3429 
89-  3762  3817  3873  3928  3984 
89-  4316  4371  4427  4482  4538 

2373  2429  2484  2540  2595 
2929  2985  3040  3096  3151 
3484  3540  3595  3651  3706 
4039  4094  4150  4205  4261 
4593  4648  4704  4759  4814 

56 
56 

56 
55 

55 

785 

786 
787 
788 
789 

89-  4870  4925  4980  5036  5091 
89-  5423  5478  5533  5588  5644 
89-  5975  6°3°  6085  6140  6195 
89-  6526  6581  6636  6692  6747 
89-  7077  7132  7187  7242  7297 

5146  5201  5257  5312  5367 
5699  5754  5809  5864  5920 
6251  6306  6361  6416  6471 
6802  6857  6912  6967  7022 
7352  7407  7462  7517  7572 

55 
55 
55 
55 
55 

790 

791 
792 

793 
794 

89-  7627  7682  7737  7792  7847 
89-  8176  8231  8286  8341  8396 
89-  8725  8780  8835  8890  8944 
89-  9273  9328  9383  9437  9492 
8g-  9821  9871;  Q91O  098; 

7902  7957  8012  8067  8122 
8451  8506  8561  8615  8670 
8999  9054  9109  9164  9218 
9547  9602  9656  9711  9766 

55 
55 
55 
55 

ec 

7Q4- 

QO-  OO3Q 

OOQ4.   OI4.Q   O2O3   O2^8   O7I2 

rc 

jj 

795 

796 

797 
798 

799 

go-  0367  0422  0476  0531  0586 
go-  0913  0968  1022  1077  1131 
go-  1458  1513  1567  1622  1676 

go-  2OO3  2O57  21  12  2l66  2221 

go-  2547  2601  2655  2710  2764 

0640   0695   0749   0804   0859 

1186  1240  1295  1349  1404 
1731  1785  1840  1894  1948 
2275  2329  2384  2438  2492 
2818  2873  2927  2981  3036 

55 
55 
54 
54 
54 

800 

80  1 

802 
803 
804 

go-  3090  3144  3199  3253  3307 

90-  3633  3687  374i  3795  3849 
go-  4174  4229  4283  4337  4391 
go-  4716  4770  4824  4878  4932 
go-  5256  5310  5364  5418  5472 

3361  3416  3470  3524  3578 
3904  3958  4012  4066  4120 
4445  4499  4553  4607  4661 
4986  5040  5094  5148  5202 
5526  5580  5634  5688  5742 

54 
54 
54 
54 
54 

805 

90-  5796  5850  5904  5958  6012 

6066  6119  6173  6227  6281 

54 

N 

01234 

56789 

D 

LOGARITHMS   OF   NUMBERS. 


55 


N 

01234 

56789 

D 

806 

80*9 

90-  6335  6389  6443  6497  6551 
go-  6874  6927  6981  7035  7089 
go-  7411  7465  7519  7573  7626 
go-  7949  8002  8056  8  no  8163 

6604  6658  6712  6766  6820 
7143  7196  7250  7304  7358 
7680  7734  7787  7841  7895 
8217  8270  8324  8378  8431 

54 

54 
54 

54 

810 

811 

812 
812 

go-  8485  8539  8592  8646  8699 
go-  9021  9074  9128  9181  9235 
9°~  9556  9610  9663  9716  9770 

QI- 

8753  8807  8860  8914  8967 
9289  9342  9396  9449  9503 
9823  9877  9930  9984  
0037 

54 
54 
54 

e-j 

813 
814 

gi-  0091  0144  0197  0251  0304 
gi-  0624  0678  0731  0784  0838 

0358  0411  0464  0518  0571 
0891  0944  0998  1051  1104 

JO 

53 
53 

815 

816 
817 
818 
819 

gi-  1158  i2ii  1264  1317  1371 
gi-  1690  1743  1797  1850  1903 
gi-  2222  2275  2328  2381  2435 
91-  2753  2806  2859  2913  2966 
91-  3284  3337  3390  3443  3496 

1424  1477  1530  1584  1637 
1956  2009  2063  2116  2169 
2488  2541  2594  2647  2700 
3019  3072  3125  3178  3231 
3549  3602  3655  3708  3761 

53 
53 
53 
53 
53 

820 

821 
822 
823 
824 

gi-  3814  3867  3920  3973  4026 
91-  4343  4396  4449  4502  4555 
gi-  4872  4925  4977  5030  5083 
gi-  5400  5453  5505  5558  5611 
9i-  5927  598o  6033  6085  6138 

4079  4132  4184  4237  4290 
4608  4660  4713  4766  4819 
5136  5189  5241  5294  5347 
5664  5716  5769  5822  5875 
6191  6243  6296  6349  6401 

53 
53> 
53 
53 
53 

825 

826 
827 
828 
829 

gi-  6454  6507  6559  6612  6664 
gi-  6980  7033  7085  7138  7190 
gi-  7506  7558  7611  7663  7716 
gi-  8030  8083  8135  8188  8240 
gi-  8555  8607  8659  8712  8764 

6717  6770  6822  6875  6927 
7243  7295  7348  7400  7453 
7768  7820  7873  7925  7978 
8293  8345  8397  8450  8502 
8816  8869  8921  8973  9026 

53 
53 
52 
52 
52 

830 

831 

£3' 
832 

?33 

834 

gi-  9078  9130  9183  9235  9287 
gi-  9601  9653  9706  9758  9810 
g2-  
ga-  0123  0176  0228  0280  0332 
g2-  0645  0697  0749  0801  0853 
g2-  1166  1218  1270  1322  1374 

9340  9392  9444  9496  9549 
9862  9914  9967  
0019  0071 
0384  0436  0489  0541  0593 
0906  0958  1010  1062  1114 
1426  1478  1530  1582  1634 

52 
52 
52 
52 
52 
52 

835 

836 

837 
838 

839 

ga-:  1686  1738  1790  1842  1894 
g2-  2206  2258  2310  2362  2414 
g2-  2725  2777  2829  2881  2933 
ga-  3244  3296  3348  3399  3451 
92-  3762  3814  3865  3917  3969 

1946   1998   2050   2102   2154 
2466   2518   257O   2622   2674 

2985  3°37  3089  3HQ  3J92 
35°3  3555  3607  3658  37™ 
4021  4072  4124  4176  4228 

52 

52 
52 
52 
52 

840 

841 

842 

843 

844 

92-  4279  4331  4383  4434  4486 
g2-  4796  4848  4899  4951  5003 
92-  5312  5364  5415  5467  5518 
92-  5828  5879  5931  5982  6034 
g2-  6342  6394  6445  6497  6548 

4538  4589  4641  4693  4744 
5054  5106  5157  5209  5261 
5570  5621  5673  5725  5776 
6085  6137  6188  6240  6291 
6600  6651  6702  6754  6805 

52 
52 
52 
5i 
5i 

845 

846 
847 
848 
849 

g2-  6857  6908  6959  7011  7062 
92-  7370  7422  7473  7524  7576 
92-  7883  7935  7986  8037  8088 
g2-  8396  8447  8498  8549  8601 
g2-  8908  8959  9010  9061  9112 

7114  7165  7216  7268  7319 
7627  7678  7730  7781  7832 
8140  8191  8242  8293  8345 
8652  8703  8754  8805  8857 
9163  9215  9266  9317  9368 

5i 
5i 
5i 
5i 
51 

850 

851 

g2-  9419  9470  9521  9572  9623 
92-  993O  9981  

9674  9725  9776  9827  9879 

5i 

Cl 

N 

01234 

56789 

D 

MATHEMATICAL   TABLES. 


N 

01234 

56789 

D 

851 

852 
853 
854 

93-  0032  0083  0134 
93-  0440  0491  0542  0592  0643 
93-  0949  1000  1051  i  102  1153 
93-  1458  1509  1560  1610  1661 

0185  0236  0287  0338  0389 
0694  0745  0796  0847  0898 
1203  1254  1305  1356  1407 
1712  1763  1814  1865  1915 

51 

5i 
5i 
5i 

855 

856 

857 
858 

859 

93-  1966  2017  2068  2118  2169 
93-  2474  2524  2575  2626  2677 
93-  2981  303*1  3082  3133  3183 
93-  3487  3538  3589  3639  3690 
93-  3993  4044  4094  4H5  4195 

2220   2271   2322   2372   2423 
2727   2778   2829   2879   2930 

3234  3285  3335  3386  3437 
3740  3791  3841  3892  3943 
4246  4296  4347  4397  4448 

5i 

51 
5i 
51 
5i 

860 

861 
862 
863 
864 

93-  4498  4549  4599  4650  4700 
93-  5°°3  5°54  5I04  5154  5205 
93-  55°7  5558  5608  5658  5709 
93-  6011  6061  6ni  6162  6212 
93-  6514  6564  6614  6665  6715 

4751  4801  4852  4902  4953 
5255  5306  5356  5406  5457 
5759  5809  5860  5910  5960 
6262  6313  6363  6413  6463 
6765  6815  6865  6916  6966 

5o 
50 
50 
50 
5o 

865 

866 
867 
868 
869 

93-  7016  7066  7117  7167  7217 
93-  75i8  7568  7618  7668  7718 
93-  8019  8069  8119  8169  8219 
93-  8520  8570  8620  8670  8720 
93-  9020  9070  9120  9170  9220 

7267  7317  7367  7418  7468 
7769  7819  7869  7919  7969 
8269  8319  8370  8420  8470 
8770  8820  8870  8920  8970 
9270  9320  9369  9419  9469 

50 
50 
50 
50 
50 

870 

871 

872 

873 
874 

93-  95*9  9569  96i9  9669  9719 
94-  0018  0068  01  18  0168  0218 
94-  0516  0566  0616  0666  0716 
94-  1014  1064  1114  1163  1213 
94-  1511  1561  1611  1660  1710 

9769  9819  9869  9918  9968 
0267  0317  0367  0417  0467 
0765  0815  0865  0915  0964 
1263  1313  1362  1412  1462 
1760  1809  1859  1909  1958 

50 
50 
5° 
5o 
50 

875 

876 

877 
878 
879 

94-  2008  2058  2107  2157  2207 
94-  2504  2554  2603  2653  2702 
94-  3000  3049  3099  3148  3198 
94-  3495  3544  3593  3643  3692 
94-  3989  4038  4088  4137  4186 

2256  2306  2355  2405  2455 
2752  2801  2851  2901  2950 
3247  3297  3346  3396  3445 
3742  3791  3841  3890  3939 
4236  4285  4335  4384  4433 

5o 
50 
49 
49 
49 

880 

881 
882 
883 
884 

94-  4483  4532  4581  4631  4680 
94-.  4976  5025  5074  5124  5173 
94-  5469  55i8  5567  5616  5665 
94-  5961  6010  6059  6108  6157 
94-  6452  6501  6551  6600  6649 

4729  4779  4828  4877  4927 
5222  5272  5321  5370  5419 
5715  5764  5813  5862  5912 
6207  6256  6305  6354  6403 
6698  6747  6796  6845  6894 

49 
49 
49 
49 
49 

885 

886 
887 
888 
889 

94-  6943  6992  7041  7090  7140 
94-  7434  7483  7532  7581  7630 
94-  7924  7973  8022  8070  8119 
94-  8413  8462  8511  8560  8609 
94-  8902  8951  8999  9048  9097 

7189  7238  7287  7336  7385 
7679  7728  7777  7826  7875 
8168  8217  8266  8315  8364 
8657  8706  8755  8804  8853 
9146  9195  9244  9292  9341 

49 
49 
49 
49 
49 

890 

891 

8qi 

94-  9390  9439  9488  9536  9585 
94-  9878  9926  9975  

Q^"  OO2J.  OO7^ 

9634  9683  9731  9780  9829 

OI2I   OI7O   O2I9   0267   0316 

49 

49 
4Q 

892 

893 
894 

95-  0365  0414  0462  0511  0560 
95-  0851  0900  0949  0997  1046 

95-  1338  1386  1435  ^83  1532 

0608   0657   0706   0754   0803 
1095   1143   1192   I24O   1289 
1580   1629   1677   1726   1775 

49 
49 
49 

895 

896 

897 

95-  1823  1872  1920  1969  2017 
95-  2308  2356  2405  2453  2502 
95-  2792  2841  2889  2938  2986 

2066   2114   2163   221  I   2260 
255°   2599   2647   2696   2744 
3034   3083   3131   3180   3228 

48 
48 

48 

N 

01234 

56789 

D 

LOGARITHMS  OF  NUMBERS. 


57 


N 

o    i    2    3.   4 

56789 

D 

898 
899 

95-  3276  3325  3373  342i  3470 
95-  376o  3808  3856  3905  3953 

3518  3566  3615  3663  37H 
4001  4049  4098  4146  4194 

48 
48 

900 

901 
902 

903 
904 

95-  4243  4291  4339  4387  4435 
95-  4725  4773  4821  4869  4918 
95-  5207  5255  5303  5351  5399 
95-  5688  5736  5784  5832  588o 
95-  6168  6216  6265  6313  6361 

4484  4532  4580  4628  4677 
4966  5014  5062  5110  5158 
5447  5495  5543  5592  5640 
5928  5976  6024  6072  6120 
6409  6457  6505  6553  6601 

48 
48 
48 
48 
48 

905 

906 
907 
908 
909 

95-  6649  6697  6745  6793  6840 
95-  7128  7176  7224  7272  7320 

95-  7607  7655  7703  775  i  7799 
95-  8086  8134  8181  8229  8277 
95-  8564  8612  8659  8707  8755 

6888  6936  6984  7032  7080 
7368  7416  7464  7512  7559 
7847  7894  7942  7990  8038 
8325  8373  8421  8468  8516 
8803  8850  8898  8946  8994 

48 
48 
48 
48 
48 

910 

911 
912 
912 

9i3 
914 

95-  9041  9089  9137  9185  9232 
95-  9518  9566  9614  9661  9709 

95-9995  ••  •••••• 
96-  0042  0090  0138  0185 
96-  0471  0518  0566  0613  0661 
96-  0946  0994  1041  1089  1136 

9280  9328  9375  9423  9471 
9757  9804  9852  9900  9947 

0233  0280  0328  0376  0423 
0709  0756  0804  0851  0899 
1184  1231  1279  1326  1374 

48 
48 
48 
48 
48 
47 

915 

916 
917 
918 
919 

96-  1421  1469  1516  1563  1611 
96-  1895  1943  1990  2038  2085 
96-  2369  2417  2464  2511  2559 
96-  2843  2890  2937  '2985  3032 
96-  3316  3363  3410  3457  3504 

1658  1706  1753  1801  1848 
2132  2180  .2227  2275  2322 
2606  2653  2701  2748  2795 
3079  3126  3174  3221  3268 
3552  3599  3646  3693  374i 

47 
47 
47 
47 
47 

920 

921 
922 

923 
924 

96-  3788  3835  3882  3929  3977 
96-  4260  4307  4354  4401  4448 
96-  4731  ,  4778  4825  4872  4919 
96-  5202  5249  5296  5343  5390 
96-  5672  5719  5766  5813  5860 

4024  4071  4118  4165  4212 
4495  4542  4590  4637  4684 
4966  5013  5061  5108  5155 
5437  5484  5531  5578  5625 
5907  5954  6001  6048  6095 

47 
47 
47 
47 
47 

925 

926 
927 
928 
929 

96-  6142  6189  6236  6283  6329 
96-  6611  6658  6705  6752  6799 
96-  7080  7127  7173  7220  7267 
96-  7548  7595  7642  7688  7735 
96-  8016  8062  8109  8156  8203 

6376  6423  6470  6517  6564 
6845  6892  6939  6986  7033 
7314  7361  7408  7454  7501 
7782  7829  7875  7922  7969 
8249  8296  8343  8390  8436 

47 
47 
47 
47 
47 

930 

931 
932 

0-7  -3 

96-  8483  8530  8576  8623  8670 
96-  8950  8996  9043  9090  9136 
96-  9416  9463  9509  9556  9602 

q6-  0882  QQ28  007^ 

8716  8763  8810  8856  8903 
9183  9229  9276  9323  9369 
9649  9695  9742  9789  9835 

47 
47 
47 
4.7 

933 
934 

97-  OO2I  OO68 

97-  0347  0393  0440  0486  0533 

0114  0161  0207  0254  0300 
0579  0626  0672  07^9  0765 

47 
46 

935 

936 
937 
938 
939 

97-  0812  0858  0904  0951  0997 
97-  1276  1322  1369  1415  1461 
97-  1740  1786  1832  1879  1925 
97-  2203  2249  2295  2342  2388 
97-  2666  2712  2758  2804  2851 

1044  1090  1137  1183  1229 
1508  1554  1601  1647  1693 
1971  2018  2064  21  10  2157 
2434  2481  2527  2573  2619 
2897  2943  2989  3035  3082 

46 
46 

46 
46 
46 

940 

941 
942 
943 

97-  3128  3174  3220  3266  3313 
97-  3590  3636  3682  3728  3774 
97-  4051  4097  4143  4189  4235 
97-  4512  4558  4604  4650  4696 

3359  3405  345  J  3497  3543 
3820  3866  3913  3959  4005 
4281  4327  4374  4420  4466 
4742  4788  4834  4880  4926 

46 
46 
46 
46 

N 

01234 

56789 

D 

MATHEMATICAL   TABLES. 


N 

01234 

56789 

D 

944 

97-  4972  5018  5064  5110  5156 

5202  5248  5294  5340  5386 

46 

945 

946 

948 
949 

97-  5432  5478  5524  557°  5616 
97-  5891  5937  5983  6029  6075 
97-  6350  6396  6442  6488  6533 
97-  6808  6854  6900  6946  6992 
97-  7266  7312  7358  7403  7449 

5662  5707  5753  5799  5845 

OI2I   6167   6212   6258   6304 
6579   6625   6671   6717   6763 
7037   7083   7129   7175   7220 

7495  754i  7586  7632  7678 

46 

46 
46 

46 
46 

950 

95i 
952 
953 
954 

97-  7724  7769  7815  7861  7906 
97-  8181  8226  8272  8317  8363 
97-  8637  8683  8728  8774  8819 
97-  9093  9138  9184  9230  9275 
97-  9548  9594  9639  9685  973° 

7952  7998  8043  8089  8135 
8409  8454  8500  8546  8591 
8865  8911  8956  9002  9047 
9321  9366  9412  9457  9503 
9776  9821  9867  9912  9958 

46 
46 

46 
46 
46 

955 

956 
957 
958 
959 

98-  0003  0049  0094  0140  0185 
98-  0458  0503  0549  0594  0640 
98-  0912  0957  1003  1048  1093 
98-  1366  1411  1456  1501  1547 

98-  1819  1864  1909  1954  2000 

0231  0276  0322  0367  0412 
0685  0730  0776  0821  0867 
1139  1184  1229  1275  I32o 
1592  1637  1683  1728  1773 
2045  2090  2135  2181  2226 

45 
45 
45 
45 
45 

960 

961 
962 

963 
964 

98-  2271  2316  2362  2407  2452 
98-  2723  2769  2814  2859  2904 

98-  3175  3220  3265  3310  3356 
98-  3626  3671  3716  3762  '3807 

98-  4077  4122  4167  4212  4257 

2497  2543  2588  2633  2678 
2949  2994  3040  3085  3130 
3401  3446  3491  3536  3581 
3852  3897  3942  3987  4032 
4302  4347  4392  4437  4482 

45 

45 
45 
45 
45 

965 

966 
967 
968 
969 

98-  4527  4572  4617  4662  4707 

98-  4977  5°22  5°67  5  "2  5J57 
98-  5426  5471  5516  5561  5606 
98-  5875  5920  5965  6010  6055 
98-  6324  6369  6413  6458  6503 

4752  4797  4842  4887  4932 
5202  5247  5292  5337  5382 
5651  5696  5741  5786  5830 
6100  6144  6189  6234  6279 
6548  6593  6637  6682  6727 

45 
45 
45 
45 
45 

970 

971 

972 
973 
974 

98-  6772  6817  6861  6906  6951 
98-  7219  7264  7309  7353  7398 
98-  7666  7711  7756  7800  7845 
98-  8113  8157  8202  8247  8291 
98-  8559  8604  8648  8693  8737 

6996  7040  7085  7130  7175 
7443  7488  7532  7577  7622 
7890  7934  7979  8024  8068 
8336  8381  8425  8470  8514 
8782  8826  8871  8916  8960 

45 
45 
45 
45 
45 

975 

976 
077 

98-  9005  9049  9094  9138  9183 
98-  945°  9494  9539  9583  9628 

08-  9891;  QQ3Q  QQ8-? 

9227  9272  9316  9361  9405 
9672  9717  9761  9806  9850 

45 
44 

A/\ 

977 

978 

979 

99-  OO28  0072 
99-  0339  0383  0428  0472  0516 
99-  0783  0827  0871  0916  0960 

0117  01  6  i  0206  0250  0294 
0561  0605  0650  0694  0738 
1004  1049  1093  1137  1182 

44 

44 
44 

980 

981 
982 
983 
984 

99-  1226  1270  1315  1359  1403 
99-  1669  1713  1758  1802  1846 

99-  21  I  I  2156  2200  2244  2288 

99-  2554  2598  2642  2686  2730 
99-  2995  3039  3083  3127  3172 

1448  1492  1536  1580  1625 
1890  1935  1979  2023  2067 
2333  2377  2421  2465  2509 
2774  2819  2863  2907  2951 
3216  3260  3304  3348  3392 

44 
44 
44 
44 
44 

985 

986 

$ 

989 

99-  3436  348o  3524  3568  3613 
99-  3877  3921  3965  4009  4053 
99-  4317  4361  4405  4449  4493 

99-  4757  4801  4845  4889  4933 
99-  5196  5240  5284  5328  5372 

3657  3701  3745  3789  3833 
4097  4141  4185  4229  4273 
4537  4581  4625  4669  4713 
4977  5021  5065  5108  5152 
5416  5460  5504  5547  5591 

44 
44 
44 
44 
44 

N 

01234 

56789 

D 

i 

LOGARITHMS   OF   NUMBERS. 


59 


N 

01234 

56789 

D 

990 

991 
992 
993 
994 

99-  5635  5679  5723  5767  58" 
99-  6074  6117  6161  6205  6249 
99-  6512  6555  6599  6643  6687 
99-  6949  6993  7037  7080  7124 
99-  7386  743°  7474  75  i  7  756  i 

5854  5898  5942  5986  6030 
6293  6337  6380  6424  6468 
6731  6774  6818  6862  6906 
7168  7212  7255  7299  7343 
7605  7648  7692  7736  7779 

44 
44 
44 
44 
44 

995 

996 
997 
998 
999 

99-  7823  7867  7910  7954  7998 
99-  8259  8303  8347  8390  8434 
99-  8695  8739  8782  8826  8869 
99-  9I3I  9r74  9218  9261  9305 
99-  9565  9609  9652  9696  9739 

8041  8085  8129  8172  8216 
8477  8521  8564  8608  8652 
8913  8956  9000  9043  9087 
9348  9392  9435  9479  9522 
9783  9826  9870  9913  9957 

44 
44 
44 
44 
43 

N 

01234 

56789 

D 

6o 


MATHEMATICAL   TABLES. 


TABLE    No.    II.— HYPERBOLIC   LOGARITHMS    OF    NUMBERS 

FROM    I.OI    TO    3O. 


Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

I.OI 

.0099 

I.36 

•3075 

I.7I 

•5365 

2.06 

.7227 

1.02 

.0198 

i-37 

.3148 

1.72 

•5423 

2.07 

•7275 

1.03 

.0296 

1.38 

.3221 

i-73 

.5481 

2.08 

•7324 

I.O4 

.0392 

i-39 

•3293 

1.74 

•5539 

2.09 

•7372 

1.05 

..0488 

1.40 

•3365 

i-75 

•5596 

2.10 

.7419 

1.  06 

.0583 

1.41 

•3436 

1.76 

•5653 

2.  II 

.7467 

1.07 

.0677 

1.42 

•3507 

1.77 

•5710 

2.12 

•75J4 

i.  08 

.0770 

1-43 

•3577 

1.78 

.5766 

2.13 

•7561 

1.09 

.0862 

1.44 

.3646 

1.79 

•5822 

2.14 

.7608 

1.  10 

•°953 

i-45 

.3716 

i.  80 

.5878 

2.15 

•7655 

i.  ii 

.1044 

1.46 

.3784 

1.81 

•5933 

2.16 

.7701 

1.  12 

•ii33 

1.47 

•3853 

1.82 

.5988 

2.17 

•7747 

I-I3 

.1222 

1.48 

.3920 

1.83 

.6043 

2.18 

•7793 

I.I4 

.1310 

1.49 

.3988 

1.84 

.6098 

2.19 

•7839 

l-*5 

.1398 

1.50 

•4055 

1.85 

.6152 

2.20 

.7885 

1.16 

.1484 

I-5I 

.4121 

1.86 

.6206 

2.21 

•7930 

1.17 

•1570 

1.52 

.4187 

1.87 

.6259 

2.22 

•7975 

1.18 

•1655 

i-53 

•4253 

1.88 

•6313 

2.23 

.8020 

1.19 

.1740 

1-54 

.4318 

1.89 

.6366 

2.24 

.8065 

1.20 

.1323 

i-55 

•4383 

1.90 

.6419 

2.25 

.8109 

1.  21 

.1906 

1.56 

•4447 

1.91 

.6471 

2.26 

•8154 

1.22 

.1988 

i-57 

•4511 

1.92 

•6523 

2.27 

.8198 

1.23 

.2O70 

1.58 

•4574 

i-93 

•6575 

2.28 

.8242 

1.24 

.2151 

i-59 

•4637 

1.94 

.6627 

2.29 

.8286 

1.25 

.2231 

i.  60 

.4700 

I-95 

.6678 

2.30 

•8329 

1.26 

.2311 

1.61 

.4762 

1.96 

.6729 

2.31 

•8372 

1.27 

.2390 

1.62 

.4824 

1.97 

.6780 

2.32 

.8416 

1.28 

.2469 

1.63 

.4886 

1.98 

.6831 

2-33 

.8458 

1.29 

.2546 

1.64 

•4947 

1.99 

.6881 

2-34 

.8502 

1.30 

.2624 

1.65 

.5008 

2.0O 

.6931 

2-35 

.8544 

LSI 

.2700 

1.66 

.5068 

2.OI 

.6981 

2.36 

.8587 

1.32 

.2776 

1.67 

.5128 

2.  02 

.7031 

2-37 

.8629 

1-33 

.2852 

1.68 

.5188 

2.03 

.7080 

2.38 

.8671 

i-34 

.2927 

1.69 

•5247 

2.04 

.7129 

2-39 

•8713 

1-35 

.3001 

1.70 

•53o6 

2.05 

•7i78 

2.40 

i 

•8755 

HYPERBOLIC   LOGARITHMS   OF  NUMBERS 


61 


Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

2.41 

.8796 

2.8l 

1.0332 

3.21 

1.1663 

3.6l 

1.2837 

2.42 

.8838 

2.82 

1.0367 

3.22 

1.1694 

3-62 

1.2865 

2-43 

.8879 

2.83 

1.0403 

3-23 

•1725 

3-63 

1.2892 

2.44 

.8920 

2.84 

1.0438 

3-24 

•I756 

3-64 

1.2920 

2-45 

.8961 

2-85 

1-0473 

3-25 

.1787 

3-65 

1.2947 

2.46 

.9002 

2.86 

1.0508 

3.26 

.1817 

3.66 

1-2975 

2.47 

.9042 

2.87 

1-0543 

3-27 

.1848 

3-67 

1.3002 

2.48 

.9083 

2.88 

1.0578 

3.28 

.1878 

3.68 

1.3029 

2.49 

.9123 

2.89 

1.0613 

3-29 

.1909 

3.69 

1-3056 

2.50 

.9163 

2.90 

1.0647 

3-30 

I-I939 

3.70 

1.3083 

2-51 

.9203 

2.91 

1.0682 

3-31 

1.1969 

3-71 

I.3IIO 

2.52 

.9243 

2.92 

1.0716 

3-32 

1.1999 

3-72 

I-3I37 

2-53 

.9282 

2-93 

1.0750 

3-33 

1.2030 

3-73 

1.3164 

2-54 

.9322 

2-94 

1.0784 

3-34 

1.2060 

3-74 

I.3I9I 

2-55 

.9361 

2-95 

I.  O8l8 

3-35 

1.2090 

3-75 

I.32I8 

2.56 

.9400 

2.96 

1.0852 

3-36 

1.2119 

3.76 

L3244 

2.57 

•9439 

2-97 

1.  0886 

3-37 

1.2149 

3-77 

1.3271 

2.58 

.9478 

2.98 

1.0919 

3-38 

1.2179 

3-78 

1.3297 

2-59 

•951? 

2-99 

1-0953 

3-39 

1.2208 

3-79 

L3324 

2.60 

•9555 

3.00 

1.0986 

3-4° 

1.2238 

3.80 

I-335° 

2.61 

•9594 

3.01 

I.IOI9 

3-4i 

1.2267 

3-8i 

I-3376 

2.62 

.9632 

3.02 

I-I053 

3-42 

1.2296 

3.82 

1-3403 

2.63 

.9670 

3-°3 

I.I086 

3-43 

1.2326 

3-83 

1-3429 

2.64 

.9708 

3-°4 

I.III9 

3-44 

r-2355 

3-84 

I-3455 

2.65 

.9746 

3-°5 

I.II5I 

3-45 

1.2384 

3-85 

1.3481 

2.66 

•9783 

3.06 

1.1184 

3:46 

1.2413 

3-86 

1.3507 

2.67 

.9821 

3-°7 

I.I2I7 

3-47 

1.2442 

3-87 

L3533 

2.68 

.9858 

3.08 

I.I249 

3-48 

1.2470 

3-88 

-1.3558 

2.69 

•9895 

3-°9 

I.I282 

3-49 

1.2499 

3-89 

1-3584 

2.70 

•9933 

3.10 

I.I3I4 

3-5o 

1.2528 

3-9° 

1.3610 

2.71 

.9969 

3-11 

1.1346 

3-5i 

i-2556 

3-91 

1-3635 

2.72 

i.  0006 

3.12 

I.I378 

3-52 

1-2585 

3-92 

1.3661 

2-73 

1.0043 

3-i3 

I.I4IO 

3-53 

1.2613 

3-93 

1.3686 

2.74 

1.0080 

3-!4 

I.I442 

3-54 

1.2641 

3-94 

1.3712 

2-75 

1.0116 

3-15 

I.I474 

3-55 

1.2669 

3-95 

1-3737  . 

2.76 

1.0152 

3.16 

1.1506 

3.56 

1.2698 

3-96 

1.3762 

2.77 

1.0188 

3-i7 

I-I537 

3-57 

1.2726 

3-97 

1.3788 

2.78 

1.0225 

3.18 

1.1569 

3.58 

1.2754 

3-98" 

1-3813 

2-79 

1.0260 

3-x9 

I.  l6oO 

3-59 

1.2782 

3-99 

1-3838 

2.80 

1.0296 

3.20 

1.1632 

3.60 

1.2809 

4.00 

1-3863 

MATHEMATICAL   TABLES. 


Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

4.01 
4.02 

4-03 
4-04 
4-05 

1.3888 

L3938 
1.3962 
1.3987 

4.41 
4.42 

4-43 

4-44 
4-45 

1.4839 
1.4861 
1.4884 
1.4907 
1.4929 

4.8l 
4.82 
4.83 
4.84 
4-85 

I-5707 
1.5728 

L5748 
L5769 
1-5790 

5-22 
5.23 
5-24 
5-25 

1.6506 

1.6525 
1.6514 

1-6563 
1.6582 

4.06 
4.07 
4.08 

4.09 
4-10 

I.40I2 
1.4036 
1.4061 
1.4085 
I.4IIO 

4.46 

4-47 
4.48 
4-49 
4-5° 

I-4951 
1.4974 
1.4996 
1.5019 
1.5041 

4.86 
4-87 
4.88 
4.89 
4.90 

1.5810 
L583I 
L585I 
1.5872 
1.5892 

5-26 

5-27 
5-28 

5-30 

1.  66oi 
1.6620 
1.6639 
1.6658 
1.6677 

4.II 
4.12 

4-13 
4.14 

4.15 

L4I34 

L4I59 
1.4183 
1.4207 
1.4231 

4-5  1 
4.52 
4.53 
4-54 
4-55 

1-5063 
1.5085 
1.5107 
1.5129 

4.91 
4.92 

4-93 
4.94 

4.95 

I-59I3 

1-5933 
1-5953 

1-5974 
T-5994 

5.31 

5.32 

5-33 
5-34 
5-35 

1.6696 
1.6715 

1.6734 
1.6752 
1.6771 

4.16 
4.17 
4.l8 
4.19 
4.2O 

1.4255 
1.4279 

1.4303 
1.4327 

I-4351 

4-56 
4.57 
4.58 

4-59 
4.60 

I-5I95 
1.5217 

1-5239 
1.5261 

4-96 
4-97 
4.98 
4.99 

5-°o 

1.6014 
1.6034 
1.6054 
1.6074 
1.6094 

5.36 

5-37 
5-38 
5-39 
5-40 

1.6790 
1.  6808 
1.6827 
1.6845 
1.6864 

4.21 

4.22 

4.23 
4.24 

4.25 

1-4375 
1.4398 
1.4422 
1.4446 
1.4469 

4.61 
4.62 

4-63 
4.64 

4-65 

1.5282 

1-5304 
1-5326 

I-5347 
J-5369 

5.01 
5-02 
5-°3 
5-°4 
5-05 

1.6114 
1.6134 
1.6154 
1.6174 
1.6194 

5-4i 
5-42 
5-43 
5-44 
5-45 

1.6882 
1.6901 
1.6919 
1.6938 
1.6956 

4.26 
4.27 
4.28 

4.29 

4-30 

1-4493 
1.4516 

1.4540 

1.4563 
1.4586 

4.66 

4-67 
4.68 

4-69 

4.70 

1.5412 

L5433 
1-5454 
L5476 

5.06 

5-07 
5.08 

5-09 

5.10 

1.6214 
1.6233 
1-6253 
1.6273 
1.6292 

5-46 
5-47 
5-48 
5-49 
5-5° 

1.6974 
1.6993 
I.7OII 
1.7029 
1.7047 

4.31 
4.32 
4-33 
4.34 

4-35 

1.4609 

1.4633 
1.4656 
1.4679 
1.4702 

4.71 
4-72 
4-73 
4-74 
4-75 

L5497 
1.5539 

5-11 

5-12 

1.6312 
1.6332 

1-6351 
1.6371 
1.6390 

5-51 
S-S2 
5-53 
5-54 
5-55 

1.7066 
1.7084 
I.7I02 
I.7I20 
I.7I38 

4-36 
4-37 
4-38 

4-39 
4.40 

1.4725 
1.4748 
1.4770 

1-4793 
1.4816 

4-76 
4-77 
4.78 

4.79 
4.80 

1.5602 
1.5623 
1.5644 

1.5665 
1.5686 

vo  t-00  O\  O 

M  H  M  M  M 

to  to  to  to  to 

1.6409 
1.6429 
1.6448 
1.6467 
1.6487 

5.56 
5-57 
5.58 

5-59 
5.60 

I.7I56 
I.7I74 
I.7I92 
I.72IO 
1.7228 

HYPERBOLIC  LOGARITHMS   OF  NUMBERS. 


Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

5.61 
5-62 
5.63 

5-64 
5.65 

.7246 
.7263 
.728l 
.7299 
.7317 

6.01 
6.  02 
6.03 
6.04 
6.05 

1-7934 

L7951 
1.7967 
1.7984 
1.8001 

6.4I 
6.42 

6-43 
6.44 

6-45 

1.8579 
1.8594 
1.  86lO 
1.8625 
1.8641 

6.81 
6.82 
6.83 
6.84 
6.85 

1.9184 
1.9199 
1.9213 
1.9228 
1.9242 

5.66 

5.67 
5.68 

5-69 
5-70 

•7334 
•7352 
•7370 
.7387 
I-7405 

6.06 
6.07 
6.08 
6.09 
6.10 

1.8017 
1.8034 
1.8050 
1.8066 

1.8083 

6.46 

6.47 
6.48 
6.49 
6.50 

1.8656 
1.8672 
1.8687 
1.8703 
1.8718 

6.86 
6.87 
6.88 
6.89 
6.90 

I-9257 
1.9272 
1.9286 
.9301 
-93^5 

5-7i 

5-72 
5-73 
5-74 
5-75 

1.7422 
1.7440 
1-7457 

1-7475 
1.7492 

6.ii 

6.12 

6.13 
6.14 

6.15 

1.8099 
1.8116 
1.8132 
1.8148 
1.8165 

6.51 
6.52 
6-53 
6-54 
6-55 

1.8733 
1.8749 
1.8764 
1.8779 
1-8795 

6.91 
6.92 

6-93 
6.94 

6-95 

•933° 
•9344 
•9359 
•9373 
•9387 

5.76 

5-77 
5.78 

5-79 
5.80 

i-75°9 
i.7527 
1-7544 
1.7561 

1-7579 

6.16 
6.17 
6.18 
6.19 
6.  20 

1.8181 
1.8197 
1.8213 
1.8229 
1.8245 

6.56 

6-57 
6.58 

6-59 
6.60 

1.  88lO 
1.8825 
1.8840 
1.8856 
1.8871 

6.96 
6.97 
6.98 

6.99 

7.00 

.9402 
.9416 
.9430 
•9445 
-9459 

5-81 
5-82 
5-83 
5-84 
5-85 

1.7596 
1.7613 
1.7630 
1.7647 
1.7664 

6.21 
6.22 
6.23 
6.24 
6.25 

1.8262 
1.8278 
1.8294 
1.8310 
1.8326 

6.61 
6.62 
6.63 
6.64 
6.65 

1.8886 
1.8901 
1.8916 
1.8931 
1.8946 

7.01 

7.02 

7-03 
7.04 

7.05 

•9473 
.9488 

-95°2 
.9516 

•9530 

5-86 

5.87 
5.88 

5-89 
5-90 

1.7681 
1.7699 
1.7716 
1-7733 

1-775° 

6.26 
6.27 
6.28 
6.29 
6.30 

1.8342 
1-8358 
1.8374 
1.8390 
1.8405 

6.66 
6.67 
6.68 
6.69 
6.70 

1.8961 
1.8976 
1.8991 
1.9006 
1.9021 

7.06 
7.07 
7.08 

7.09 
7.10 

1-9544 
1-9559 
J-9573 
1-9587 
1.9601 

5.9i 
5-92 
5-93 
5-94 
5-95 

1.7766 

1-7783 
1.7800 
1.7817 
1-7834 

6.3I 
6.32 

6-33 
6-34 
6-35 

1.8421 
1.8437 
1.8453 
1.8469 
1.8485 

6.71 
6.72 

6-73 
6-74 
6.75 

1.9036 
1.9051 
1.9066 
1.9081 
I.9095 

7.11 
7.12 

7.13 
7.14 
7.15 

1.9615 
1.9629 
1.9643 

1.9657 
1.9671 

5-96 
5-97 
5.98 

5-99 
6.00 
I 

1.7851 
1.7867 
1.7884 
1.7901 
1.7918 

6.36 
6-37 
6.38 

6-39 
6.40 

1.8500 
1.8516 
1-8532 
1.8547 
1.8563 

6.76 

6-77 
6.78 

6-79 
6.80 

1.9110 
1.9125 
1.9140 

i.9i55 
1.9169 

7.16 
7.17 
7.18 
7.19 
7.20 

1.9685 
1.9699 

i.97i3 
1.9727 
1.9741 

64 


MATHEMATICAL  TABLES. 


Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

Logarithm. 

7.21 

7.22 

7.23 
7.24 

7.25 

1-9755 
1.9769 
1.9782 
1.9796 
1.9810 

7.6l 
7.62 

7-63 

7.64 

7.65 

2.0295 
2.0308 
2.0321 

2.0334 
2.0347 

8.01 

8.02 

8.03 
8.04 
8.05 

2.0807 
2.0819 
2.0832 
2.0844 
2.0857 

8.41 
8.42 

8-43 
8.44 

8-45 

2.1294 
2.1306 
2.1318 
2.1330 
2.1342 

7.26 

7.27 
7.28 
7.29 
7-30 

1.9824 
1.9838 
1.9851 
1.9865 
1.9879 

7.66 
7.67 
7.68 
7.69 

7.70 

2.0360 
2.0373 
2.0386 
2.0399 
2.0412 

8.06 
8.07 
8.08 
8.09 
8.10 

2.0869 
2.0882 
2.0894 
2.0906 
2.0919 

8.46 

8-47 
8.48 
8.49 
8.50 

2.1353 
2.1365 
2.1377 
2.1389 
2.1401 

7-31 
7.32 

7-33 
7-34 
7-35 

1.9892 
1.9906 
1.9920 

1-9933 
1.9947 

7.71 

7.72 

7-73 
7-74 
7-75 

2.0425 
2.0438 
2.0451 
2.0464 
2.0477 

8.ii 

8.12 

8.13 
8.14 
8.15 

2.0931 
2.0943 
2.0956 
2.0968 
2.0980 

8.51 
8.52 
8.53 
8.54 
8-55 

2.1412 
2.1424 
2.1436 
2.1448 
2-1459 

7.36 
7-37 

7.33 

7-39 

7.40 

1.9961 
1.9974 
1.9988 

2.0001 
2.0015 

7.76 

7-77 
7.78 

7-79 

7.80 

2.0490 
2.0503 
2.0516 
2.0528 
2.0541 

8.16 
8.17 
8.18 
8.19 

8.20 

2.0992 
2.1005 
2.IOI7 
2.1029 
2.I04I 

8.56 

8.57 
8.58 

8-59 
8.60 

2.1471 
2.1483 
2.1494 
2.1506 
2.1518 

7.41 
7.42 

7-43 
7-44 
7-45 

2.0028 
2.0042 
2.0055 
2.0069 
2.0082 

7.81 
7.82 

7-83 

7.84 

7.85 

2.0554 
2.0567 
2.0580 
2.0592 
2.0605 

8.21 
8.22 
8.23 
8.24 
8.25 

2.1054 
2.1066 
2.1078 
2.1090 
2.II02 

8.61 
8.62 
8.63 
8.64 
8.65 

2.1529 
2.1541 

2.I552 
2.1564 
2.1576 

7.46 

7-47 
7.48 

7-49 
7-5° 

2.0096 
2.0IO9 
2.0122 
2.0136 
2.0149 

7.86 
7.87 
7.88 
7.89 
7.90 

2.o6l8 
2.0631 
2.0643 
2.0656 
2.0669 

8.26 

8.27 
8.28 
8.29 
8.30 

2.III4 
2.II26 
2.1138 
2.II50 
2.1163 

8.66 
8.67 
8.68 
8.69 
8.70 

2.1587 
2.1599 

2.1610 
2.1622 

2.1633 

7-5i 
7-52 
7-53 
7-54 
7-55 

2.0l62 
2.0176 
2.0189 
2.0202 
2.0215 

7.91 
7.92 

7-93 
7-94 
7-95 

2.0681 
2.0694 
2.0707 
2.0719 
2.0732 

8.3I 
8.32 
8-33 

8.34 
8.35 

2.II75 
2.II87 
2.1199 
2.I2II 
2.1223 

8.71 
8.72 

8-73 
8.74 

8-75 

2.1645 

2.1656 

2.1668 
2.1679 

2.1691 

7.56 

7-57 
7.58 

7-59 
7.60 

2.0229 
2.0242 
2.0255 
2.0268 
2.028l 

7.96 

7-97 
7.98 

7-99 
8.00 

2.0744 
2.0757 
2.0769 
2.0782 
2.0794 

8.36 

8-37 
8.38 

8-39 
8.40 

2.1235 
2.1247 
2.1258 
2.I27O 
2.1282 

8.76 

8.77 
8.78 
8.79 
8.80 

2.1702 

2.1713 

2.1725 
2.1736 
2.1748 

HYPERBOLIC   LOGARITHMS   OF   NUMBERS. 


Number. 

Logarithm. 

Number. 

Logarithm. 

Number. 

•  Logarithm. 

Number. 

Logarithm. 

8.81 

2.1759 

9.II 

2.2094 

9.41 

2.2418 

9.71 

2.2732 

8.82 

2.1770 

9.12 

2.2105 

9.42 

2.2428 

9.72 

2.2742 

8.83 

2.1782 

9.13 

2.2Il6 

9-43 

2.2439 

9-73 

2.2752 

8.84 

2.1793 

9.14 

2.2127 

9.44 

2.2450 

9-74 

2.2762 

8.85 

2.1804 

9.15 

2.2138 

9-45 

2.2460 

9-75 

2.2773 

8.86 

2.l8l5 

9.16 

2.2148 

9.46 

2.2471 

9.76 

2.2783 

8.87 

2.1827 

9.17 

2.2159 

9.47 

2.2481 

9-77 

2.2793 

8.88 

2.1838 

9.18 

2.2170 

9.48 

2.2492 

9.78 

2.2803 

8.89 

2.1849 

9.19 

2.2l8l 

9-49 

2.2502 

9-79 

2.2814 

8.90 

2.1861 

9.20 

2.2192 

9-5° 

2.2513 

9.80 

2.2824 

8.91 

2.1872 

9.2I 

2.2203 

9-51 

2.2523 

9.81 

2.2834 

8.92 

2.1883 

9.22 

2.2214 

9-52 

2.2534 

9.82 

2.2844 

8-93 

2.1894 

9.23 

2.2225 

9-53 

2.2544 

9-83 

2.2854 

8.94 

2.1905 

9.24 

2.2235 

9-54 

2-2555 

9.84 

2.2865 

8-95 

2.1917 

9.25 

2.2246 

9-55 

2-2565 

9-85 

2.2875 

8.96 

2.1928 

9.26 

2.2257 

9.56 

2.2576 

9.86 

2.2885 

8.97 

2.1939 

9.27 

2.2268 

9.57 

2.2586 

9.87 

2.2895 

8.98 

2.1950 

9.28 

2.2279 

9.58 

2.2597 

9.88 

2.2905 

8.99 

2.1961 

9.29 

2.2289 

9-59 

2.2607 

9.89 

2.2915 

9.00 

2.1972 

9.30 

2.2300 

9.60 

2.26l8 

9.90 

2.2925 

9.01 

2.1983 

9.31 

2.23II 

9.61 

2.2628 

9.91 

2-2935 

9.02 

2.1994 

9.32 

2.2322 

9.62 

2.2638 

9.92 

2.2946 

9-03 

2.2006 

9-33 

2.2332 

9.63 

2.2649 

9-93 

2.2956 

9.04 

2.2017 

9-34 

2.2343 

9.64 

2.2659 

9-94 

2.2966 

9-05 

2.2028 

9-35 

2.2354 

9.65 

2.2670 

9-95 

2.2976 

9to6 

2.2039 

9.36 

2.2364 

9.66 

2.2680 

9.96 

2.2986 

9.07 

2.2050 

9-37 

2.2375 

9.67 

2.2690 

9-97 

2.2996 

9.08 

2.2061 

9.38 

2.2386 

9.68 

2.2701 

9.98 

2.3006 

9.09 

2.2072 

9-39 

2.2396 

9.69 

2.27II 

9.99 

2.3016 

9.10 

2.2083 

9.40 

2.2407 

9.70 

2.2721 

IO.OO 

2.3026 

10.25 

2.3279 

12.75 

2-5455 

I5-SQ 

2.7408 

21.0 

3-0445 

10.50 

2-35I3 

13.00 

2.5649 

16.0 

2.7726 

22.0 

3.09II 

iQ-75 

2-3749 

13-25 

2.5840 

16.5 

2.8034 

23.0 

3-1355 

II.  OO 

2.3979 

13.5° 

2.6027 

17.0 

2.8332 

24.0 

3.I78I 

11.25 

2.4201 

13.75 

2.6211 

J7-5 

2.8621 

25.0 

3.2l89 

11.50 

2.4430 

14.00 

2.6391 

18.0 

2.8904 

26.0 

3.2581 

11.75 

2.4636 

14-25 

2.6567 

18.5 

2.9173 

27.0 

3-2958 

12.00 

2.4849 

14.50 

2.6740 

19.0 

2-9444 

28.0 

3-3322 

12.25 

2.5052 

14-75 

2.6913 

19-5 

2.9703 

29.0 

3.3673 

12.50 

2.5262 

15.00 

2.7081 

2O.O 

2-9957 

30.0 

3.4012 

66 


MATHEMATICAL   TABLES. 


TABLE  No.  III.— NUMBERS,  OR  DIAMETERS  OF  CIRCLES,  CIR- 
CUMFERENCES, AREAS,  SQUARES,  CUBES,  SQUARE  ROOTS, 
AND  CUBE  ROOTS. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

I 

3.I4I6 

0.7854 

I 

I 

1.  000 

1.  000 

2 

6.28 

3.14 

4 

8 

I.4I4 

1.2^0 

3 

9.42 

o-  *-  1 
7.07 

T- 

9 

27 

T"   T^ 

1.732 

D  ? 
1.442 

,  .  12X7 

•  12X7 

16 

64 

2.  OOO 

I  ^87 

5 

•  *"  j  1 

"'  o  1 
19.63 

25 

I25 

2.236 

1.709 

6 

...  18.85 

28.27 

16 

.  ...  216 

2.44Q 

I  8l7 

•  / 

1  1  S 

*•***/ 

7 

21.99 

38.48 

49 

343 

2.645 

I.9I2 

8 

...  25.13 

50.27 

......  64 

512 

2.828 

2.  OOO 

9 

28.27 

63.62 

81 

729 

3.000 

2.080 

10 

..  31.42 

78.154 

100 

1,000 

3.162 

2  I  ^4 

ii 

0    , 

34.56 

/    O  ^ 

95.03 

121 

O 

3-3l6 

2.223 

12 

...  37.70 

113.10 

144 

1^728 

3.464 

2.289 

13 

40.84 

132.73 

169 

2,197 

3-605 

2-351 

14 

.  .  43.Q8 

1^3  04 

.  106 

2,744 

3.741 

2.4IO 

•*•  *T 
15 

to  y  - 

47.12 

176.71 

7 

225 

*j  /  *r*r 

3,375 

O'  1  ^ 

3.872 

Wi.i{.i  W 

2.466 

16 

...  50.26 

201.06 

4,096 

4.000 

2.519 

17 

53-41 

226.98 

289 

4,913 

4.123 

2.571 

18 

..  c6.  s  z 

..2^4.47 

324 

e  832 

4.242 

2.62O 

J   J  J 

OT^  *T  / 

o  T- 

•  •  D,^O^ 

19 

59.69 

283.53 

36l 

6,859 

4.358 

2.668 

20 

6^  83 

3  IzL  1  6 

4OO 

8  ooo 

A  472 

2  7  T  A. 

tt*itl  * 

•*•  /  x^f 

21 

65.97 

346.36 

441 

9,261 

4.582 

2.758 

22 

...  69.11 

380.13 

484 

10,648 

4.690 

2.802 

23 

72.26 

415.48 

529 

12,167 

4-795 

2.843 

24 

...  75.40 

452.39 

576 

13,824 

4.898 

2.884 

25 

78.54 

490.87 

625 

15.625 

5.0OO 

2.924 

26 

...  81.68 

530.93 

676 

17,576 

5-°99 

2.962 

27 

84.82 

572.56 

729 

19,683 

5.196 

3.000 

28 

..87.96 

615.75 

784 

21,952 

5.291 

3.036 

29 

91.11 

660.52 

841 

24,389 

5.385 

3.072 

30 

...  94.25 

706.86 

900 

27,000 

5-477 

3.107 

31 

97-39 

754-77 

961 

29,791 

5-567 

3.141 

32 

100.53 

804.25 

...  1,024 

32,768 

5-656 

3,174 

33 

103.67 

855.30 

,089 

35,937 

5-744 

3.207 

34 

106.81 

907.92 

...   ,I56 

39,304 

5-830 

3-239 

35 

109.96 

962.11 

,225 

42,875 

5.916 

3.271 

36 

113.10 

...  1017.88 

...   ,296 

46,656 

6.  ooo 

3-301 

37 

116.24 

1075.21 

,369 

50,653 

6.082 

3-332 

38 

119.38 

...  1134.11 

...   ,444 

54,872 

6.164 

3-361 

39 

122.52 

1194.59 

,521 

59,3i9 

6.244 

3-391 

40 

125.66 

...  1256.64 

,600 

64,000 

6.324 

3-4I9 

41 

128.80 

1320.25 

,681 

68,921 

6.403 

3-448 

42 

w-n 

...  1385.44 

...  ,764 

74,o88 

6.480 

3-476 

NUMBERS,   OR  DIAMETERS  OF  CIRCLES,  &c. 


67 


Number, 
or 
Diameter,  j 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

43 

1-35-09 

I452.2O 

1,849 

79,507 

6-557 

3.503 

44 

138.23 

...  I520.53 

...  1,936 

85,184 

6.633 

3-530 

45 

MI.37 

I590-43 

2,025 

91,125 

6.708 

3.556 

46. 

I44-51 

...  1661.90 

...  2,116 

97,336 

6.782 

3.583 

47 

147-65 

1734.94 

2,209 

103,823 

6.855 

3.608 

48 

150.80 

...  1809.56 

...  2,304 

H0,592 

6.928 

3-634 

49 

I53.94 

1885.74 

2,401 

117,649 

7-000 

3.659 

5° 

157.08 

...  1963.50 

...  2,500 

125,000 

7.071 

3.684 

51 

160.22 

2O42.82 

2,601 

132,651 

7.I4I 

3.708 

52 

163.36 

...  2123.72 

...  2,704 

I4O,6o8 

7.2II 

3.732 

53 

166.50 

2206.18 

2,809 

148,877 

7.280 

3.756 

54 

169.65 

...  2290.22 

...  2,916 

157,464 

7.348 

3-779 

55 

172.79 

2375^3 

3,025 

166,375 

7.416 

3.802 

56 

175-93 

...  2463.01 

...  3.136 

I75,6l6 

7.483 

3.825 

57 

179.07 

255I-76 

3?249 

185,193 

7-549 

3.848 

58 

182.21 

...  2642.08 

.-•  3>364 

i95,112 

7-6i5 

3.870 

59 

185.35 

2733.97 

3,481 

205,379 

7.681 

3.892 

60 

188.50 

...  2827.43 

...  3,600 

216,000 

7-745 

3-9J4 

61 

191.64 

2922.47 

3,721 

226,981 

7.810 

3.936 

62 

194.78 

...  3019.07 

...  3,844 

238,328 

7-874 

3-957 

63 

197.92 

3II7-25 

3,969 

250,047 

7-937 

3-979 

64 

201.06 

...  3216.99 

...  4,096 

...  262,144 

8.000 

4.000 

65 

204.20 

33l8-3I 

4,225 

274,625 

8.062 

4.020 

66 

207.34 

...  3421.19 

...  4,356 

287,496 

8.124 

4.041 

67 

210.49 

3525-65 

4,489 

300,763 

8.185 

4.061 

68 

213.63 

...  3631.68 

...  4,624 

3J4,432 

8.246 

4.081 

69 

216.77 

3739-28 

4,761 

328,509 

8.306 

4.101 

70 

219.91 

.  3848.45 

...  4,900 

343,00° 

8.366 

4.121 

7i 

223.05 

3959.19 

5,04! 

357,9n 

8.426 

4.140 

72 

226.19 

...  4071-5° 

...  5,184 

373,248 

8.485 

4.160 

73 

229.34 

4185-39 

5,329 

389,017 

8.544 

4.179 

74 

232.48 

...  4300.84 

...  5,476 

405,224 

8.602 

4.198 

75 

235.62 

4417.86 

5,625 

421,875 

8.660 

4.217 

76 

238.76 

...  4536.46 

•••  5,776 

•••438,976 

8.717 

4-235 

77 

241.90 

4656.63 

5,929 

456,533 

8-744 

4-254 

78 

245.04 

...  4/78.36 

...  6,084 

474,552 

8.831 

4.272 

79 

248.19 

4901.67 

6,241 

493,039 

8.888 

4.290 

So 

25r-33 

...  5026.55 

...  6,400 

512,000 

8.044 

4.308 

81 

254.47 

5i53-oo 

6,561 

53M4i 

X  I  ^ 

9.000 

•  \J 

4.326 

82 

257.61 

...  5281.02 

...  6,724 

551,368 

9-055 

4-344 

83 

260.75 

5410.61 

6,889 

571,787 

9.110 

4.362 

84 

263.89 

•••  5541-77 

...  7,056 

592,704 

9.165 

4-379 

85 

267.03 

5674-5° 

7,225 

614,125 

9.219 

4.396 

86 

270.18 

...  5808.80 

...  7,396 

636,056 

9-273 

4.414 

87 

273.32 

5944-68 

7,569 

658,503 

9-327 

4-43  1 

88 

276.46 

...  6082.12 

...  7,744 

681,472 

9.380 

4-447 

89 

279.60 

6221.14 

7,921 

704,969 

9-433 

4.461 

go 

282.74 

...  6361.73 

...  8,100 

729,000 

9.486 

4.481 

68 


MATHEMATICAL   TABLES. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

91 

285.88 

6503.88 

8528l 

753,571 

9-539 

4-497 

92 

289.03 

...  6647.61 

...  8,464 

778,688 

9-59i 

4.514 

93 

292.17 

6792.91 

8,649 

804,357 

9-643 

4-530 

94 

295-31 

...  6939.78 

...  8,836 

830,584 

9-695 

4.546 

95 

298.45 

7088.22 

9,025 

857,375 

9.746 

4-562 

96 

3oi-59 

...  7238.23 

...  9,216 

884,736 

9-797 

4.578 

97 

304.73 

7389.81 

9,409 

912,673 

9.848 

4-594 

08 

^07.88 

.  7=542.06 

.  0,604 

..  04.1,102 

0.800 

4.610 

? 
99 

O   / 

311.02 

/  O  »    -/ 

7697.69 

...    y  y  WV«f 

9,801 

yf-*-,  *  y** 
970,299 

y*  ss 
9-949 

T..  **:*** 

4.626 

IOO 

314.16 

...  7853.98 

...10,000 

,000,000 

10.000 

4.641 

IOI 

3*7-3° 

8011.85 

10,201 

.030,301 

10.049 

4-657 

102 

320.41 

...  8171.28 

...10,404 

,061,208 

10.099 

4.672 

103 

323-58 

8332.29 

10,609 

,092,727 

10.148 

4.687 

IO4 

326.73 

...  8494.87 

..  .10,816 

...   ,124,864 

10.198 

4.702 

105 

329.87 

8659.01 

11,025 

^57,625 

10.246 

4.7I7 

106 

333-01 

•  8824.73 

...11,236 

,191,016 

10.295 

4-732 

107 

336.15 

8992.02 

n,449 

,225,043 

10.344 

4-747 

108 

339.29 

...  9160.88 

...11,664 

...   ,259,712 

10.392 

4.762 

109 

342-43 

933L32 

11,881 

.295,029 

10.440 

4.776 

no 

345-57 

•••  9503-32 

...12,100 

...   ,331,000 

10.488 

4.791 

in 

348.72 

9676.89 

12,321 

I>367,631 

10.535 

4.805 

112 

351.86 

...  9852.03 

...12,544 

...  1,404,928 

10.583 

4.820 

"3 

355-oo 

10028.75 

12,769 

1,442,897 

10.630 

4-834 

114 

358.14 

...10207.03 

...12,996 

...  1,481,544 

10.677 

4.848 

JI5 

361.28 

10386.89 

13.225 

^520,875 

10.723 

4.862 

116 

364-42 

...10568.32 

...13.456 

...  1,560,896 

10.770 

4.876 

117 

367-57 

10751.32 

13.689 

1,601,613 

10.816 

4.890 

118 

370.71 

...10935.88 

...13,924 

...  1,643,032 

10.862 

4-904 

119 

373.85 

11122.02 

14,161 

1,685,159 

10.908 

4.918 

I2O 

376.99 

...11309.73 

...14,400 

...  1,728,000 

10.954 

4.932 

121 

380.13 

11499.01 

14,641 

I,77I,56i 

II.OOO 

4.946 

122 

383-27 

...11689.87 

...14,884 

...  1,815,848 

11.045 

4-959 

I23 

386.42 

11882.29 

i5>129 

1,860,867 

11.090 

4-973 

124 

389.56 

...12076.28 

...15.376 

...  1,906,624 

11-135 

4.986 

125 

392.70 

12271.85 

15.625 

I,953,I25 

11.180 

5.00° 

126 

395.84 

...12468.98 

...15,876 

...  2,000,376 

11.224 

5-013 

127 

398.98 

12667.69 

16,129 

2,048,383 

11.269 

5.026 

128 

402.12 

...12867.96 

...16,384 

...  2,097,152 

11-313 

5-039 

I29 

405.26 

13069.81 

16,641 

2,146,689 

n-357 

S-oS2 

I30 

408.41 

•••13273-23 

...16,900 

...  2,197,000 

11.401 

5-065 

I3I 

411-55 

13478.22 

17,161 

2,248,091 

n.445 

5-078 

132 

414.69 

..13684.78 

...17,424 

...  2,299,968 

11.489 

5-09I 

133 

417-83 

13892.91 

17,689 

2,352,637 

n-532 

5.104 

!34 

420.97 

...I4I02.6l 

...17,956 

...  2,406,104 

n-575 

$•"7 

J35 

424.11 

14313.88 

18,225 

2,460,375 

11.618 

5.129 

136 

427.26 

...14526.72 

...18,496 

•••  2,515,456 

11.661 

5-U2 

137 

430.40 

14741.14 

18,769 

2,57i,353 

11.704 

5-J55 

138 

433-54 

...14957.12 

...19,044 

...  2,620,872 

11.747 

5-T67 

NUMBERS,    OR  DIAMETERS   OF   CIRCLES,   &c. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

139 

436.68 

15174.68 

19.321 

2,685,619 

11.789 

5.180 

140 

439.82 

...15393.80 

...19,600 

...  2,744,000 

11.832 

5.192 

141 

442.96 

15614.50 

I9,88l 

2,803,221 

11.874 

5.204 

142 

446.  1  1 

...15836.77 

...20,l64 

...  2,863,288 

II.9I6 

5.217 

143 

449.25 

l6o6o.6l 

20,449 

2,924,207 

11.958 

5.229 

144 

452-39 

...16286.02 

...20,736 

•••  2,985,984 

I2.OOO 

5.241 

145 

455-53 

16513.00 

21,025 

3,048,625 

12.041 

5-253 

146 

458.67 

...16741.55 

...21,316 

...  3,112,136 

12.083 

5-265 

147 

461.81 

16971.67 

21,609 

3,176,523 

12.124 

5-277 

148 

464.96 

...17203.36 

...21,904 

...  3,241,792 

12.165 

5.289 

149 

468.10 

17436.62 

22,201 

3,307,949 

I2.2O6 

5.301 

150 

471.24 

...17671.46 

...22,500 

•••  3,375,ooo 

12.247 

5.313 

151 

474.38 

17907.86 

22,801 

3,442,95! 

12.288 

5.325 

J52 

477-52 

...18145.84 

...23,104 

...  3,5^,808 

12.328 

5-336 

J53 

480.66 

18385.39 

23,409 

3,58i,577 

12.369 

5-348 

J54 

483.80 

...18626.50 

...23,716 

...  3,652,264 

12.409 

5.360 

J55 

486.95 

18869.19 

24,025 

3,723,875 

12.449 

5-371 

156 

490.09 

...19113.45 

---24,336 

...  3,796,416 

12.489 

5.383 

i57 

493-23 

19359.28 

24,649 

3,869,893 

12.529 

5-394 

158 

496.37 

...19606.68 

...24,964 

•••  3>944,312 

12.569 

5.406 

!59 

499-5  1 

19855.65 

25,28l 

4,019,679 

12.609 

5-4I7 

160 

502.65 

...2OI06.I9 

...25,600 

...  4,096,000 

12.649 

5.428 

161 

505.80 

20358.34 

25,921 

4,173,281 

12.688 

5-440 

162 

508.94 

...20611.99 

...26,244 

•••  4,251,528 

12.727 

5-451 

163 

512.08 

20867.24 

26,569 

4,330,747 

12.767 

5.462 

164 

5J5.22 

...21124.07 

...26,896 

...  4,410,944 

12.806 

5-473 

165 

518.36 

21382.46 

27,225 

4,492,125 

12.845 

5-484 

1  66 

521.5° 

...21642.43 

•••27,556 

...  4,574,296- 

12.884 

5-495 

167 

524.65 

21903.97 

27,889 

4,657,463 

12.922 

5-506 

168 

527.79 

...22167.08 

...28,224 

...  4,741,632 

12.961 

5.5I7 

169 

530.93 

22431.76 

28,561 

4,826,809 

13.000 

5-528 

170 

534.07 

...22698.01 

...28,900 

...  4,913,000 

13-038 

5-539 

171 

537-21 

22965.83 

29,241 

5,000,21  1 

13.076 

5-550 

172 

540.35 

...23235.22 

...29,584 

...  5,088,448 

13.114 

5-561 

173 

543.50 

23506.18 

29,929 

5,177,717 

I3^S2 

5-572 

174 

546.64 

...23778.71 

...30,276 

...  5,268,024 

13.190 

5-582 

175 

549.78 

24052.82 

30,025 

5,359,375 

13.228 

5-593 

176 

552-92 

...24328.49 

...30,976 

•••  5,45^776 

13.266 

5-604 

177 

556.o6 

24605.79 

31,329 

5,545,233 

13-304 

5.614 

178 

559-20 

...24884.56 

...31,684 

•••  5,639,752 

i3.34i 

5-625 

179 

562.34 

25164.94 

32,041 

5,735,339 

13-379 

5.635 

180 

565.49 

...25446.90 

...32,400 

...  5,832,000 

13.416 

5.646 

181 

568.63 

25730.43 

32,76l 

5,929,741 

13-453 

5.656 

182 

571-77 

...26015.53 

•••33,I24 

...  6,028,568 

13.490 

5.667 

183 

574-91 

26302.20 

33,489 

6,128,487 

13-527 

5.677 

184 

578.05 

...26590.44 

-  -.33,856 

...  6,229,504 

13-564 

5.687 

185 

581.19 

26880.25 

34,225 

6,331*625 

13.601 

5.698 

186 

584-34 

...27171.63 

•••34,596 

...  6,434,856 

13-638 

5.7o8 

MATHEMATICAL   TABLES. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

I87 

587.48 

27464.59 

34,969 

6,539,203 

13.674 

5.718 

188 

590.62 

...27759.11 

•••35,344 

...  6,644,672 

13.711 

5.728 

189 

593-76 

28055.21 

35.721 

6,751,269 

13.747 

5.738 

190 

596.90 

...28352.87 

...36,100 

...  6,859,000 

13.784 

5.748 

191 

60O.O4 

28652.11 

36,481 

6,967,871 

13.820 

5.758 

192 

603.19 

...28952.92 

...36,864 

...  7,077,888 

13.856 

5-768 

!93 

606.33 

29255-30 

37,249 

7,189,057 

13.892 

5.778 

194 

609.47 

...29559.26 

...37,636 

...  7,301,384 

13.928 

5-788 

195 

612.61 

29864.77 

38,025 

754M,875 

13.964 

5-798 

196 

6i5.75 

...30171.86 

...38,416 

•••  7,529,536 

I4.OOO 

5.808 

197 

618.89 

30480.52 

38,809 

7,645,373 

14.035 

5-8l8 

198 

622.03 

...30790.75 

...39,204 

...  7,762,392 

14.071 

5.828 

199 

625.18 

3H02.55 

39,601 

7,880,599 

14.106 

5.838 

200 

628.32 

•••3I4i5-93 

...40,000 

...  8,OOO,OOO 

14.142 

5.848 

201 

631.46 

31730.87 

40,401 

8,120,601 

14.177 

5.857 

202 

634.60 

...32047.39 

...40,804 

...  8,242,408 

14.212 

5.867 

203 

637-74 

32365-47 

41,209 

8,365,427 

14.247 

5.877 

204 

640.88 

...32685.13 

...41,616 

...  8,489,664 

14.282 

5.886 

205 

644.03 

33006.36 

42,025 

8,615,125 

I4.3I7 

5.896 

206 

647.17 

...33329.16 

...42,436 

...  8,741,816 

14.352 

5.905 

207 

650-31 

33653.53 

42,849 

8,869,743 

14.387 

5.915 

208 

653-45 

•••33979.47 

...43,264 

...  8,998,912 

14.422 

5-924 

209 

656-59 

34306.98 

43,681 

9,123,329 

14.456 

5-934 

210 

659.73 

...34636.06 

...44,100 

...  9,26l,000 

14.491 

5-943 

211 

662.88 

34966.71 

44,52i 

9,393,931 

I4-525 

5-953 

212 

666.02 

..35298.94 

...44,944 

...  9,528,128 

14.560 

5-962 

2I3 

669.16 

35632.73 

45.369 

9,663,597 

14-594 

5-972 

214 

672.30 

..35968.09 

...45,796 

...  9,800,344 

14.628 

5.981 

215 

675.44 

36305-03 

46,225 

9,938,375 

14.662 

5-990 

216 

678.58 

..36643.61 

...46,656 

...10,077,696 

14.696 

6.000 

217 

681.73 

36983.61 

47,089 

10,218,313 

14.730 

6.009 

218 

684.87 

••37325-26 

...47.524 

...10,360,232 

14.764 

6.018 

219 

688.01 

37668.48 

47,961 

!0,503,459 

14.798 

6.027 

220 

691.15 

..38013.27 

...48,400 

...10,648,000 

14.832 

6.036 

221 

694.29 

38359-63 

48,841 

10,793,861 

14.866 

6.045 

222 

697.43 

..38707.56 

...49,284 

...10,941,048 

14.899 

6.055 

223 

700.57 

39057.07 

49,729 

11,089,567 

J4-933 

6.064 

224 

703.72 

..39408.14 

...50,176 

...11,239,424 

14.966 

6.073 

225 

706.86 

39760.78 

50,625 

11,390,625 

15.000 

6.082 

226 

710.00 

..40115.00 

...51,076 

•••n,  543,176 

!5.033 

6.091 

227 

7I3-I4 

40470.78 

5^529 

11,697,083 

15.066 

6.100 

228 

716.28 

..40828.14 

...51,984 

...11,852,352 

15.099 

6.109 

229 

719.42 

41187.07 

52,44i 

12,008,989 

I5-I32 

6.118 

230 

722.57 

••41547.56 

...52,900 

...12,167,000 

15-165 

6.126 

23I 

725-71 

41909.63 

53.361 

12,326,391 

15.198 

6.135 

232 

728.85 

-.42273-27 

••53,824 

...12,487,168 

J5-23i 

6.144 

233 

73!-99 

42638.48 

54,289 

12,649,337 

15.264 

6.153 

234 

735-I3 

..43005.26 

•••54,756 

...12,812,904 

15-297 

6.162 

NUMBERS,    OR   DIAMETERS   OF  CIRCLES,   &c. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

235 

738.27 

43373-61 

55.225 

12,977,875 

!5-329 

6.171 

236 

741.42 

•••43743-54 

...55,696 

...13,144,256 

15.362 

6.179 

237 

744.56 

44II5-03 

56,169 

i3>312.°53 

J5-394 

6.188 

238 

747.70 

...44488.09 

.••56,644 

...13,481,272 

15-427 

6.197 

239 

750.84 

44862.73 

57.121 

13,651,919 

I5-459 

6.205 

240 

753.98 

•••45238.93 

...57,600 

...13,824,000 

I5-491 

6.214 

241 

757.12 

45616.71 

58,081 

i3,997,52i 

i5-524 

6.223 

242 

760.26 

...45996.06 

.-•58,564 

...14,172,488 

6.231 

243 

763.4I 

46376.98 

59.049 

14,348,907 

15-588 

6.240 

244 

766.55 

...46759.47 

•••59.536 

...14,526,784 

15.620 

6.248 

245 

769.69 

47143.52 

60,025 

14,706,125 

15.652 

6.257 

246 

772.83 

...47529.16 

...60,516 

...14,886,936 

15.684 

6.265 

247 

775-97 

47916.36 

61,009 

15,069,223 

15.716 

6.274 

248 

779.11 

...48305.13 

...61,504 

...15,252,992 

I5-748 

6.282 

249 

782.26 

48695.47 

62,001 

15,438,249 

!5-779 

6.291 

250 

785.40 

...49087.39 

...62,500 

...15,625,000 

15.811 

6.299 

251 

788.54 

49480.87 

63,001 

15,813,251 

15.842 

6.307 

252 

791.68 

...49875.92 

•  •  -63,504 

...16,003,008 

15.874 

6.316 

253 

794.82 

50272.55 

64,009 

16,194,277 

6.324 

254 

797.96 

...50670.75 

...64,516 

...16,387,064 

15-937 

6-333 

255 

801.11 

51070.52 

65,025 

16,581,375 

15.968 

6.341 

256 

804.25 

...51471.86 

•  ••65,536 

...16,777,216 

16.000 

6-349 

257 

807.39 

51874.76 

66,049 

i6,974,593 

16.031 

6.357 

258 

810.53 

...52279.24 

...66,564 

•••17,173,512 

16.062 

6.366 

259 

813.67 

52685.29 

67,081 

17,373,979 

16.093 

6-374 

260 

816.81 

...53092.96 

...67,600 

•••17,576,000 

16.124 

6.382 

26l 

819.96 

53502.11 

68,121 

I7,779,58r 

16.155 

6.390 

262 

823.10 

...53912.87 

...68,644 

...17,984,728 

16^186 

6.398 

263 

82.6.24 

54325.21 

69,169 

18,191,447 

16.217 

6.406 

264 

829.38 

•••54739-I1 

...69,696 

•••18,399,744 

16.248 

6.415 

265 

832.52 

55J54-59 

70,225 

18,609,625 

16.278 

6.423 

266 

835.66 

-••55571-63 

...70,756 

...18,821,096 

16.309 

6.431 

267 

838.80 

55990.25 

71,289 

19,034,163 

16.340 

6-439 

268 

841.95 

...56410.44 

...71,824 

...19,248,832 

16.370 

6.447 

269 

845.09 

56832.20 

72,361 

19,465,109 

16.401 

6-455 

270 

848.23 

•••57255-53 

...72,900 

...19,683,000 

16.431 

6.463 

271 

85L37 

57680.43 

73,441 

19,902,511 

16.462 

6.471 

272 

854-51 

...58106.90 

•••73.984 

...20,123,648 

16.492 

6-479 

273 

857.65 

58534-94 

74,529 

20,346,417 

16.522 

6.487 

274 

860.80 

•••58964.55 

...75,076 

...20,570,824 

16.552 

6-495 

275 

863.94 

59395-74 

75,625 

20,796,875 

16.583 

6.502 

276 

867.08 

...59828.49 

...76,176 

...21,024,576 

16.613 

6.510 

277 

870.22 

60262.82 

76,729 

21,253,933 

16.643 

6.518 

278 

873.36 

...60698.72 

...77,284 

...21,484,952 

16.673 

6.526 

279 

876.50 

61136.18 

77,84i 

21,717,639 

16.703 

6-534 

280 

879.65 

...61575.22 

...78,400 

...21,952,000 

16.733 

6-542 

28l 

882.79 

62015.82 

78,961 

22,188,041 

16.763 

6-549 

282 

885-93 

...62458.00 

...79,524 

...22,425,768 

16.792 

6-557 

MATHEMATICAL   TABLES. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

283 

889.07 

62901.75 

80,089 

22,665,187 

16.822 

6.565 

284 

892.21 

...63347.07 

...80,656 

...22,906,304 

16.852 

6-573 

285 

895-35 

63793-97 

81,225 

23,149,125 

16.881 

6.580 

286 

898.49 

...64242.43 

...81,796 

••23,393>656 

16.911 

6.588 

287 

901.64 

64692.46 

82,369 

23,639,903 

16.941 

6.596 

288 

904.78 

...65144.07 

...82,944 

...23,887,872 

16.970 

6.603 

289 

907.92 

65597-24 

83,521 

24,137,569 

17.000 

6.611 

2QO 

911.06 

...66051.99 

...84,100 

...24,389,000 

17.029 

6.619 

291 

914.20 

66508.30 

84,681 

24,642,171 

17-059 

6.627 

292 

9*7-34 

...66966.19 

...85,264 

...24,897,088 

17.088 

6.634 

293 

920.49 

67425.65 

85,849 

25^53,757 

17.117 

6.642 

294 

923.63 

...67886.68 

...86,436 

...25,412,184 

17.146 

6.649 

295 

926.77 

68349.28 

87,025 

25>672,375 

17.176 

6.657 

296 

929.91 

...68813.45 

...87,616 

•••25*934,336 

17.205 

6.664 

297 

933-05 

69279.19 

88,209 

26,198,073 

17-234 

6.672 

298 

936.19 

...69746.50 

...88,804 

•••26,463,592 

17.263 

6.679 

299 

939-34 

70215.38 

89,401 

26,730,899 

17.292 

6.687 

300 

942.48 

...70685.83 

...90,000 

.  .  .27,OOO,OOO 

17.320 

6.694 

3OI 

945.62 

71157.86 

90,601 

27,270,901 

*7.349 

6.702 

302 

948.76 

...71631.45 

...91,204 

•--27,543,608 

17-378 

6.709 

3°3 

951.90 

72106.62 

91,809 

27,818,127 

17.407 

6.717 

3°4 

955-°4 

•--72583-36 

...92,416 

...28,094,464 

I7-436 

6.724 

305 

958.19 

73061.66 

93>025 

28,372,625 

17.464 

6.731 

306 

96l-33 

•••73541-54 

...93,636 

...28,652,616 

!7-493 

6.739 

307 

964.47 

74022.99 

94,249 

28,934,443 

17.521 

6.746 

308 

967.61 

...74506.01 

...94,864 

...29,2l8,II2 

17.549 

6-753 

309 

970-75 

74990.60 

95.481 

29,503,629 

17.578 

6.761 

310 

973-89 

...75476.76 

...96,100 

...29,791,000 

17.607 

6.768 

3H 

977.03 

75964-50 

96,721 

30,080,231 

17-635 

6.775 

312 

980.18 

...76453.80 

•••97,344 

...30,371,328 

17.663 

6.782 

3i3 

983-32 

76944-67 

97,969 

30,664,297 

17.692 

6.789 

3U 

986.46 

...77437.12 

...98,596 

...30,959,144 

17.720 

6.797 

3i5 

989.60 

7793I-I3 

99,225 

3I,255,875 

17.748 

6.804 

316 

992-74 

...78426.72 

...99,856 

•••31,554,496 

17.776 

6.8n 

3i7 

995.88 

78923.88 

100,489 

3r,855,oi3 

17.804 

6.8i8 

3i8 

999.03 

...79422.60 

101,124 

...32,157,432 

17.832 

6.826 

319 

1002.17 

79922.90 

101,761 

32,461,759 

17.860 

6.833 

320 

1005.31 

...80424.77 

102,400 

...32,768,000 

17.888 

6.839 

321 

1008.45 

80928.21 

103,041 

33,076,161 

17.916 

6.847 

322 

1011.59 

...81433.22 

103,684 

...33,386,248 

17.944 

6.854 

323 

1014.73 

81939.80 

104,329 

33,698,267 

17.972 

6.861 

324 

1017.88 

...82447.96 

104,976 

...34,012,224 

18.000 

6.868 

325 

IO2I.O2 

82957.68 

105,625 

34,328,125 

18.028 

6.875 

326 

I024.I6 

...83468.98 

106,276 

...34,645,976 

18.055 

6.882 

327 

1027.30 

83981.84 

106,929 

34,965,783 

18.083 

6.889 

328 

1030.44 

...84496.28 

107,584 

•••35>287,552 

18.111 

6.896 

329 

1033.58 

85012.28 

108,241 

35,611,289 

18.138 

6.903 

330 

1036.73 

...85529.86 

108,900 

•••35,937,000 

18.166  6.910 

NUMBERS,    OR  DIAMETERS  OF   CIRCLES,   &c. 


73 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

331 

1039.87 

86049.01 

109,561 

36,264,691 

18.193 

6.917 

332 

1043.01 

...86569.73 

IIO,224 

...36,594,368 

18.221 

6.924 

333 

1046.15 

87092.02 

110,889 

36,926,037 

18.248 

6.931 

334 

1049.29 

...87615.88 

in,556 

•••37,259,704 

18.276 

6.938 

335 

1052.43 

88141.31 

112,225 

37,595^375 

18.303 

6-945 

336 

1055.57 

...88668.31 

112,896 

•••37,933,056 

18.330 

6.952 

337 

1058.72 

89196.88 

113.569 

38,272,753 

18-357 

6-959 

338 

1061.86 

...89727.03 

114,244 

...38,614,472 

18.385 

6.966 

339 

IO65.0O 

90258.74 

114,921 

38,958,219 

18.412 

6-973 

340 

IO68.I4 

...90792.03 

115,600 

...39,304,000 

18.439 

6.979 

34i 

1071.28 

91326.88 

116,281 

39,651,821 

18.466 

6.986 

342 

1074.42 

...91863.31 

116,964 

...40,001,688 

18.493 

6-993 

343 

1077.57 

92401.31 

117,649 

40,353,607 

18.520 

7.000 

344 

1080.71 

...92940.88 

118,336 

•••40,707,584 

18.547 

7.007 

345 

1083.85 

93482.02 

119,025 

41,063,625 

18.574 

7.014 

346 

1086.99 

...94024.73 

119,716 

...41,421,736 

18.601 

7.020 

347 

1090.13 

94569.01 

120,409 

41,781,923 

18.628 

7.027 

348 

1093.27 

...95114.86 

121,104 

...42,144,192 

18.655 

7.034 

349 

1096.42 

95662.28 

121,801 

42,508,549 

18.681 

7.040 

350 

1099.56 

...96211.28 

122,500 

...42,875,000 

18.708 

7.047 

35i 

1102.70 

96761.84 

123,201 

43,243,5s1 

18.735 

7.054 

352 

1105.84 

...97314.76 

123,904 

...43,614,208 

18.762 

7.061 

353 

1108.98 

97867.68 

124,609 

43,986,977 

18.788 

7.067 

354 

1  1  12.  12 

...98422.96 

i25,3l6 

...44,361,864 

18.815 

7.074 

355 

1115.26 

98979.80 

126,025 

44,738,875 

18.842 

7.081 

356 

III8.4I 

...99538.22 

126,736 

...45,118,016 

18.868 

7.087 

357 

II2i.55 

100098.21 

127,449 

45,499,293 

18.8-94 

7.094 

358 

1124.69 

100659.27 

128,164 

...45,882,712 

18.921 

7.101 

359 

1127.83 

IOI222.9O 

128,881 

46,268,279 

18.947 

7.107 

360 

1130.97 

101787.60 

129,600 

...46,656,000 

18.974 

7.114 

361 

1134.11 

102353.87 

130,321 

47,045,881 

19.000 

7.120 

362 

1137.26 

102921.72 

131,044 

•••47,437,928 

19.026 

7.127 

363 

1140.40 

103491.13 

131,769 

47,832,147 

19.052 

7.133 

364 

1143.54 

IO4062.I2 

132,496 

...48,228,544 

19.079 

7.140 

365 

1146.68 

104634.67 

133,225 

48,627,125 

19.105 

7.146 

366 

1149.82 

105208.80 

133,956 

...49,027,896 

19.131 

7.153 

367 

1152.96 

105784.49 

134,689 

49,430,863 

I9-I57 

7-159 

368 

1156.11 

106361.76 

i35>424 

•••49,836,032 

19.183 

7.166 

369 

1159.25 

106940.60 

136,161 

50,243,409 

19.209 

7.172 

370 

1162.39 

107521.01 

136,900 

...50,653,000 

19-235 

7.179 

37i 

1165.53 

108102.99 

137,641 

51,064,811 

19.261 

7.185 

372 

1168.67 

108686.54 

138,384 

...51,478,848 

19.287 

7.192 

373 

1171.81 

109271.66 

139,129 

51,895,117 

19-313 

7.198 

374 

1174.96 

109858.35 

139,876 

...52,313,624 

19-339 

7.205 

375 

1178.10 

110446.62 

140,625 

52,734,375 

19-365 

7.211 

376 

1181.24 

111036.45 

!4i,376 

••  .53^57,376 

I9-391 

7.218 

377 

1184.38 

111627.86 

142,129 

53,582,633 

19.416 

7.224 

378 

1187.52 

II222O.83 

142,884 

...54,010,152 

19.442 

7.230 

74 


MATHEMATICAL   TABLES: 


Number, 
or 

|  Diameter. 

Circum-       Circular 
ference.        Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

379 

1190.66 

112815.38 

143,641 

54,439.939 

19.468 

7-237 

380 

II93.80 

113411.49 

144,400 

...54,872,000 

*9-493 

7-243 

38i 

1196.95 

114009.18 

I45,l6l 

55,306,341 

19-519 

7.249 

382 

1200.09 

114608.44 

145.924 

...55,742,968 

!9-545 

7.256 

383 

1203.23 

115209.27 

146,689 

56,181,887 

J9-57o 

7.262 

384 

1206.37 

115811.67 

147,456 

...56,623,104 

19.596 

7.268 

385 

1209.51 

116415.64 

148,225 

57,066,625 

19.621 

7-275 

386 

1212.65 

117021.18 

148,996 

...57,512,456 

19.647 

7.28l 

387 

1215.80 

117628.30 

149,769 

57,960,603 

19.672 

7.287 

388 

1218.94 

118236.98 

I5°>544 

...58,411,072 

19.698 

7.294 

389 

1222.08 

118847.24 

15^321 

58,863,869 

19.723 

7.299 

390 

1225.22 

119459.06 

152,100 

•••59,3I9,°oo 

19.748 

7.306 

39i 

1228.36 

120072.46 

152,881 

59,776,471 

19.774 

7-312 

392 

1231.50 

120687.42 

J53,664 

...60,236,288 

19.799 

7-319 

393 

1234.65 

121303.96 

1  54,449 

60,698,457 

19.824 

7.325 

394 

1237.79 

121922.07 

I55>236 

...61,162,984 

19.849 

7.331 

395 

1240.93 

122541.75 

156,025 

61,629,875 

19.875 

7-337 

396 

1244.07 

123163.00 

156,816 

...62,099,136 

19.899 

7-343 

397 

1247.21 

123785.82 

157,609 

62,570,773 

19.925 

7-349 

398 

1250.35 

124410.21 

158,404 

•••63,044,792 

19.949 

7.356 

399 

1253-49 

125036.17 

159,201 

63,52I,I99 

19-975 

7.362 

400 

1256.64 

125663.71 

160,000 

.  ..64,000,000 

20.000 

7-368 

401 

1259.78 

126292.81 

160,801 

64,481,201 

20.025 

7-374 

402 

1262.92 

126923.48 

161,604 

...64,964,808 

20.049 

7.380 

403 

1266.06 

127553-73 

162,409 

65,450,827 

20.075 

7.386 

404 

1269.20 

128189.55 

163,216 

•••65,939,264 

20.099 

7-392 

405 

1272.34 

128824.93 

164,025 

66,430,125 

20.125 

7-399 

406 

1275.49 

129461.89 

164,836 

...66,923,416 

2O.I49 

7-405 

407 

1278.63 

130100.42 

165,649 

67,4i9>143 

20.174 

7.411 

408 

I28I.77 

130740.52 

166,464 

...67,911,312 

20.199 

7.417 

409 

1284.91 

131382.19 

167,281 

68,417,929 

2O.224 

7.422 

410 

1288.05 

132025.43 

168,100 

...68,921,000 

20.248 

7.429 

411 

1291.19 

132670.24 

168,921 

69,426,531 

20.273 

7-434 

412 

1294.34 

1333*6.63 

169,744 

•••69,934,528 

20.298 

7.441 

4i3 

1297.48 

133964-58 

170,569 

70,444,997 

20.322 

7-447 

414 

I30O.62 

134614.10 

171,396 

•••70,957,944 

20.347 

7-453 

4i5 

1303.76 

135265.20 

172,225 

71,473,375 

20.371 

7-459 

416 

1306.90 

135917.86 

^3,056 

...71,991,296 

20.396 

7-465 

417 

1310.04 

136572.10 

173,889 

72,511,713 

20.421 

7.471 

418 

I3I3-I9 

137227.91 

174,724 

...73,034,632 

20.445 

7-477 

419 

I3I6.33 

137885.29 

I75»561 

73,560,059 

20.469 

7-483 

420 

i3J9-47 

138544.24 

176,400 

...74,088,000 

20.494 

7.489 

421 

1322.61 

139204.70 

177,241 

74,618,461 

20.518 

7-495 

422 

I325-75 

139866.85 

178,084 

•••75,I5I,448 

20.543 

7-501 

423 

1328.89 

140530.51 

178,929 

75,686,967 

20.567 

7-507 

424 

1332-03 

I4II95.74 

179,776 

...76,225,024 

20.591 

7-5*3 

425 

1335-iS 

141862.54  . 

180,625 

76,765,625 

20.6l5 

7-518 

426 

1338.32 

142530.92 

181,476 

...77,308,776 

20.639 

7-524 

NUMBERS,    OR   DIAMETERS   OF   CIRCLES,   &c. 


75 


'  Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

427 

1341.46 

143200.86 

182,329 

77,854,483 

20.664 

7-530 

428 

1344.60 

143872.38 

183,184 

...78,402,752 

20.688 

7.536 

429 

1347-74 

144545.46 

184,041 

78,953,589 

20.712 

7-542 

430 

1550.88 

145220.12 

184,900 

...79,507,000 

20.736 

7.548 

43? 

I354-03 

145896.35 

185,761 

80,062,991 

20.760 

7-554 

432 

1357.17 

I46574.I5 

186,624 

...80,621,568 

20.785 

7-559 

433 

1360.31 

147253-52 

187,489 

81,182,737 

20.809  7.565 

434 

I363-45 

147934.46 

188,356 

...81,746,504 

20.833  7-571 

435 

1366.59 

148616.97 

189,225 

82,312,875 

20.857 

7-577 

436 

I369-73 

149301.05 

190,096 

...82,881,856 

20.881 

7-583 

437 

1372.88 

149986.70    190,969 

83,453,453 

20.904 

7-588 

438 

1376.02 

I50673.93  i  19^844 

...84,027,672 

20.928 

7-594 

439 

I379.I6 

151362.72    I92J2I 

84,604,519 

20.952 

7.600 

440 

1382.30 

152053.08  !  193,600 

...85,184,000 

20.976 

7.606 

441 

I385-44 

152745.02  !  194,481 

85,766,121 

21.000 

7.612 

442 

1388.58 

153438-53  ;;  195,364 

...86,350,388 

21.024 

7.617 

443 

i39I-73 

154133.60  196,249 

86,938,307 

21.047 

7.623 

444 

1394.87 

154830.25  197,136 

...87,528,384 

21.071 

7.629 

445 

1398.01 

155528.47 

198,025 

88,121,125 

21.095 

7.635 

446 

1401.15 

156228.26 

198,916 

...88,716,536 

21.119 

7.640 

447 

1404.29 

156929.62 

199,809 

89,314,623 

21.142 

7.646 

448 

1407.43 

157632.55 

2OO,7O4 

••-89>9I5,392 

21.166 

7.652 

449 

1410.57 

158337.06 

201,601 

90,518,849 

21.189 

7-657 

450 

1413.72 

i59°43-I3 

202,5OO 

...91,125,000 

21.213 

7-663 

451 

1416.86 

159750.77 

203,401 

9i,733,85i 

21.237 

7.669 

452 

1420.00 

160459.99 

204,304 

...92,345,408 

21.260 

7-674 

453 

1423.14 

161170.77 

205,209 

92,959,677 

21.284 

7.680 

454 

1426.28 

161883.13 

2O6,IO6 

...93,576,664 

21.307 

7.686 

455 

1429.42 

162597.06 

207,025 

94,196,375 

21.  331 

7.691 

456 

1432.57 

163312.55 

207,936 

...94,818,816 

21-354 

7.697 

457 

i435-7i 

164029.62 

208,849 

95,443,993 

21.377 

7-703 

458 

1438-85 

164748.26 

209,764 

...96,071,912 

21.401 

7.708 

459 

1441.99 

165468.47 

2IO,68l 

96,702,579 

21.424 

7.714 

460 

i445-I3 

166190.25 

211,600 

...97,336,000 

21.447 

7.719 

461 

1448.27 

166913.60 

212,521 

97,972,181 

21.471 

7.725 

462 

1451.42 

167638.53 

2I3>444 

...98,611,128 

21.494 

-  7-731 

463 

1454.56 

168365.02 

214,369 

99,252,847 

2i.5J7  !  7.736 

464 

1457.70 

169093.08   215,296 

•••99,897,345 

21.541 

7.742 

465 

1460.84 

169822.72   216,225 

100,544,625 

21.564 

7-747 

466 

1463.98 

170553-92   217,156 

101,194,696 

21.587 

7-753 

467 

1467.12 

171286.70   218,089 

101,847,563 

21.610  ;  7.758 

468 

1470.26 

172021.05   219,024 

102,503,232 

21.633 

7-764 

469 

i473'4i 

172756.97   219,961 

103,161,709 

21.656 

7.769 

470 

1476.55 

1  73494-45  1  220,900 

103,823,000 

21.679 

7-775 

471 

1479.69 

I74233-51  l!  221,841 

104,487,111 

21.702 

7.780 

472 

1482.83 

174974.14 

222,784 

105,154,048 

21.725 

7.786 

473 

1485.97 

175716.35 

223,729 

105,823,817 

21.749 

7.791 

474 

1489.11 

176460.12 

224,676 

106,496,424 

21.771 

7-797 

MATHEMATICAL   TABLES. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

475 

1492.26 

177205.46 

225,625 

107,171,875 

21.794 

7.802 

476 

1495.40 

177952.37 

226,576 

107,850,176 

2I.8I7 

7.808 

477 

1498.54 

178700.86 

227,529 

108,531,333 

21.840 

7.813 

478 

1501.68 

179450.91 

228,484 

109.215,352 

21.863 

7.819 

479 

1504.82 

180202.54 

229,441 

109,902,239 

21.886 

7.824 

480 

1507.96 

180955.74 

230,400 

110,592,000 

21.909 

7.830 

481 

I5II.II 

181710.50 

231,361 

111,284,641 

21.932 

7.835 

482 

i5J4-25 

182466.84 

232,324 

III,98o,l68 

21-954 

7.840 

483 

1517-39 

183224.75 

233.289 

112,678,587 

21.977 

7.846 

484 

I520-53 

183984.23 

234,256 

113.379.904 

22.000 

7.851 

485 

i  J523.67 

184745.28 

235^25 

114,084,125 

22.023 

7.857 

486 

j  1526.81 

185507.90 

236,196 

114,791,256 

22.045 

7.862 

487 

|  1529.96 

186272.10 

237,169 

ii5.5OI>3°3 

22.069 

7.868 

488 

1533-10 

187037.86 

238,144 

116,214,272 

22.091 

7.873 

489 

!  i536-24 

187805.19 

239,121 

116,936,169 

22.113 

7.878 

490 

I539.38 

188574.10 

240,100 

117,649,000 

22.136 

7.884 

491 

i542-52 

i89344.57 

24I,o8l 

1*8,370,771 

22.158 

7.889 

492 

1545.66 

190116.62 

242,064 

119*095,488 

22.l8l 

7.894 

493 

1548.80 

190890.24 

243,049 

119.823,157 

22.2O4 

7.899 

494 

I55I-9S 

191665.43 

244,036 

120,553,784 

22.226 

7.905 

495 

:  !555-o9 

192442.19 

2455025 

121,287,375 

22.248 

7.910 

496 

!  !558.23 

193220.51 

246,016 

122,023,936 

22.271 

7.9I5 

497 

1561.37 

194000.42 

247,009 

122,763,473 

22.293 

7.921 

498 

1564-51 

194781.89 

248,004 

I23>5°5.992 

22.3l6 

7.926 

499 

1567-65 

!95564-93 

249,001 

124,251,499 

22.338 

7.932 

500 

1570.80 

i96349.54 

250,000 

125,000,000 

22.361 

7-937 

501 

I573.94 

197135-72 

251,001 

125,751,501 

22.383 

7-942 

502 

1577.08 

197923.48 

252,004 

126,506,008 

22.405 

7-947 

5°3 

1580.22 

198712.80 

253.009 

127,263,527 

22.428 

7-953 

5°4 

1583-36 

199503-70   254,016 

128,024,864 

22.449 

7.958 

5°5 

1586.50 

200296.17 

255.025 

128,787,625 

22.472 

7-963 

506 

i589.65 

201090.20 

256,036 

129.554,216 

22.494 

7-969 

5°7 

1592.79 

201885.81 

257.049 

130,323.843 

22.517 

7-974 

508 

I595.93 

202682.99 

258,064 

131,096,512 

22-539 

7-979 

5°9 

1599.07 

203481.74 

259,081 

131,872,229 

22.561 

7-984 

510 

1602.21 

204282.06 

260,100 

132,651,000 

22.583 

7.989 

511 

1605.35 

205083.95 

261,121 

133.432,831 

22.605 

7-995 

512 

1608.49 

205887.42 

262,144 

134,217,728 

22.627 

8.000 

5J3 

1611.64 

206692.45 

263,169 

i35.oo5.697 

22.649 

8.005 

5J4 

1614.78 

207499.05 

264,196 

135.796,744 

22.671 

8.010 

5I5 

1617.92 

208307.23 

265,225 

136,590,875 

22.694 

8.016 

5i6 

1621.06 

209116.97 

266,256 

137,388,096 

22.716 

8.021 

5J7 

1624.20 

209928.29 

267,289 

138,188,413 

22.738 

8.026 

5i8 

1627.34 

210741.18 

-  268,324 

138,991,832 

22-759 

8.031 

5J9 

1630.49 

211555-63 

269,361 

139.798,359 

22.782 

8.036 

520 

1633-63 

212371.66 

270,400 

140,608,000 

22.803 

8.041 

521 

1636.77 

213189.26 

271,441 

141,420,761 

22.825 

8.047 

522   1639.91 

214008.43 

272,484 

142,236,648 

22.847 

8.052 

NUMBERS,    OR  DIAMETERS   OF  CIRCLES,   &c. 


77 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

523 

1643.05 

214829.17 

273,529 

143,055,667 

22.869 

8.057 

524 

1646.19 

215651.49 

274,576 

143,877,824 

22.891 

8.062 

525 

1649.34 

2I6475-37 

275,625 

144,703,125 

22.913 

8.067 

526 

1652.48 

217300.82 

276,676 

145,531,576 

22-935 

8.072 

S2? 

1655.62 

218127.85 

277,729 

146,363,183 

22.956 

8.077 

528 

1658.76 

218956.44 

278,784 

147,197,952 

22.978 

8.082 

529 

1661.90 

219786.61 

279,841 

148,035,889 

23.000 

8.087 

530 

1665.04 

220618.32 

280,900 

148,877,000 

23.022 

8-093 

531 

1668.19 

221451.65 

281,961 

149,721,291 

23-043 

8.098 

532 

1671.33 

222286.53 

283,024 

150,568,768 

23.065 

8.103 

533 

1674.47 

223122.98 

284,089 

i5I,4i9>437 

23.087 

8.108 

534 

1677.61 

223961.00 

285,156 

152,273,304 

23.108 

8.II3 

535 

1680.75 

224800.59 

286,225 

!53,i3o,375 

23.130 

8.II8 

536 

1683.89 

225641.75 

287,296 

153,990,656 

23.152 

8.123 

537 

1687.04 

226484.48 

288,369 

154,854,153 

23«I73 

8.128 

538 

1690.18 

227328.77 

289,444 

155,720,872 

23«I95 

8.133 

539 

1693.32 

228174.66 

290,521 

156,590,819 

23.216 

8.138 

540 

1696.46 

229022.IO 

291,600 

157,464,000 

23.238 

8.143 

54i 

1699.60 

229871.12 

292,681 

158,340,421 

23-259 

8.148 

542 

1702.74 

230721.71 

293,764 

159,220,088 

23.281 

8.153 

543 

1705.88 

231573.86 

294,849 

160,103,007 

23.302 

8.158 

544 

1709.03 

232427.59 

295,936 

160,989,184 

23-324 

8.163 

545 

1712.17 

233282.89 

297,025 

161,878,625 

23-345 

8.168 

546 

I7I5-31 

234139.76 

298,116 

162,771,336 

23.367 

8.173 

547 

1718.45 

234998.20 

299,209 

163,667,323 

23-388 

8.178 

548 

1721.59 

235858.21 

300,304 

164,566,592 

23.409 

8-183 

549 

1724.73 

236719.79 

301,401 

165,469,149 

23-431 

8.188 

550 

1727.88 

237582.94 

302,500 

166,375,000 

23-452 

8.193 

55i 

1731.02 

238447.67 

303,601 

167,284,151 

23-473 

8.198 

552 

1734.16 

239313.96 

304,704 

168,196,608 

23-495 

8.203 

553 

I737-30 

240181.83 

305,809 

169,112,377 

23-516 

8.208 

554 

1740.44 

241051.26 

306,916 

170,031,464 

23-537 

8.213 

555 

I743-58 

241922.27 

308,025 

i7o,953,875 

23-558 

8.218 

556 

1746.73 

242794.85 

309,136 

171,879,616 

23-579 

8.223 

557 

1749.87 

243668.99 

310,249 

172,808,693 

23.601 

8.228 

558 

1753.00 

244544.61 

3H,364 

173,741,112 

23.622 

8.233 

559 

1756-15 

245422.00 

3I2,48l 

174,676,879 

23-643 

8.238 

560 

1759.29 

246300.86 

313,600 

175,616,000 

23.664 

8.242 

56i 

1762.43 

247181.30 

3!4,72i 

176,558,481 

23-685 

8.247 

562 

1765-57 

248062.30 

3i5>844 

177,504,328 

23.706 

8.252 

563 

1768.72 

248946.87 

316,969 

178,453,547 

23.728 

8.257 

564 

1771.86 

249832.01 

318,096 

179,406,144 

23.749 

8.262 

565 

1775.00 

2507l8.73 

319,225 

180,362,125 

23.769 

8.267 

566 

1778.14 

251607.01 

320,356 

181,321,496 

23.791 

8.272 

567 

1781.28 

252496.87 

321,489 

182,284,263 

23.812 

8.277 

568 

1784.42 

253388.30 

322,624 

183,250,432 

23.833 

8.282 

569 

1787.57 

254281.30 

323,761 

184,220,009 

23-854 

8.286 

570 

1790.71 

255I75-86 

324,900 

185,193,000 

23-875 

8.291 

MATHEMATICAL  TABLES. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

571 

I793-85 

256072.00 

326,041 

186,169,411 

23.896 

8.296 

572 

1796.99 

256969.71 

327,184 

187,149,248 

23.916 

8.301 

573 

I80O.I3 

257868.99 

328,329 

188,132,51-7 

23-937 

8.306 

574 

1803.27 

258769.85 

329,476 

189,119,224 

23-958 

8.3II 

575 

1806.42 

259672.27 

330?625 

190,109,375 

23-979 

8.315 

576 

1809.56 

260576.26 

33^776 

191,102,976 

24.000 

8.320 

577 

l8l2.70 

261481.83 

332,929 

192,100,033 

24.021 

8.325 

573 

1815.84 

262388.96 

334,084 

!93,ioo,552 

24.042 

8-330 

579 

1818.98 

263297.67 

335,241 

194,104,539 

24.062 

8-335 

58o 

I822.I2 

264207.94 

336,400 

195,112,000 

24.083 

8-339 

58i 

1825.26 

265119.79 

337,56i 

196,122,941 

24.104 

8-344 

582 

1828.41 

266033.21 

338,724 

197,137,368 

24.125 

8-349 

583 

I83L55 

266948.20 

339,889 

198,155,287 

24.145 

8-354 

584 

1834.69 

267864.76 

341,056 

199,176,704 

24.166 

8-359 

585 

1837.83 

268782.80 

342,225 

200,201,625 

24.187 

8-363 

586 

1840.97 

269702.59 

343,396 

201,230,056 

24.207 

8.368 

587 

1844.11 

270623.86 

344,569 

202,262,003 

24.228 

8-373 

588 

1847.26 

271546.70 

345,744 

203,297,472 

24.249 

8.378 

589 

1850.40 

272471.12 

346,921 

204,336,469 

24.269 

8-382 

590 

I853.54 

273397.  10 

348,ioo 

205,  379>000 

24.289 

8.387 

59i 

1856.68 

274324.66 

349,281 

206,425,071 

24.310 

8.392 

592 

1859.82 

275253-78 

3505464 

207,474,688 

24.331 

8-397 

593 

1862.96 

276184.48 

35^649 

208,527,857 

24-351 

8.401 

594 

l866.  1  1 

277116.75 

352,836 

209,584,584 

24.372 

8.406 

595 

1869.25 

278050.59 

354,025 

210,644,875 

24-393 

8.411 

596 

l872.39 

278985.99 

355,216 

211,708,736 

24-413 

8.415 

597 

I875-53 

279922.97 

356,409 

212,776,173 

24-433 

8.420 

598 

1878.67 

280861.53 

357,604 

213,847,192 

24.454 

8.425 

599 

1881.81 

281801.65 

358,8oi 

214,921,799 

24.474 

8.429 

600 

1884.96 

282743.34 

360,000 

216,000,000 

24-495 

8-434 

601 

1888.10 

283686.60 

361,201 

217,081,801 

24-5*5 

8-439 

602 

1891.24 

284631.44 

362,404 

218,167,208 

24-536 

8.444 

603 

1894.38 

285577.84 

363,609 

219,256,227 

24-556 

8.448 

604 

-1897.52 

286525.82 

364,816 

220,348,864 

24.576 

8.453 

605 

1900.66 

287475.36 

366,025 

221,445,125 

24-597 

8.458 

606 

1903.80 

288426.48 

367,236 

222,545,016 

24.617 

8.462 

607 

1906.95 

289379.17 

368,449 

223,648,543 

24.637 

8.467 

608 

1910.09 

290333.43 

369,664 

224,755,712 

24.658 

8.472 

609 

1913.23 

291289.26 

370,881 

225,866,529 

24.678 

8.476 

610 

1916.37 

292246.66 

3.72,100 

226,981,000 

24.698 

8.481 

6n 

1919.51 

293205.63 

373,321 

228,099,131 

24.718 

8-485 

612 

1922.65 

294166.17 

374,544 

229,220,928 

24-739 

8.490 

6i3 

1925.80 

295128.28 

375.769 

230,346,397 

24.758 

8.495 

614 

1928.94 

296091.97 

376,996 

23^475,544 

24.779 

8.499 

615 

1932.08 

297057.22 

378,225 

232,608,375 

24.799 

8.504 

616 

1935.22 

298024.05 

379,456 

233,744,896 

24.819 

8.509 

617 

1938.36 

298992.44 

380,689 

234,885,113 

24.839 

8-513 

618 

1941.50 

299962.41 

381,924 

236,029,032 

24.859 

8.518 

NUMBERS,    OR  DIAMETERS   OF  CIRCLES,   &c. 


79 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

619 

1944.65 

300933.95 

383^61 

237,1^6,659 

24.879 

8.522 

620 

1947.79 

301907.05 

384,400 

238,628,000 

24.899 

8.527 

621 

19S°-93 

302881.73 

385,641 

239,483,061 

24.919 

8-532 

622 

1954.07 

303857.98 

386,884 

240,641,848 

24-939 

8-536 

623 

1957.21 

304835.80 

388,I29 

241,804,367 

24-959 

8.541 

624 

1960.35 

305815.20 

389?376 

242,970,624 

24.980 

8-545 

625 

1963.50 

306796.16 

390,625 

244,140,625 

25.000 

8-549 

626 

1966.64 

307778.69 

391,876 

245,3J4>376 

25.019 

8-554 

627 

1969.78 

308762.79 

393,129 

246,491,883 

25.040 

8.559 

628 

1972.92 

309748.47 

394,384 

247,673,152 

25-059 

8.563 

629 

1976.06 

3I0735-7I 

395,641 

248,858,189 

25.079 

8.568 

630 

1979.20 

3IJ724-53 

396,900 

250,047,000 

25.099 

8-573 

63I 

1982.34 

312714.92 

398,l6l 

25^239,591 

25.119 

8-577 

632 

1985.49 

313706.88 

399.424 

252,435,968 

25-I39 

8.582 

633 

1988.63 

314700.40 

400,689 

253,636,137 

25-!59 

8.586 

634 

1991.77 

315695-5° 

401,956 

254,840,104 

25-!79 

8.591 

635 

1994.91 

316692.17 

403,225 

256,047,875 

25-i99 

8-595 

636 

1998.05 

317690.42 

404,496 

257,259,456 

25-219 

8-599 

637 

2001.19 

318690.23 

405,769 

258,474,853 

25-239 

8.604 

638 

2004.34 

319691.61 

407,044 

259,694,072 

25-259 

8.609 

639 

2007.48 

320694.56 

408,321 

260,917,119 

25-278 

8.613 

640 

2010.62 

321699.09 

409,600 

262,144,000 

25-298 

8.618 

641 

2013.76 

322705.18 

4IO,88l 

263,374,721 

25-318 

8.622 

642 

2016.90 

323712.85 

412,164 

264,609,288 

25-338 

8.627 

643 

2020.04 

324722.09 

413,449 

265,847,707 

25-357 

8.631 

644 

2023.19 

325732.89 

414,736 

267,089,984 

25-377 

8.636 

645 

2026.33 

326745.27 

416,025 

268,836,125 

25-397 

8.640 

646 

2029.47 

327759.22 

417,316 

269,586,136 

25.416 

8.644 

647 

2032.61 

328774.74 

418,609 

270,840,023 

25-436 

8.649 

648 

2035-75 

329791.83 

419,904 

272,097,792 

25-456 

8.653 

649 

2038.89 

330810.49 

421,201 

273,359,449 

25-475 

8.658 

650 

2042.04 

331830.72 

422,500 

274,625,000 

25-495 

8.662 

651 

2045.18 

332852-53 

423,801 

275,894,451 

25-5!5 

8.667 

652 

2048.32 

333875-90 

425,104 

277,167,808 

25-534 

8.671 

653 

2051.46 

334900.85 

426,409 

278,445,077 

25-554 

8.676 

654 

2054.60 

335927-36 

427,716 

279,726,264 

25-573 

8.680 

655 

2057.74 

336955-45 

429,025 

281,011,375 

25-593 

8.684 

656 

2060.88 

337985-!o 

430,336 

282,800,416 

25.612 

8.689 

657 

2064.03 

339016.33 

431,649 

283,593,393 

25-632 

8.693 

658 

2067.17 

340049.13 

432,964 

284,890,312 

25-651 

8.698 

659 

2070.31 

341083.50 

434,28l 

286,191,179 

25.671 

8.702 

660 

2073-45 

342119.44 

435,600 

287,496,000 

25.690 

8.706 

661 

2076.59 

343156.95 

436,921 

288,804,781 

25.710 

8.711 

662 

2079.73 

344196.03 

438,244 

290,117,528 

25.720 

8.715 

663 

2082.88 

345236-69 

439,569 

291,434,247 

25-749 

8.719 

664 

2086.02 

346278.91 

440,896 

292,754,944 

25-768 

8.724 

665 

2089.16 

347322.70 

442,225 

294,079,625 

25-787 

8.728 

666 

2092.30 

348368.07 

443,556 

295,408,296 

25-807 

8-733 

8o 


MATHEMATICAL  TABLES. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

667 

2095.44 

349415.00 

444,889 

296,740,963 

25.826 

8-737 

668 

2098.58 

350463.5I 

446,224 

298,077,632 

25.846 

8.742 

669 

2101-73   35I5I3-59 

447,561 

299,418,309 

25.865 

8.746 

670 

2104.87 

352565-24 

448,900 

300,763,000 

25.884 

8.750 

671 

2108.  oi 

3536l8-43 

450,241 

302,111,711 

25.904 

8.753 

672 

2III.I5 

354673-24 

451,584 

303,464,448 

25-923 

8.759 

673 

2114.29 

355729.60 

452,929 

304,821,217 

25.942 

8.763 

674 

2117.43 

356787.54 

454,276 

306,182,024 

25.961 

8.768 

675 

2120.58 

357847.04 

455,625 

307,546,875 

25.981 

8.772 

676 

2123.72 

358908.11 

456,976 

308,915,776 

26.OOO 

8.776 

677 

2126.86 

359970.75 

458,329 

310,288,733 

26.019 

8.781 

678 

2130.00 

361034.97 

459,684 

311,665,752 

26.038 

8.785 

679 

2I33-I4 

362100.75 

461,041 

313,046,839 

26.058 

8.789 

680 

2136.28 

363168.11 

462,400 

314,432,000 

26.077 

8.794 

681 

2139.42 

364237.04 

463,761 

315,821,241 

26.096 

8.798 

682 

2142.57 

365307.54 

465,124 

317,214,568 

26.115 

8.802 

683 

2145.71 

366379.60 

466,489 

318,611,987 

26.134 

8.807 

684 

2148.85 

367453-24 

467,856 

320,013,504 

26.153 

8.811 

685 

2151.99 

368528.45 

469,225 

321,419,125 

26.172 

8.815 

686 

2I55-I3 

369605.23 

470,596 

322,828,856 

26.192 

8.819 

687 

2158.27 

370683.59 

471,969 

324,242,703 

26.211 

8.824 

688 

2l6l.42 

37I763-51 

473,344   325,660,672 

26.229 

8.828 

689 

2164.56 

372845.00 

474,721 

327,082,769 

26.249 

8.832 

690 

2167.70 

373928.07 

476,100 

328,509,000 

26.268 

8.836 

691 

2170.84 

375012.70 

477,481 

329,939,371 

26.287 

8.841 

692 

2173.98 

376098.91 

478,864 

331,373,888 

26.306 

8.845 

693 

2177.12 

377186.68 

480,249 

332,812,557 

26.325 

8.849 

694 

2l8o.27 

378276.03 

481,636 

334,255,384 

26.344 

8-853 

695 

2183.41 

379366.95 

483,025 

335,702,375 

26.363 

8.858 

696 

2186.55 

380459.44 

484,416 

337,153,536 

26.382 

8.862 

697 

2189.69 

38i553.50 

485,809 

338,608,873 

26.401 

8.866 

698 

2192.83 

382649.43 

487,204 

340,068,392 

26.419 

8.870 

699 

2195.97 

383746.33 

488,601 

341,532,099 

26.439 

8-875 

700 

2199.12 

384845.10 

490,000 

343,000,000 

26.457 

8.879 

701 

2202.26 

385945-44 

491,401 

344,472,101 

26.476 

8.883 

702 

2205.40 

387047-36 

492,804 

345,948,088 

26.495 

8.887 

703 

2208.54 

388150.84 

494,209 

347,428,927 

26.514 

8.892 

704 

2211.68 

389255-90 

495,616 

348,913,664 

26.533 

8.896 

705 

2214.82 

390362.52 

497,025 

350,402,625 

26.552 

8.900 

706 

2217.96 

391470.32 

498,436 

351,895,816 

26.571 

8.904 

707 

2221.  II 

392580.49 

499,849 

353,393,243 

26.589 

8.908 

708 

2224.25 

393691.83 

501,264 

354,894,912 

26.608 

8.913 

709 

2227.39 

394804.74 

502,681 

356,400,829 

26.627 

8.917 

710 

2230.53 

3959I9-21 

504,100 

357,911,000 

26.644 

8.921 

711 

2233.67 

397035-27 

505,521 

359,425,431 

26.664 

8.925 

712 

2236.81 

398152.89 

5o6,944 

360,944,128 

26.683 

8.929 

7i3 

2239.96 

399272.08 

508,369 

362,467,097 

26.702 

8-934 

7i4 

2243.10   400392.84 

509,796 

363,994,344 

26.721 

8.938 

NUMBERS,    OR  DIAMETERS   OF  CIRCLES,   &c. 


8l 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

715 

^246.24 

401515.18 

511,225 

365.525.875 

26.739 

8.942 

7l6 

2249.38 

402639.08 

512,656 

367,061,696 

26.758 

8.946 

717 

2252.52 

403764.56 

514,089 

368,601,813 

26.777 

8.950 

7l8 

2255.66 

404891.60 

515.524 

370,146,232 

26.795 

8-954 

719 

2258.81 

4O6O2O.22 

516,961 

371,694,959 

26.814 

8-959 

720 

2261.95 

407150.41 

5l8,400 

373,248,000 

26.833 

8.963 

721 

2265.09 

408282.17 

519,841 

374,805,361 

26.851 

8.967 

722 

2268.23 

409415.50 

521,284 

376,367,048 

26.870 

8.971 

723 

2271.37 

410550.40 

522,729 

377,933,067 

26.889 

8-975 

724 

2274.51 

411686.87 

524,176 

379.503,424 

26.907 

8.979 

725 

2277.66 

412824.91 

525.625 

381,078,125 

26.926 

8.983 

726 

2280.80 

413964.52 

527,076 

382,657,176 

26.944 

8.988 

727 

2283.94 

4i5I05-7* 

528,529 

384,240,583 

26.963 

8.992 

728 

2287.08 

416248.46 

529,984 

385,828,352 

26.991 

8.996 

729 

2290.22 

417392.79 

531.441 

387,420,489 

27.000 

9.000 

730 

2293.36 

418538.68 

532,900 

389,017,000 

27.018 

9.004 

731 

2296.50 

419686.15 

534,361 

390,617,891 

27.037 

9.008 

732 

2299.65 

420835.19 

535.824 

392,223,168 

27.055 

9.012 

733 

2302.79 

421985.79 

537.289 

393.832,837 

27.074 

9.016 

734 

2305-93 

423137-97 

538,756 

395,446,904 

27.092 

9.020 

735 

2309.07 

424291.72 

540,225 

397,065,375 

27.111 

9.023 

736 

2312.21 

425447.04 

541,696 

398,688,256 

27.129 

9.029 

737 

2315-35 

426603.93 

543.169 

400,315,553 

27.148 

9-033 

738 

2318.50 

427762.40 

544,644 

401,947,272 

27.166 

9-037 

739 

2321.64 

428922.43 

546,121 

403,583.4i9 

27.184 

9.041 

740 

2324.78 

430084.03 

547.600 

405,224,000 

27.203 

9-045 

741 

2327.92 

431247.21 

549,081 

406,869,021 

27.221 

9.049 

742 

2331.06 

432411.95 

550.564 

408,518,488 

27.239 

9-053 

743 

2334.20 

433578.27 

552,049 

410,172,407 

27.258 

9-057 

744 

2337.35 

434746.16 

553.536 

411,830,784 

27.276 

9.061 

745 

2340.49 

4359I5-62 

555.025 

4i3.493.625 

27.295 

9.065 

746 

2343-63 

437086.64 

556,5  16 

415,160,936 

27-3I3 

9.069 

747 

2346.77 

438259.24 

558,009 

416,832,723 

27-331 

9.073 

748 

2349.91 

439433-41 

559.504 

418,508,992 

27-349 

9.077 

749 

2353.05 

440609.16 

561,001 

420,189,749 

27.368 

9.081 

750 

2356.20 

441786.47 

562,500 

421,875,000 

27.386 

9.086 

75i 

2359-34 

442965-35 

564,001 

423,564,75! 

27.404 

9.089 

752 

2362.48 

444145.80 

565.504 

424,525,900 

27.423 

9.094 

753 

2365.62 

445327-83 

567,009 

426,957,777 

27.441 

9.098 

754 

2368.76 

446511.42 

568,516 

428,661,064 

27.459 

9.102 

755 

2371.90 

447696.59 

570,025 

430,368,875 

27.477 

9.106 

756 

2375.04 

448883.32 

571,536 

432,081,216 

27-495 

9.109 

757 

2378.19 

450071.63 

573.049 

433»798,o93 

27-5I4 

9.114 

758 

238i-33 

451261.51 

574,564 

435.5I9>512 

27-532 

9.118 

759 

2384.47 

452452.96 

576,o8i 

437,245,479 

27.549 

9.122 

760 

2387.61 

453645.98 

577,600 

438,976,000 

27.568 

9.126 

761 

239o.75 

454840.57 

579.T2i 

440,711,081 

27.586 

9.129 

762 

2393-89 

456036.73 

580,644 

442,450,728 

27.604 

9.134 

82 


MATHEMATICAL  TABLES. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

763 

2397.04 

457234.46 

582,169 

444,  1  94,947 

27.622 

9.138 

764 

240O.I8 

458433.77 

583,696 

445,943,744 

27.640 

9.142 

765 

2403.32 

459634-64 

585,225 

447,697,125 

27.659 

9.146 

766 

2406.46 

460837.08 

586,756 

449,455,096 

27.677 

9.149 

767 

2409.60 

462041.10 

588,289 

.451,217,663 

27.695 

9.154 

768 

2412.74 

463246.69 

589,824 

452,984,832 

27.713 

9.158 

769 

2415.89 

464453.84 

59^361 

454,756,609 

27.731 

9.162 

770 

2419.03 

465662.57 

592,900 

456,533,ooo 

27.749 

9.166 

771 

2422.17 

466872.87 

594,441 

458,3I4,on 

27.767 

9.169 

772 

2425.31 

468084.74 

595,984 

460,099,648 

27.785 

9-173 

773 

2428.45 

469298.18 

597,529 

461,889,917 

27.803 

9.177 

774 

243L59 

47°5I3-I9 

599,076 

463,684,824 

27.821 

9.l8l 

775 

2434-73 

471729.77 

600,625 

465,484,375 

27.839 

9.185 

776 

2437.88 

472947.92 

602,176 

467,288,576 

27-857 

9.189 

777 

2441.02 

474167.65 

603,729 

469,097,433 

27.875 

9.193 

778 

2444.16 

475388.94 

605,284 

470,910,952 

27.893 

9.197 

779 

2447.30 

476611.81 

606,841 

472,729,139 

27.910 

9.2OI 

780 

2450.44 

477836.24 

608,400 

474,552,ooo 

27.928 

9.205 

78i 

2453-58 

479062.25 

609,961 

476,379,541 

27.946 

9.209 

782 

2456-73 

480289.83 

611,524 

478,211,768 

27.964 

9.213 

783 

2459.87 

481518.97 

613,089 

480,048,687 

27.982 

9.217 

784 

2463.01 

482749.69 

614,656 

481,890,304 

28.000 

9.221 

785 

2466.15 

483981.98 

616,225 

483,736,025 

28.017 

9.225 

786 

2469.29 

485215.84 

617,796 

485,587,656 

28.036 

9.229 

787 

2472.43 

486451.28 

619,369 

487,443,403 

28.053 

9-233 

788 

2475-58 

487688.28 

620,944 

489,303,872 

28.071 

9-237 

789 

2478.72 

488926.85 

622,521 

491,169,069 

28.089 

9.240 

790 

2481.86 

490166.99 

624,100 

493,039,ooo 

28.107 

9-244 

791 

2485.00 

491408.71 

625,681 

494,913,671 

28.125 

9.248 

792 

2488.14 

492651.99 

627,264 

496,793,o88 

28.142 

9.252 

793 

2491.28 

493896.85 

628,849 

498,677,257 

28.160 

9.256 

794 

2494.43 

495143.28 

630,436 

500,566,184 

28.178 

9.260 

795 

2497-57 

496391.27 

632,025 

502,459,875 

28.196 

9.264 

796 

2500.71 

497640.84 

633,6l6 

504,358,336 

28.213 

9.268 

797 

2503-85 

498891.98 

635,209 

506,261,573 

28.231 

9.271 

798 

2506.99 

500144.69 

636,804 

508,169,592 

28.249 

9-275 

799 

2510.13 

501398.97 

638,401 

510,082,399 

28.266 

9.279 

800 

2513.27 

502654.82 

640,000 

512,000,000 

28.284 

9.283 

801 

2516.42 

5Q39I2-25 

641,601 

513,922,401 

28.302 

9.287 

802 

2519.56 

505171.24 

643,204 

515,849,608 

28.319 

9.291 

803 

2522.70 

506431.80 

644,809 

517,781,627 

28.337 

9.295 

804 

2525-84 

507693.94 

646,416 

519,718,464 

28.355 

9.299 

805 

2528.98 

508957.65 

648,025 

521,660,125 

28.372 

9.302 

806 

2532.12 

510222.92 

649,636 

523,606,616 

28.390 

9.306 

807 

2535-27 

511489.77 

651,249 

525,557,943 

28.408 

9.310 

808 

2538.41 

512758.19 

652,864 

527,514,112 

28.425 

9-3J4 

809 

254L55 

514028.19 

654,481 

529,474,129 

28.443 

9.318 

810 

2544.09 

515299.74 

656,100 

531,441,000 

28.460 

9.321 

NUMBERS,    OR  DIAMETERS   OF   CIRCLES,   &c. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

811 

2547.83 

516572.86 

657,721 

533,4H,73I 

28.478 

9.325 

812 

2550.97 

5I7847-57 

659,344 

535,387,328 

28.496 

813 

2554.12 

519123.84 

660,969 

537,366,797 

28.513 

9-333 

814 

2557.26 

520401.68 

662,596 

539,353,144 

28.531 

9-337 

815 

2560.40 

52l68l.IO 

664,225 

541,343,375 

28.548 

9.341 

816 

2563.54 

522962.08 

665,856 

543,338,496 

28.566 

9-345 

817 

2566.68 

524244.63 

667,489 

545,338,513 

28.583 

9.348 

818 

2569.82 

525528.76 

669,124 

547,343,432 

28.601 

9.352 

819 

2572.96 

526814.46 

670,761 

549,353,259 

28.6l8 

9-356 

820 

2576.11 

528101.73 

672,400 

551,368,000 

28.636 

9.360 

821 

2579.25 

529390.56 

674,041 

553,387,661 

28.653 

9-364 

822 

2582.39 

530680.97 

675,684 

555,412,248 

28.670 

9.367 

823 

2585.53 

531972.95 

677,329 

557,441,767 

28.688 

9.371 

824 

2588.67 

533266.50 

678,976 

559,476,224 

28.705 

9-375 

825 

2591.81 

534561.63 

680,625 

561,515,625 

28.723 

9-379 

826 

2594.96 

535858.32 

682,276 

563,559,976 

28.740 

9-383 

827 

2598.10 

537156.58 

683,929 

565,609,283 

28.758 

9.386 

828 

26OI.24 

538456.41 

685,584 

567,663,552 

28.775 

9-390 

829 

2604.38 

539757-82 

687,241 

569,722,789 

28.792 

9-394 

830 

2607.52 

541060.79 

688,900 

571,787,000 

28.810 

831 

2610.66 

542365.34 

690,561 

573,856,191 

28.827 

9.401 

832 

2613.81 

543671.46 

692,224 

575,930,368 

28.844 

9-405 

833 

2616.95 

544979.15 

693,889 

578,009,537 

28.862 

9.409 

834 

2620.O9 

546288.40 

695,556 

580,093,704 

28.879 

9-4I3 

835 

2623.23 

547599-23 

697,225 

582,182,875 

28.896 

9.417 

836 

2626.37 

548911.63 

698,896 

584,277,056 

28.914 

9.420 

837 

2629.51 

550225.61 

700,569 

586,376,253 

28.931 

9.424 

838 

2632.64 

55I54LI5 

702,244 

588,480,472 

28.948 

9.428 

839 

2635.80 

552858.26 

703,921 

590,589,719 

28.965 

9-432 

840 

2638.94 

554176.94 

705,600 

592,704,000 

28.983 

9-435 

841 

2642.08 

555497-20 

707,281 

594,823,321 

29.000 

9-439 

842 

2645.22 

556819.02 

708,964 

596,947,688 

29.017 

9-443 

843 

2648.36 

558142.42 

710,649 

599,077,107 

29.034 

9-447 

844 

2651.50 

559467.39 

712,336 

601,211,584 

29.052 

9-450 

845 

2654.65 

560793.92 

714,025 

603,351,125 

29.069 

9-454 

846 

2657.79 

562122.03 

605,495,736 

29.086 

9.458 

847 

2660.93 

563451.71 

717,409 

607,645,423 

29.103 

9.461 

848 

2664.07 

564782.96 

719,104 

609,800,192 

29.120 

9-465 

849 

2667.21 

566115.78 

720,801 

611,960,049 

29.I38 

850 

2670.35 

56745°.][7 

722,500 

6l4,I25,OOO 

9-473 

851 

2673.50 

568786.14 

724,201 

616,295,051 

29.172 

9-476 

852 

2676.64 

570123.67 

725,904 

618,470,208 

29.189 

9.480 

853 

2679.78 

571462.77 

727,609 

620,650,477 

29.206 

9.483 

854 

2682.92 

572803.45 

729,316 

622,835,864 

29.223 

9.487 

855 

2686.06 

574145.69 

73I>025 

625,026,375 

29.240 

9.491 

856 

2689.20 

575489.51 

732,736 

627,222,Ol6 

29.257 

9-495 

857 

2692.35 

576834.90 

734,449 

629,422,793 

29.274 

9-499 

858 

2695.49 

578181.85 

736,164 

631,628,712 

29.292 

9.502 

84 


MATHEMATICAL  TABLES. 


Number, 
or 
Diameter. 

Circum-  ' 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

859 

2698.63 

579530-38 

737,881 

633,839,779 

29.309 

9.506 

860 

2701.77 

580880.48 

739,600 

636,056,000 

29.326 

9-509 

861 

2704.91 

582232.15 

74I532I 

638,277,381 

29-343 

9.513 

862 

2708.05 

583585-39 

743,044 

640,503,928 

29.360 

9.517 

863 

2711.19 

584940.21 

7445769 

642,735,647 

29-377 

9.520 

864 

2714.34 

586296.59 

746,496 

644,972,544 

29.394 

9.524 

865 

2717.45 

587654.54 

748,225 

647,214,625 

29.411 

9.528 

866 

2720.62 

589014.07 

749,956 

649,461,896 

29.428 

9.532 

867 

2723.76 

59°375-16 

751,689 

651,714,363 

29-445 

9-535 

868 

2726.90 

59I737-83 

753,424 

653,972,032 

29.462 

9-539 

869 

2730.04 

593102.06 

755,!6i 

656,234,909 

29.479 

9-543 

870 

2733-I9 

594467.87 

756,900 

658,503,000 

29.496 

9-546 

871 

2/36.33 

595835-25 

758,641 

660,776,311 

29-5I3 

9-550 

872 

2739-47 

597204.20 

760,384 

663,054,848 

29.529 

9-554 

873 

2742.61 

598574.72 

762,129 

665,338,617 

29.546 

9-557 

874 

2745-75 

599946.81 

763,876 

667,627,624 

29-563 

9.561 

875 

2748.89 

601320.47 

765,625 

669,921,875 

29.580 

9-565 

876 

2752.04 

602695.70 

767,376 

672,221,376 

29-597 

9.568 

877 

2755-18 

604072.50 

769,129 

674,526,133 

29.614 

9-572 

878 

2758.32 

605450.88 

770,884 

676,836,152 

29.631 

9-575 

879 

2761.46 

606830.82 

772,641 

679,I5I,439 

29.648 

9-579 

880 

2764.60 

608212.34 

774,400 

681,472,000 

29.665 

9-583 

881 

2767.74 

609595-42 

776,161 

683,797,841 

29.682 

9-586 

882 

2770.89 

610980.08 

777,924 

686,128,968 

29.698 

9-590 

883 

2774.03 

612366.31 

779,689 

688,465,387 

29-715 

9-594 

884 

2777.17 

613754.11 

781,456 

690,807,104 

29.732 

9-597 

885 

2780.31 

615143.48 

783,225 

693,154,125 

29-749 

9.601 

886 

2783-45 

616534.42 

784,996 

695,506,456 

29.766 

9.604 

887 

2786.59 

617926.93 

786,769 

697,864,103 

29.782 

9.608 

888 

2789.73 

619321.01 

788,544 

700,227,072 

29.799 

9.612 

889 

2792.88 

620716.66 

790,321 

702,595,369 

29.816 

9.615 

890 

2796.02 

622113.89 

792,100 

704,969,000 

29-833 

9.619 

891 

2799.16 

623512.68 

793,881 

707,347,971 

29.850 

9.623 

892 

2802.30 

624913.04 

795,664 

709,732,288 

29.866 

9.626 

893 

2805.44 

626314.98 

797,449 

712,121,957 

29-883 

9-630 

894 

2808.58 

627718.49 

799,236 

714,516,984 

29.900 

9-633 

895 

2811.73 

629123.56 

801,025 

716,917,375 

29.916 

9-637 

896 

2814.87 

630530.21 

802,816 

7i9>323,136 

29-933 

9.640 

897 

2818.01 

631938.43 

804,609 

721,734,273 

29.950 

9-644 

898 

2821.15 

633348.22 

806,404 

724,150,792 

29.967 

9.648 

899 

2824.29 

634759-58 

808,201 

726,572,699 

29.983 

9.651 

goo 

2827.43 

636172.51 

810,000 

729,000,000 

30.000 

9.655 

901 

2830.58 

637587-oi 

811,804 

731>432,7oi 

30.017 

9.658 

902 

2833.72 

639003.09 

813,604 

733,870,808 

30.033 

9.662 

903 

2836.86 

640420.73 

815,409 

736,3*4,327 

30.050 

9.666 

904 

2840.00 

641839.95 

817,216 

738,763,264 

30.066 

9.669 

905 

2843.14 

643260.73 

819,025 

741,217,625 

30.083 

9-673 

906 

2846.28 

644683.09 

820,836 

743,677,4i6 

30.100 

9.676 

NUMBERS,   OR  DIAMETERS  OF  CIRCLES,  &c. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

907 

2849.43 

646107.01 

822,649 

746,142,643 

30.1l6 

9.680 

908 

2852.57 

647532.51 

824,464 

748,613,312 

30.133 

9.683 

909 

2855-7I 

648959.58 

826,281 

751,089,429 

30.150 

9.687 

gio 

2858.85 

650388.21 

828,100 

753,571,000 

30.163 

9.690 

911 

2861.99 

651818.43 

829,921 

756,058,031 

30.183 

9.694 

912 

2865.13 

653250.21 

831,744 

758,550,528 

30.199 

9.698 

9i3 

2868.27 

654683.56 

833,569 

761,048,497 

30.216 

9.701 

914 

2871.42 

656118.48 

835,396 

763,551,944 

30.232 

•9-705 

9i5 

2874.56 

657554.98 

837,225 

766,060,875 

30.249 

9.708 

916 

2877.70 

658993.04 

839,056 

768,575,296 

30.265 

9.712 

917 

2880.84 

660432.68 

840,889 

771,095,213 

30.282 

9-7I5 

918 

2883.98 

661873.88 

842,724 

773,620,632 

30.298 

9.7l8 

919 

2887.12 

663316.66 

844,561 

776,151,559 

30-3I5 

9.722 

920 

2890.27 

664761.01 

846,400 

778,688,000 

30-331 

9.726 

921 

2893.41 

666206.92 

848,241 

781,229,961 

30-348 

9.729 

922 

2896.55 

667654.41 

850,084 

783,777,448 

30-364 

9-733 

923 

2899.69 

669103.47 

851,929 

786,330,467 

30-381 

9-736 

924 

2902.83 

670554.10 

853,776 

788,889,024 

30.397 

9.740 

925 

2905.97 

672006.30 

855,625 

791,453,125 

30.414 

9-743 

926 

2909.12 

673460.08 

857,476 

794,022,776 

30-430 

9-747 

927 

2912.26   674915.42 

859,329 

796,597,983 

30-447 

9-75° 

928 

2915.40 

6763/2.33 

861,184 

799,^8,752 

30-463 

9-754 

929 

2918.54 

677830.82 

863,041 

801,765,089 

30-479 

9.757 

930 

2921.68 

679290.87 

864,900 

804,357,000 

30.496 

9.761 

93i 

2924.82 

680752.50 

866,761 

806,954,491 

30.512 

9.764 

932 

2927.96 

682215.69 

868,624 

809,557,568 

30-529 

9.768 

933 

2931.11 

683680.46 

870,489 

812,166,237 

30-545 

9.771 

934 

2934-25 

685146.80 

872,356 

814,780,504 

30.561 

9-775 

935 

2937-39 

686614.71 

874,225 

817,400,375 

30-578 

9.778 

936 

2940-53 

688084.19 

876,096 

820,025,856 

30-594 

9-783 

937 

2943.67 

689555.24 

877,969 

822,656,953 

30.610 

9-785 

938 

2946.81 

691027.86 

879,844 

825,293,672 

30.627 

9.789 

939 

2949.96 

692502.05 

88l,72I 

827,936,019 

30-643 

9.792 

940 

2953.10 

693977.82 

883,600 

830,584,000 

30-659 

9.796 

941 

2956.24 

695455^5 

885,481 

833'237,62I 

30.676 

9-799 

942 

2959-38 

696934.06 

887,364 

835,896,888 

30.692 

9.803 

943 

2962.52 

698414.53 

889,249 

838,561,807 

30.708 

9.806 

944 

2965.66 

699896.58 

891,136 

841,232,384 

30.724 

9.810 

945 

2968.81 

701380.28 

893,025 

843,908,625 

30-74I 

9-813 

946 

2971.95 

702865.38 

894,916 

846,590,536 

30-757 

9.817 

947 

2975.09 

704352.14 

896,809 

849,278,123 

30.773 

9.820 

948 

2978.23 

705840.47 

898,704 

851,971,392 

30.790 

9.823 

949 

2981.37 

707330.37 

900,601 

854,670,349 

30.806 

9.827 

950 

2984.51 

708821.84 

902,500 

857,375,000 

30.822 

9.830 

951 

2987.66 

710314.88 

904,401 

860,085,351 

30-838 

9.834 

952 

2990.80!  711809.58 

906,304 

862,801,408 

30.854 

9.837 

953 

2993-94   7I3305-68 

908,209 

865,523,177 

30.871 

9.841 

954 

2997.08   714803.48 

9IO,Il6    868,250,664 

30.887 

9.844 

86 


MATHEMATICAL   TABLES. 


Number, 
or 
Diameter. 

Circum- 
ference. 

Circular 
Area. 

Square. 

Cube. 

Square 
Root. 

Cube 
Root. 

955 

3000.22 

716302.76 

912,025 

870,983,875 

30.903 

9.848 

956 

3003.36 

717803.66 

9J3,936 

873,722,8l6 

30.919 

9.851 

957 

3006.50 

719306.12 

9J5,849 

876,467,493 

30.935 

9.854 

958 

3009.65 

72o8l0.l6 

917,764 

879,217,912 

30-951 

9.858 

959 

3012.79 

722315.77 

919,681 

881,974,079 

30.968 

9.861 

960 

30l5-93 

723822.95 

921,600 

884,736,000 

30.984 

9.865 

961 

3019.07 

72533L70 

923,521 

887,503,681 

31.000 

9.868 

962 

3022.21 

726842.02 

9255444 

890,277,128 

3I.OI6 

9.872 

963 

3025.35 

728353.91 

927,369 

893,056,347 

31.032 

9-875 

964 

3028.50 

729867.37 

929,296 

895,841,344 

31.048 

9.878 

965 

3031.64 

731382.40 

931,225 

898,632,125 

31.064 

9.881 

966 

3034.78 

732899.01 

933^56 

901,428,696 

3I.O80 

9.885 

967 

3037.92 

734417.18 

935>o89 

904,231,063 

31.097 

9.889 

968 

3041.06 

735936.93 

937,024 

907,039,232 

SJ-iiS 

9.892 

969 

3044.20 

737458.25 

938,961 

909,853,209 

31.129 

9-895 

970 

3047-35 

738981.13 

940,900 

912,673,000 

31.145 

9.899 

971 

3050.49 

740505.59 

942,841 

915,498,611 

31.161 

9.902 

972 

3053.63 

742031.62 

944,784 

918,330,048 

31.177 

9.906 

973 

3056.77 

743559-22 

946,729 

921,167,317 

3LI93 

9.909 

974 

3059-91 

745088.39 

948,676 

924,010,424 

31.209 

9.912 

975 

3063.05 

746619.13 

950,625 

926,859,375 

3L225 

9.916 

976 

3066.19 

748151.44 

952,576 

929,714,176 

31.241 

9.919 

977 

3069.34 

749685.32 

954,529 

932,574,833 

3I-257 

9-923 

978 

3072.48 

751220.78 

956,484 

935,441,352 

3I-273 

9.926 

979 

3075.62 

752757.8o 

958,441 

938,313,739 

31.289 

9.929 

980 

3078.76 

754296.40 

960,400 

941,192,000 

31-305 

9-933 

981 

3081.90 

755836.56 

962,361 

944,076,141 

3l.32i 

9-936 

982 

3085.04 

757378.30 

964,324 

946,966,168 

3*-337 

9.940 

983 

3088.19 

758921.61 

966,289 

949,862,087 

3T-353 

9-943 

984 

309L33 

760466.48 

968,256 

952,763,904 

3I-369 

9.946 

•985 

3094.47 

762012.93 

970,225 

955,671,625 

3L385 

9-950 

986 

3097.61 

763560.95 

972,196 

958,585,256 

31.401 

9-953 

987 

3100.75 

765110.54 

974,169 

961,504,803 

31.416 

9-956 

988 

3103.89 

766661.71 

976,144 

964,430,272 

3M32 

9.960 

989 

3107.04 

768214.44 

978,121 

967,361,669 

31.448 

9-963 

990 

3IIO.I8 

769768.74 

980,100 

970,299,000 

31.464 

9.966 

991 

3II3.32 

771324.61 

982,081 

973,242,271 

31.480 

9-970 

992 

3116.46 

772882.06 

984,064 

976,191,488 

31.496 

9-973 

993 

3II9.60 

774441.07 

986,049 

979,146,657 

3i-512 

9-977 

994 

3122.74 

776001.66 

988,036 

982,107,784 

3L528 

9.980 

995 

3I25.89 

777563.82 

990,025 

985,074,875 

31-544 

9-983 

996 

3129.03 

779127.54 

992,016 

988,047,936 

31-559 

9.987 

997 

3132.17 

780692.84 

994,009 

991,026,973 

31-575 

9-990 

998 

3135.31 

782259.71 

996,004 

994,011,992 

3I-591 

9-993 

999 

3138.45 

783828.15 

998,001 

997,002,999 

31.607 

9-997 

1000 

3141.60 

785398.16 

1,000,000 

1,000,000,000 

31.623 

IO.OOO 

CIRCLES: — DIAMETER,   CIRCUMFERENCE,  &C. 


TABLE   No.   IV.     CIRCLES:— DIAMETER,    CIRCUMFERENCE, 
AREA,  AND   SIDE   OF  EQUAL   SQUARE. 


Side  of 

Side  of 

Diameter. 

Circum- 
ference. 

Area. 

Equal  Square 
(Square  Root 

Diameter. 

Circum- 
ference. 

Area. 

Equal  Square 
(Square  Root 

of  Area). 

of  Area). 

3 

9.4248 

7.0686 

2.6586 

'/i6 

.1963 

.00307 

•0553 

3  x/i6 

9.62II 

7.3662 

2.7140 

% 

.3927 

.01227 

.1107 

31A 

9.8175 

7.6699 

2.7694 

3/i6 

.5890 

.02761 

.l66l 

33/i6 

10.014 

7.9798 

2.8248 

X 

.7854 

.04909 

.2215 

3X 

IO.2IO 

8.2957 

2.8801 

5/i6 

.9817 

.07670 

.2770 

35/i6 

10.406 

8.6  1  80 

2-9355 

% 

.1781 

.1104 

•3323 

33/s 

10.602 

8.9462 

2.9909 

7/i6 

•3744 

.1503 

.3877 

37/i6 

10.799 

9.2807 

3.0463 

% 

.5708 

.1963 

•4431 

3/2 

10.995 

9.6211 

3.IOI7 

9/i6 

.7771 

.2485 

.4984 

39/i6 

II.I9I 

9.9680 

3.!57i 

# 

•9635 

.3068 

•5539 

3% 

11.388 

10.320 

3.2124 

"/i6 

2.1598 

.3712 

.6092 

3IJ/i6 

11.584 

10.679 

3.2678 

* 

2.3562 

.4418 

.6646 

3X 

II.78I 

11.044 

3-3232 

•Sfo 

2.5525 

.5185 

.7200 

3'3/i6 

11.977 

11.416 

3.3786 

# 

2.7489 

.6013 

•7754 

37/s 

12.173 

n.793 

3-4340 

'5/16 

2.9452 

.6903 

.8308 

3I5/i6 

12.369 

12.177 

3.4894 

i 

3.1416 

.7854 

.8862 

4 

12.566 

12.566 

3-5448 

I   z/i6 

3-3379 

.8866 

.9416 

4  J/i6 

12.762 

12.962 

3.6002 

I# 

3-5343 

.9940 

.9969 

4l/s 

12.959 

13-364 

3-6555 

I   3/l6 

3.7306 

1.1075 

1.0524 

43/i6 

I3.I55 

13.772 

3.7109 

iX 

3.9270 

1.2271 

1.1017 

4X 

!3-35i 

14.186 

3.7663 

I   S/i6 

4-1233 

1-3530 

1.1631 

45/i6 

13-547 

14.606 

3.8217 

I# 

4.3197 

1.4848 

1.2185 

43A 

13.744 

15-033 

3.8771 

I   7/i6 

4.5160 

1.6229 

1.2739 

47/i6 

13.940 

15.465 

3-9325 

I# 

4.7124 

1.7671 

1.3293 

4/2 

14.137 

15.904 

3.9880 

I   9/i6 

4.9087 

1.9175 

1-3847 

49/i6 

14-333 

16.349 

4.0434 

I# 

5.1051 

2.0739 

1.4401 

4K 

14.529 

16.800 

4-0987 

I"/i6 

5-3014 

2.2365 

1-4955 

4TI/i6 

14.725 

17.257 

4.1541 

If* 

5.4978 

2.4052 

1.5508 

4X 

14.922 

17.720 

4.2095 

1*3/16 

5.6941 

2.5800 

1.6062 

4x3/i6 

15.119 

18.190 

4.2648 

I# 

5.8905 

2.7611 

1.6616 

47/8 

T5-3i5 

18.665 

4.3202 

1  15/16 

6.0868 

2.9483 

1.7170 

4J5/i6 

15.511 

19.147 

4.3756 

2 

6.2832 

3.1416 

1.7724 

5 

15.708 

19-635 

4.4310 

2  '/i6 

6.4795 

3.338o 

1.8278 

5   Vx6 

15.904 

20.  1  29 

4.4864 

2^ 

6.6759 

3-5465 

1.8831 

S/B 

16.100 

20.629 

4.5417 

2  3/l6 

6.8722 

37584 

1.9385 

5  3/i6 

16.296 

21.135 

4-5971 

2X 

7.0686 

3.9760 

1-9939 

5X 

16.493 

21.647 

4.6525 

2  S/l6 

7.2649 

4.2000 

2.0493 

5  5/i6 

16.689 

22.166 

4.7079 

2^ 

7.4613 

44302 

2.1047 

53/s 

16.886 

22.690 

4.7633 

2  7/l6 

7.6576 

4.7066 

2.1601 

5  7/i6 

17.082 

23.221 

4.8187 

2^ 

7.8540 

4.9087 

2.2155 

S/2 

17.278 

23-758 

4.8741 

2  9/l6 

8.0503 

5-J573 

2.2709 

5  9/i6 

17.474 

24.301 

4.9295 

2^ 

8.2467 

5.4119 

2.3262 

S5/s 

17.671 

24.850 

4.9848 

2"/i6 

8.4430 

5.6723 

2.3816 

5x?/«6 

17.867 

25.406 

5.0402 

2X 

8.6394 

5-9395 

2.4370 

5X 

18.064 

25.967 

5.0956 

2I3/l6 

8.8357 

6.2126 

2.4924 

513/i6 

18.261 

26.535 

5.1510 

2^ 

9.0321 

6.4918 

2.5478 

57/s 

18.457 

27.108 

5.2064 

2*S/,6 

9.2284 

6.7772 

2.6032 

5  l  S/i  6 

18.653 

27.688 

5.2618 

88 


MATHEMATICAL   TABLES. 


Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

6 

6/8 

w 

6/8 

6/2 

6/8 

6% 

6/8 

18.849 
19.242 

I9-635 
20.027 
20.420 
20.813 
21.205 
21.598 

28.274 
29.464 
30.679 

31.919 
33.183 
34-471 
35784 
37.122 

5.3I72 
5.4280 
5.5388 
5.6495 
5.7603 
5.87II 
5.9819 
6.0927 

12 
12/8 
12% 
12/8 

12& 

37.699 
38.091 
38.484 
38.877 
39.270 
39.662 
40.055 
40.448 

113.097 
115.466 
117.859 
120.276 
I22.7l8 
125.184 
127.676 
130.192 

10.634 
10.745 
10.856 
10.966 
11.077 
II.I88 
11.299 
11.409 

NM  NW  No,  N»  NUJ  N-  Nw 
OoN  -P\  0§N  N\  0§N  -PS  OON 

21.991 
22.383 
22.776 
23.169 
23.562 
23-954 
24-347 
24.740 

38.484 
39.871 
41.282 
42.718 
44.178 
45.663 
47.173 
48.707 

6.2034 
6.3142 
6.4350 
6.5358 
6.6465 

6-7573 
6.8681 

6.9789 

13 

li\x 

l3/£ 
i3>6 

40.840 

41.626 
42.018 
42.411 
42.804 
43-197 
43.589 

132.732 
135.297 
137.886 
140.500 

I43.I39 
145.802 
148.489 
I5I.20I 

11.520 
11.631 
11.742 
11.853 
11.963 
12.074 
I2.I85 
12.296 

NOO  \-4-NOONN  NOO  X.4.NOO 

X  -^  X  i-N  X  ^  X 

oooooooooocooooo 

25.132 
25.515 
25.918 
26.310 
26.703 
27.096 
27.489 
27.881 

50.265 
51.848 
53.456 
55.088 

56.745 
58.426 
60.  1  32 
61.862 

7.0897 
7.2005 
7.3II2 
7.4220 
7.5328 
7.6436 

77544 
7.8651 

H 
14% 

43.982 

44-375 
44.767 
45.160 
45-553 
45-945 
46.338 
46.731 

153.938 
156.699 
159.485 
162.295 
165.130 
167.989 
170.873 
173782 

12.406 
12.517 
12.628 
12.739 
12.850 
12.960 
13.071 
I3.I82 

NOO  Vj-NOO\N  NpO\-4.NOO 

X^tX^xX^tX 

ONONONONONONONON 

28.274 
28.667 
29.059 
29.452 
29.845 
30.237 
30.630 
31.023 

63.617 
65.396 
67.200 
69.029 
70.882 
72.759 
74.662 
76.588 

7.9760 
8.0866 
8.1974 
8.3081 
8.4190 
8.5297 
8.6405 
8.7513 

15 
I5# 

47.124 
47.516 
47.909 
48.302 
48.694 
49.087 
49.480 
49.872 

176.715 
179.672 
182.654 
185.661 
188.692 
191.748 
194.828 
197-933 

I3-293 
I3-403 
I3-5I4 
13.625 

13736 
13.847 
13-957 
14.068 

10 
10/8 

io)4 
io/2 

10% 

3I.4I6 
31.808 
32.201 

32.594 
32.986 

33-379 
33-772 
34.164 

78.540 

8o.5I5 
82.516 
84.540 
86.590 
88.664 
90.762 
92.885 

8.8620 
8.9728 
9.0836 
9-1943 
9.3051 
9.4159 

9.5267 
9-6375 

16 

16/8 

\6/8 

50.265 
50.658 
51.051 

5M43 
51.836 
52.229 
52.621 
53.014 

201.062 
204.216 
207.394 
210.597 
213.825 
217.077 
220.353 
223.654 

14.179 
14.290 
14.400 
14.511 
14.622 
14732 
14.843 
14.954 

II 

H/8 
11/8 

H/8 

34.558 
34-950 
35-343 
35-735 
36.128 

36.521 

36.913 
37.306 

95.033 
97-205 
99.402 
101.623 
103.869 
106.139 
108.434 
110.753 

97482 
9.8590 
9.9698 
10.080 
10.191 
10.302 
10.413 
10.523 

17 

I'jYz 
17^ 

53407 
53-799 
54.192 

54.585 
54.978 

55.370 
55763 
56.156 

226.980 
230.330 

233.705 
237.104 
240.528 

243.977 
247.450 
250.947 

15.065 
15.176 
15.286 

15-397 
15.508 
I5.6I9 
I5-730 
15.840 

CIRCLES: — DIAMETER,  CIRCUMFERENCE,  &c. 


89 


Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

IS 

i8>£ 

i$X 

8^ 
8K 
8^ 
8^ 
8^ 

56.548 
56.941 

57-334 
57.726 
58.119 
58.512 
58.905 
59.297 

254.469 
258.016 
261.587 
265.182 
268.803 
272.447 
276.117 
279.811 

I5.95I 
16.062 

16.173 
16.283 

16.394 
16.505 

16.616 
16.727 

24 

24l/& 
24X 
24^ 

24X 

24^ 
24^ 
247/& 

75-398 

75-791 
76.183 
76.576 
76.969 
77.361 

77-754 
78.147 

452.390 

457-115 
461.864 
466.638 
471.436 
476.259 
481.106 
485.978 

21.268 

21-379 
21490 
21.601 
21.712 
21.822 

21-933 

22.044 

19 
'9# 
19* 
i9# 
i9# 
19% 
19* 
197/8 

59.690 
60.083 
60.475 
60.868 
61.261 
61.653 
62.046 
62.439 

283.529 
287.272 
291.039 
294.831 
298.648 
302.489 

306.355 
310.245 

16.837 
16.948 
17.060 
17.170 
17.280 

I7-39I 
17.502 
17.613 

25 

25^ 

25X 
25  3A 

25  X 
25^ 
25^ 
25^ 

78.540 
78.932 
79.325 
79718 

8o.no 
80.503 
80.896 
81.288 

490.875 
495796 
500.741 
505.711 
510.706 

5I5-725 
520.769 

525.837 

22.155 
22.265 
22.376 
22.487 
22.598 
22.709 
22.819 
22.930 

20 
20^ 
20X 
20^ 
20^ 
20^ 
20^ 
207/& 

62.832 
63.224 
63.617 
64.010 
64.402 
64.795 
65.188 
65.580 

314.160 
318.099 
322.063 
326.05  1 
330.064 

334-101 
338.163 
342.250 

17.724 
17-834 
17-945 
18.056 
18.167 
18.277 
18.388 
18.499 

26 
26^ 

26^ 
26^ 

26^ 
26^ 

26^ 
26^ 

81.681 
82.074 
82.467 
82.859 
83-252 
83.645 
84.037 
84.430 

530.930 
536.047 
541.189 

546.356 
551.547 
556.762 
562.002 
567.267 

23.041 
23.152 
23.062 

23.373 
23.484 

23-595 
23.708 
23.816 

21 
2Il/i 
21* 
21^ 
21% 
21# 
2t# 
2IJ/8 

65-973 
66.366 
66.759 
67.151 
67.544 
67.937 
68.329 
68.722 

346-361 
350.497 
354.657 
358.841 
363.051 
367.284 

371-543 
375.826 

18.610 
18.721 
18.831 
18.942 

19-053 
19.164 

19.274 
19-385 

27 

27^ 

27X 

27^ 

27% 
27tt 

27% 
27  7A 

84.823 
85.215 
85.608 
86.001 
86.394 
86.786 
87.179 
87.572 

572.556 
577.870 
583.208 
588.571 
593-958 
599-370 
604.807 
610.268 

23-927 
24.038 
24.149 
24.259 
24.370 
24.481 
24.592 
24.703 

22 
22>£ 
22% 
22% 
22% 
22% 
22% 
227/& 

69.115 
69.507 
69.900 
70.293 
70.686 
71.078 
71.471 
71.864 

380.133 
384.465 
388.822 
393-203 
397.608 
402.038 
406.493 
410.972 

19.496 
19.607 
19.718 
19.828 

19-939 
20.050 
20.161 
20.271 

28 
28^ 

28X 
28H 

28^ 
28^ 
28% 

2Sj/8 

87.964 

88.357 
88.750 
89.142 

89-535 
89.928 
90.321 
90.713 

6I5-753 
621.263 
626.798 

632.357 
637.941 

643-594 
649.182 
654.839 

24.813 
24.924 

25.035 
25.146 
25.256 

25.367 
25-478 

25.589 

23 
23^ 
23X 
23^ 

23K 

23^ 
23^ 
23^ 

72.256 
72.649 
73.042 

73434 
73.827 
74.220 

74.613 
75.005 

415476 
420.004 
424.557 
429.135 
433-731 
438-363 
443.014 

447.699 

20.382 
20.493 
20.604 
20.715 
20.825 
20.936 
21.047 
21.158 

29 
29^ 

29X 

29^ 

29% 

29% 
29% 
29^ 

91.106 
91.499 
91.891 
92.284 

92.677 
93.069 
93462 
93.855 

660.521 

666.227 
671.958 
677.714 

683.494 
689.298 
695.128 
700.981 

25.699 
25.810 
25.921 
26.032 
26.143 
26.253 
26.364 
26.478 

MATHEMATICAL   TABLES. 


Diameter 

Circum- 
ference. 

Area. 

Side  of 
Equal  Squar 
(Square  Roo 
of  Area). 

Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

30 

3ol/8 
3oX 

303/8 

30/2 
30% 
30% 

3o7/s 

94.248 
94.640 

95-033 
95426 
95.818 
96.21  1 
96.604 
96.996 

706.860 
712.762 
718.690 
724.641 
730.6l8 
736.619 
742.644 
748.694 

26.586 
26.696 
26.807 
26.918 
27.029 
27.139 
27.250 
27.361 

36 
36/8 

36X 
36M 
36^ 
36^ 

36X 
36?£ 

113.097 
II3490 
113.883 
114.275 
114.668 
II5.06I 

H5-453 
1  1  5.846 

1017.88 
1024.95 
1032.06 
1039.19 
1046.35 
1053.52 
1060.73 
1067.95 

3L903 
32.014 
32.124 
32.235 
32.349 
32.457 
32.567 
32.678 

3i 

3*l/8 
3iX 
3'# 

3iX 
3i^ 
3iX 
3i# 

97.389 
97782 
98.175 
98.567 
98.968 

99-353 
99-745 
100.138 

754.769 
760.868 
766.992 
773.140 

779-3I3 
785.510 
791.732 
797.978 

27.472 

27.583 
27.693 
27.804 
27.915 
28.026 
28.136 
28.247 

37/8 
37X 
373A 
37^ 
37*3 
37X 
37^ 

116.239 
116.631 
117.024 
II74I7 
II7.8IO 
II8.202 
118.595 
118.988 

1075.21 
1082.48 
1089.79 
1097.11 
1104.46 
IIII.84 
1119.24 
1126.66 

32.789 
32.900 

33-011 
33-021 
33-232 
33-343 
33-454 
33-564 

32 

32^ 
32X 
32^ 
32^ 
32^ 
32X 
32^ 

100.531 
100.924 
101.316 
101.709 

102.102 
102.494 
102.887 
103.280 

804.249 
810.545 
816.865 
823.209 
829.578 
835.972 
842.390 
848.833 

28.358 
28.469 
28.580 
28.691 
28.801 
28.912 
29.023 
29.133 

38 

38^ 
38X 
38^ 
38K 
38^ 
38^ 
38^ 

119.380 

II9.773 
I20.I66 
120.558 
120.951 
121.344 
121.737 
122.129 

II34.II 
1141.59 
II49.08 
1156.61 
1164.15 
II7I.73 
1179.32 
1186.94 

33.675 
33.786 

33.897 
34.008 
34.118 
34.229 
34.340 
3445  J 

33 

33>£ 
33X 
333/8 
33% 
33% 
33% 
337A 

103.672 
104.055 
104.458 
104.850 
105.243 
105.636 
106.029 
I06.42I 

855-30 
861.79 
868.30 
874.84 
881.41 

888.00 
894.61 
901.25 

29.244 

29-355 
29.466 

29-577 
29.687 
29.798 
29.909 
3O.020 

39 
39^ 
39X 
39^ 
39X 
39^ 
39X 
39^ 

122.522 
122.915 
123.307 
123700 
124.093 
124.485 
124.878 
125.271 

1194.59 
1202.26 
1209.95 
1217.67 
122542 
1233.18 
1240.98 
1248.79 

34.56i 
34.672 

34.783 
34.894 
35-005 

35-115 
35.226 

35-337 

34 
34^ 
34X 
34^ 
34X 
34% 
34X 
34-J/s. 

Io6.8l4 
107.207 
107.599 
107.992 
108.385 
108.777 
109.170 
109.563 

907.92 
914.61 
921.32 
928.06 
934.82 
941.60 
948.41 
955-25 

30.131 
30.241 

30.352 
30'463 
30.574 
30.684 

30.795 
30.906 

40 
40^ 

4oX 
40^ 

4oX 
40^ 
4oX 
40^ 

125.664 
126.056 
126.449 
126.842 
127.234 
127.627 
128.020 
128.412 

1256.64 
1264.50 
1272.39 
1280.31 
1288.25 
1296.21 
1304.20 
1312.21 

35-448 
35-558 
35-669 
35.78o 

35-891 
36.002 
36.112 
36.223 

35 

35^ 
35X 
353A 
35/2 
35% 
35% 
357/s 

109.956 
110.348 
II0.74I 
III.I34 
III.526 
III.9I9 
II2.3I2 
112.704 

962.11 
968.99 
975.90 
982.84 
989.80 
996.78 
1003.78 
1010.82 

31.017 
31.128 
3L238 
3L349 
31.460 

3L57I 
3I.68I 
31.792 

4i 
4iM 
4iX 
4iK 
4i  /2 

4*H 

41  % 

4*# 

128.805 
129.198 
129.591 
129.983 
130.376 
130.769 
I3I.l6l 
I3L554 

1320.25 
1328.32 
1336.40 

1344.51 
1352.65 
1360.81 
1  369.00 
1377.21 

36.334 
36.445 

3£!I! 
36.666 

36.777 
36.888 
36.999 
37.109 

CIRCLES: — DIAMETER,   CIRCUMFERENCE,  &C. 


Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

42 

42/8 

42^ 
42^ 
42/8 
42X 
42J/8 

I3I-947 
132.339 
132.732 

133.125 
I33-5I8 
133.910 

134.303 
134.696 

1385.44 
I393-70 
1401.98 
1410.29 
1418.62 
1426.98 
I435.36 
1443-77 

37.220 

37.331 

37-442 

37.663 
37-774 
37.885 
37.996 

4%/8 

48X 
48^ 
48^ 
48/8 
48X 
48^ 

150.796 
151.189 
151.582 
151.974 
152.367 
152.760 
153.153 
153.545 

1809.56 
1818.99 
1828.46 

1837.93 
1847.45 
1856.99 
1866.55 
1876.13 

42-537 
42.648 
42.759 
42.870 
42.980 
43.091 
43.202 
43.3I3 

43  / 

43/s 
43X 
43^ 
43^ 
43^ 
43X 

135.088 
135.481 

I35.874 
136.266 
136.659 
137.052 
137-445 
137.837 

1452.20 
1460.65 
1469.13 
1477.63 
1486.17 
1494.72 
1503.30 
1511.90 

38.106 
38.217 
38.328 
38.439 
38.549 
38.660 

38.771 
38.882 

49 
49^ 
49X 
49^ 
49  1A 
49/8 
49% 
49% 

153.938 
154.331 
154723 
I55.II6 
155.509 
155.901 
156.294 
I  56.687 

1885.74 
I895.37 
1905.03 
I9H.70 
1924.42 

1934.15 
1943.91 
1953.69 

43423 

43-534 
43-645 
43-756 
43-867 
43.977 
44.088 
44.199 

44 
44/8 
44X 
44^ 

44/8 
44% 

138.230 
138.623 
139.015 
139.408 
139.801 
140.193 
140.586 
140.979 

1520.53 
1529.18 
1537.86 
1546.55 
1555.28 
1564.03 
I572.8I 
I58l.6l 

38-993 
39-103 
39.214 

39-325 
39-436 
39-546 
39-657 
39-768 

$0% 

I  57.080 
157.865 
158.650 
159436 

1963.50 
1983.18 
2002.96 
2O22.84 

44.310 
44-531 
44-753 
44.974 

5JX 

Sl% 

I60.22I 
l6l.007 
161.792 
162.577 

2042.82 
2062.90 
2083.07 
2103.35 

45.196 
45-4I7 
45-639 
45-861 

45 
45^ 
45X 
45^ 
45K 
45  /8 
45% 
4S7/8 

141.372 
141.764 
142.157 
142.550 
142.942 

H3-335 
143728 
144.120 

1590.43 
1599.28 
1608.15 
1617.04 
1625.97 
1634.92 
1643.89 
1652.88 

39-879 
39.989 
40.110 
40.211 
40.322 
40.432 

40.543 
40.654 

52 
52X 
52^ 
52X 

163.363 
164.148 
164.934 
165.719 

212372 
2144.19 
2164.75 
2185.42 

46.082 
46.304 

46.525 

46.747 

53X 

166.504 
167.290 
168.075 
I68.86I 

2206.18 
2227.05 
2248.01 
2269.06 

46.968 
47.190 
47.411 
47.633 

ON  ON  ON  ON  ON  ON  ON  ON 

Ssq  SU)  So!  SM  SW  SM  S^H 

oo\  -P\  o§s  N\  oos  -P\  o^s 

144.513 
144.906 
145.299 
145.691 
146.084 
146.477 
146.869 
147.262 

l66l.90 
1670.95 

1680.01 
1689.10 
1698.23 
1707.37 
1716.54 
1725.73 

40.765 
40.876 
40.986 
41.097 
41.208 

4L3I9 
41.429 
41.540 

54 
54X 
54^ 
54% 

169.646 
170.431 
I7I.2I7 
172.002 

2290.22 
2311.48 
2332.83 
2354.28 

47.854 
48.076 
48.298 
48.519 

i 

55% 

172.788 
173.573 
I74.358 
175.144 

2375.83 
2397.48 
2419.22 
2441.07 

48.741 
48.962 
49.184 
49.405 

47 

47  l/s 
47X 
47^ 
47/2 
47/8 
47% 
477A 

147.655 
148.047 
148.440 
148.833 
149.226 
149.618 
I5O.OI  I 
1  50.404 

1734-94 
1744.18 

1753-45 
1762.73 
1772.05 
1781.39 
1790.76 
1800.14 

41.651 
41.762 

41.873 
41.983 
42.094 

42.205 
42.316 
42.427 

56 

56X 
56^ 
56X 

175.929 
176.715 
177.500 
178.285 

2463.01 

2485.05 
2507.19 
2529.42 

49.627 
49.848 
50.070 
50.291 

92 


MATHEMATICAL   TABLES. 


Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Squal  Square 
Square  Root 
of  Area). 

Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

57 

57* 

57^ 

179.071 
179.856 
180.642 
181.427 

2551.76 
2574.19 
2596.72 
2619.35 

50.513 
50.735 
50.956 

51.178 

68 

68# 

68^ 

213.628 
214.414 
215.199 
215.985 

3631.68 

3658.44 
3685.29 
3712.24 

6o.26l 
60.483 
60.704 
60.926 

58 

58X 

182.212 
182.998 
183783 

184.569 

2642.08 
2664.91 
2687.83 
2710.85 

51-399 
51.621 
51.842 
52.064 

69 

69^ 

216.770 

217-555 
2I8.34I 
219.126 

3739-28 
3766.43 

3793-67 
3821.02 

61.147 
61.369 
61.591 

61.812 

59  1 
59X 

185.354 
186.139 
186.925 
187.710 

2733-97 
2757.19 
2780.51 
2803.92 

52.285 
52.507 
52.729 
52.950 

70 
70% 

219.912 
220.097 
221.482 
222.268 

3848.45 
3875.99 
3903-63 
393I-36 

62.034 
62.255 
62.477 
62.698 

60 
6o# 

188.496 
189.281 
190.066 
190.852 

2827.43 
2851.05 
2874.76 
2898.56 

53-I72 

53-393 
53-836 

7i 
71* 

71/2 

223.053 
223.839 
224.624 
225.409 

3959-19 
3987.13 
40I5.I6 
4043.28 

62.920 
63.141 
63-363 
63.545 

ON  ON  ON  ON 

XXX  ~ 

191.637 
192.423 
193.208 
193-993 

2922.47 

2946.47 
2970.57 

2994-77 

54.048 
54.279 
54.501 
54723 

72 
72# 

226.195 
226.980 
227.766 
228.551 

4071.50 
4099.83 
4128.25 
4156.77 

63.806 
64.028 
64.249 
64.471 

ON  ON  ON  ON 
to  to  to  to 

XXX 

194.779 
195.564 
196.350 

197.135 

3019.07 

304347 
3067.96 
3092.56 

54-944 
55.166 

55.387 
55-609 

73^ 

73^ 

229.336 
230.122 
230.907 
231.693 

4185.39 
4214.11 
4242.92 
4271.83 

64.692 
64.914 
65-I35 
65.357 

ON  ON  ON  ON 
Oo  Oo  Oo  Oo 

XXX 

197.920 
198.706 
I9949I 
200.277 

3117.25 
3142.04 
3166.92 
3191.91 

55-830 
56.052 

56.273 
56.495 

74 
74* 

232.478 
233.263 
234.049 
234.834 

4300.84 

4329-95 
4359.16 

4388.47 

65.578 
65.800 

66.022 

66.243 

64  1 

64^ 

64^ 

2OI.o62 
201.847 
202.633 
203.418 

3216.99 
3242.17 
326746 
3292.83 

56.716 
56.938 
57.159 

75  i 
75# 

235.620 

236.405 
237.190 
237.976 

4417.86 

4447-37 
4476.97 
4506.67 

66.465 

66.686 
66.908 
67.129 

ON  ON  ON  ON 

«J-l  Ol  Ol  Ol 

XXX 

204.204 
204.989 
205.774 
206.560 

3318.31 
3343.88 
3369.56 
3395.33 

57.603 
57.824 
58.046 
58.267 

76  , 

76^ 
76^ 

238.761 

239-547 
240.332 
24I.II7 

4536.46 
4566.36 

4596.35 
4626.44 

67-351 
67.572 

67794 
68.016 

66 
66% 
66^ 
66^ 

207.345 
208.131 
208.916 
209.701 

3421.19 
3447.16 
3473-23 

3499-39 

58.489 
58.710 
58.932 
59-154 

77 

777/2 
77ti 

241.903 
242.688 

243-474 
244.259 

4656.63 
4686.92 
4717.30 
4747-79 

68.237 

68.459 
68.680 
68.902 

67  , 

67  1/2 
67^ 

210.487 
211.272 
212.058 
212.843 

3525.66 
3552.01 

3578.47 
3605.03 

59-375 
59-597 
59.818 
60.040 

78  , 

78K 
78^ 

245.044 
245.830 
246.615 
247401 

4778.36 
4809.05 

4839-83 
4870.70 

69.123 

69-345 
69.566 
69.788 

CIRCLES: — DIAMETER,   CIRCUMFERENCE,  &C. 


93 


Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

79 
79% 

79% 

248.186 

248^71 

249-757 
250.542 

4901.68 

493275 
4963.92 
4995.19 

70.009 
70.231 

70-453 
70.674 

9°  , 
90% 

282.744 
283.529 
284.314 

285.o99 

6361.73 
6399.12 
6432.62 
6468.16 

79-758 
79.980 
80.201 
80.423 

80 

So% 

251.328 
252.113 

252.898 
253.683 

5026.55 
5058.00 
5089.58 
5121.22 

70.896 
7I.II8 

71-339 
7I.56I 

91  i 

91  /^2 
9I^ 

285.885 
286.670 
287.456 
288.242 

6503.88 

653^68 
6573.56 
6611.52 

80.644 
80.866 
81.087 
81.308 

81 

254.469 
255.254 
256.040 
256.825 

5153.00 
5184.84 
5216.82 
5248.84 

71.782 
72.004 

72.225 
72.447 

92 

9*1A 
9*l/2 

92^ 

289.027 
289.8l2 

29o.598 

291.383 

6647.61 
6683.80 
6720.07 
6756.40 

81.530 
81.752 

81.973 
82.195 

82 
82X 

S2/2 

82^ 

257.611 
258.396 

259.l82 

259-967 

5281.02 
53I3-28 
5345-62 
5378.04 

72.668 
72.890 
73.111 

73-333 

93 
931/ 
911A 
93% 

292.l68 

292.953 
293-739 

294.524 

6792.9i 
6829.48 
6866.16 
6882.92 

82.416 
82.638 
82.859 
83.081 

00  00  00  OO 
Oo  00  Oo  Oo 

K^x 

260.752 
261.537 
262.323 

263.108 

54I0.6l 

5443-24 
5476.00 
5508.84 

73-554 
73.776 

73-997 
74-219 

94 
941/ 
94/2 

94% 

295.310 

296.o95 
296.88i 
297.666 

6939.78 
6976.72 
7013.81 
7050.92 

83.302 
83.524 
83746 
83.968 

84  1 

841! 

263.894 
264.679 
265.465 
266.250 

5541-77 
5574.80 
5607.95 
5641.16 

74.440 
74.662 
74.884 
75.106 

95 

9S1/ 
9S/2 
9S% 

298.452 

299.237 
300.022 
300.807 

7088.22 
7125.56 
7163.04 
7200.56 

84.i89 
84.411 
84.632 
84.854 

x^x 

un  u-i  un  u-i 

OO  CO  OO  OO 

267.035 
267.821 
268.606 
269.392 

5674.51 
5707.92 
5741.47 

5775-09 

75-549 
75.770 

75.992 

96 
9W 
9W 
96% 

301.593 
302.378 
302.164 
303.948 

7238.23 
7275.96 

73I3-84 
7351.72 

85.077 
85.299 
85.520 
85.742 

86 
86X 
86^ 
86^ 

270.177 
270.962 
271.748 
272.533 

5808.80 
5842.60 

5876.55 
5910.52 

76.213 

76.435 
76.656 
76.878 

97  1 

97  /2 
97% 

304.734 
305.520 
306.306 
307-090 

7389.81 
7427.96 
7474.20 
7504.52 

85.963 
86.185 
86.407 
86.628 

87  1 

87^ 

87^ 

273-319 
274.104 
274.890 
275.675 

5944-68 
5978.88 
6013.21 
6047.60 

77-099 
77.321 
77.542 
77764 

98' 
98^ 

307.876 
308.662 
309-446 
310.232 

7542.96 
7581.48 
7620.12 
7658.80 

86.850 
87.072 

87.293 
87.515 

88 

88X 
88^ 
88X 

276.460 
277.245 
278.031 
278.816 

6082.12 
6116.72 
6151.44 
6186.20 

78.207 
78.428 
78.650 

99 
991A 
99/2 
99% 

311.018 
311.802 
312.588 
313.374 

7697.69 
7736.60 
7775.64 
7814.76 

87.736 

88.180 
88.401 

100 

3H.I59 
315.730 

7853.98 
7932.72 

88.623 
89.o66 

89 
89X 
89^ 

279.602 
280.387 
281.173 
281.958 

6221.14 
6256.12 
6291.25 
6326.44 

78.871 
79-093 
79.3I5 
79-537 

101 
101% 

317.301 
318.872 

8011.85 
8091.36 

89>5o9 
89.952 

94 


MATHEMATICAL   TABLES. 


Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
Square  Root 
of  Area). 

Diameter. 

Circum- 
ference. 

Area. 

Side  of 
Equal  Square 
(Square  Root 
of  Area). 

102 

102^ 

320.442 
322.014 

8171.28 
8251.60 

90-395 
90.838 

112 
112& 

351.858 
353-430 

9852.03 
9940.20 

99.258 
99.701 

103 

I03/2 

323.584 
325.154 

8332.29 
8413.40 

91.282 
91.725 

H3 

H3/2 

355-ooo 
356.570 

10028.75 
10117.68 

IOO.I44 
100.587 

104 

104^ 

326.726 
328.296 

8494.87 
8576.76 

92.168 
92.6ll 

114 
114/2 

358.142 
359.712 

10207.03 
10296.76 

IOI.03I 
101.474 

I05 
105^ 

329.867 
33I-438 

8659.01 
8741.68 

93-054 

93-497 

H5 

"5# 

361.283 
362.854 

10386.89 
10477.40 

IOI.9I7 
102.360 

106 
io6K 

333-009 
334.580 

882473 
8908.20 

93-940 
94.383 

116 
116% 

364-425 
365.996 

10568.32 
10659.64 

102.803 
103.247 

107 

107/2 

336.150 
337722 

8992.02 
9076.24 

94.826 
95.269 

117 
"7# 

367.566 
369.138 

10751.32 
10843.40 

103.690 
104.133 

108 
io8X 

339.292 
340.862 

9160.88 
9245.92 

9l'7Il 
96.156 

118 
u8# 

370.708 
372.278 

10935.88 
11028.76 

104.576 
105.019 

109 
109  >£ 

342.434 
344.004 

933I-32 
9417.12 

96.599 
97.042 

119 

H9/2 

373-849 
375.420 

1  1  122.02 
11215.68 

105.463 
105.906 

IIO 

110% 

345-575 
347.146 

9503.32 
9589.92 

97.485 
97.928 

120 

376.991 

II30973 

106.350 

in 

HI/2 

348.717 
350.288 

9676.89 
9764.28 

98.371 
98.815 

LENGTHS   OF  CIRCULAR  ARCS. 


95 


TABLE   No.   V.— LENGTHS   OF   CIRCULAR  ARCS   FROM 
i°  TO  180°.      GIVEN,   THE   DEGREES. 

(RADIUS  =  i.) 


Degrees. 

Length. 

Degrees. 

Length. 

Degrees. 

Length. 

Degrees. 

Length. 

I 

-0175 

40 

.6981 

79 

1.3788 

117 

2.O42O 

2 

•0349 

41 

.7156 

118 

2-°595 

3 

.0524 

42 

•7330 

80 

/-> 

I-3963 

119 

2.0769 

4 

.0698 

43 

.7505 

81 

I-4I37 

6 

.0873 
.1047 

44 
45 

.7679 
.7854 

82 
83 

1.4312 
1.4486 

120 

121 

2.0944 
2.III8 

7 

.1222 

46 

.8028 

84 

1.4661 

122 

2.1293 

8 

.1396 

47 

.8203 

85 

1.4835 

I23 

2.1468 

9 

48 

•8377 

86 

1.5010 

I24 

2.1642 

40 

•8552 

'87 

1.5184 

125 

2.I8I7 

10 

-1745 

Tx 

<J  *J 

88 

1-5359 

126   |   2.1991 

ii 

.1920 

50 

.8727 

89 

1-5533 

127   |   2.  2l66 

12 

.2094 

51 

.8901 

128 

2.2340 

13 

.2269 

52 

.9076 

90 

1.5708 

I29 

2.2515 

14 

•2443 

53 

.9250 

91 

1.5882 

15 

.26l8 

54 

•9425 

92 

1.6057 

130 

2.2689 

16 

-2793 

55 

•9599 

93 

1.6232 

I3I 

2.2864 

17 

.2967 

56 

•9774 

94 

1.6406 

132 

2.3038 

18 

.3142 

57 

•9948 

95 

1.6581 

133 

2.3213 

19 

.3316 

58 

1.0123 

96 

1.6755 

134 

2.3387 

59 

1.0297 

97 

1.6930 

135 

2.3562 

20 

•3491 

98 

1.7104 

136 

2.3736 

21 

.3665 

60 

1.0472 

99 

1.7279 

137 

2.39II 

22 

.3840 

61 

1.0647 

138 

2.4086 

23 

.4014 

62 

1.0821 

IOO 

1-7453 

2.4260 

24 

.4189 

63 

1.0996 

101 

1.7628 

25 

•4363 

64 

1.1170 

102 

1.7802 

140 

2-4435 

26 

.4538 

65 

I-I345 

I03 

1-7977 

141 

2.4609 

27 

.4712 

66 

1.1519 

IO4 

1.8151 

142 

2.4784 

28 

.4887 

67 

1.1694 

105 

1.8326 

143 

2.4958 

29 

.5061 

68 

1.1868 

106 

1.8500 

144 

2.5133 

69 

1.2043 

107 

1.8675 

145 

2.5307 

30 

•5236 

108 

1.8850 

146 

2.5482 

31 

-54II 

70 

1.2217 

109 

1.9024 

147 

2.5656 

32 

.5585 

71 

1.2392 

148 

2.5831 

33 

.5760 

72 

1.2566 

no 

1.9199 

149 

2.6OO5 

34 

-5934 

73 

1.2741 

III 

1-9373 

35 

.6109 

74 

1.2915 

112 

1.9548 

15° 

2.  6l8o 

36 

.6283 

75 

1.3090 

113 

1.9722 

2.6354 

37 

-6458 

76 

1.3265 

114 

1.9897 

IS2 

2.6529 

38 

.6632 

77 

1-3439 

115 

2.0071 

153 

2.6704 

39 

.6807 

78 

1.3614 

116 

2.0246 

154 

2.6878 

96 


MATHEMATICAL  TABLES. 


Degrees. 

Length. 

Degrees. 

Length. 

Degrees. 

Length. 

Degrees. 

Length. 

155 

2.7053 

161 

2.  8lOO 

1  68 

2.932I 

174 

3.0369 

156 

2.7227 

162 

2.8274 

169 

2.9496 

175 

3-0543 

157 

2.7402 

163 

2.8449 

I76 

3.0718 

158 

2.7576 

164 

2.8623 

170 

2.9670 

177 

3.0892 

159 

2-7751 

165 

2.8798 

171 

2.9845 

I78 

3.1067 

166 

2.8972 

172 

3.0O20 

179 

3.1241 

1  6O 

2.7925 

167 

2.9147 

i73 

3.0194 

180 

3.1416 

LENGTHS   OF  CIRCULAR  ARCS. 


97 


TABLE   No.  VI.— LENGTHS   OF   CIRCULAR  ARCS,   UP   TO   A 

SEMICIRCLE.     GIVEN,   THE   HEIGHT. 

(CHORD  =  i.) 


Height. 

Length. 

Height. 

Length. 

Height. 

Length. 

Height. 

Length. 

.100 

.02646 

.140 

1.05147 

.ISO 

1.08428 

.220 

.12444 

.101 

.02698 

.141 

1.05220 

.l8l 

1.08519 

.221 

•12554 

.102 

.02752 

.142 

1.05293 

.182 

I.  o86ll 

.222 

.12664 

.103 

.02806 

.143 

1.05367 

.183 

1.08704 

.223 

.12774 

.104 

.02860 

.144 

I.0544I 

.184 

.08797 

.224 

.12885 

.105 

.02914 

•145 

1.05516 

.185 

.08890 

.225 

.12997 

.IO6 

.02970 

.146 

I-05591 

.186 

.08984 

.226 

.13108 

.107 

.03026 

.147 

1.05667 

.187 

.09079 

.227 

.13219 

.I08 

.03082 

.148 

1.05743 

.188 

•09174 

.228 

•I3331 

.ICQ 

•03I39 

.149 

1.05819 

.189 

.09269 

.229 

•13444 

.110 

.03196 

•ISO 

1.05896 

.IQO 

.09365 

.230 

•I3557 

.III 

.03254 

•J51 

1.05973 

.191 

.09461 

.231 

.13671 

.112 

.03312 

.152 

1.06051 

.192 

•09557 

.232 

.13785 

•113 

.03371 

•153 

1.06130 

•193 

.09654 

•233 

.13900 

.114 

.03430 

•  154 

1.06209 

.194 

.09752 

.234 

.14015 

•US 

.03490 

.155 

1.06288 

.195 

.09850 

•235 

1.14131 

.116 

•03551 

.156 

1.06368 

.196 

.09949 

.236 

1.14247 

.117 

1.03611 

.157 

1.06449 

.197 

.10048 

.237 

•14363 

.118 

1.03672 

.158 

1.06530 

.198 

.10147 

.238 

.14480 

.119 

1.03734 

•159 

i.  06611 

.199 

.10247 

•239 

•14597 

.I2O 

1.03797 

.160 

1.06693 

.200 

.10347 

.240 

.14714 

.121 

1.03860 

.161 

1.06775 

.2OI 

.10447 

.241 

.14832 

.122 

1.03923 

.162 

1.06858 

.202 

.10548 

.242 

•I4951 

.123 

1.03987 

.163 

1.06941 

.203 

.10650 

•243 

.15070 

.124 

1.04051 

.164 

1.07025 

.204 

.10752 

.244 

.15189 

.125 

1.04116 

.165 

1.07109 

.205 

•10855 

•245 

.i53o8 

.126 

1.04181 

.166 

1.07194 

.206 

.10958 

.246 

.15428 

.127 

1.04247 

.167 

1.07279 

.207 

.11062 

.247 

.15549 

.128 

i-043I3 

.168 

1.07365 

.208 

.11165 

.248 

.15670 

.129 

1.04380 

.169 

1.07451 

.209 

.11269 

.249 

•I5791 

•ISO 

1.04447 

.170 

1-07537 

.2IO 

.11374 

.250 

1.15912 

438 

i-045I5 

.171 

1.07624 

.211 

.11479 

.251 

1.16034 

.132 

1.04584 

.172 

1.07711 

.212 

1.11584 

.252 

1.16156 

•i33 

1.04652 

.173 

1.07799 

.213 

1.11690 

•253 

1.16279 

•i34 

1.04722 

.174 

1.07888 

.214 

I.II796 

•254 

1.16402 

•!35 

1.04792 

•*75 

1.07977 

.215 

I.II9O4 

.255 

1.16526 

.136 

1.04862 

.176 

1.08066 

.216 

I.I20II 

.256 

1.16650 

•137 

1.04932 

.177 

1.08156 

.217 

I.  I2Il8 

.257 

1.16774 

.138 

1.05003 

.178 

1.08246 

.218 

I.I2225 

.258 

1.16899 

•139 

1-05075 

.179 

1.08337 

.219 

I.I2334 

.259 

1.17024 

93 


MATHEMATICAL  TABLES. 


Height. 

Length. 

Height. 

Length. 

Height. 

Length. 

Height. 

Length. 

.260 

I.I7I50 

.307 

1.23492 

•354 

1.30634 

.4OI 

1.38496 

.261 

1.17276 

.308 

1.23636 

•355 

1.30794 

.402 

1.38671 

.262 

I.I7403 

•3°9 

1.23781 

.356 

I-30954 

•4°3 

1.38846 

.263 
.264 

1.1753° 
1.17657 

.310 
.^n 

1.23926 

I.24O7O 

•357 
•358 

M11^ 
1.31276 

.404 
•405 

I.3902I 
1.39196 

•265 

1.17784 

3 
.312 

1       T^       / 

1.24216 

•359 

I-3I437 

.406 

!•  39372 

.266 
.267 

I.I79I2 

1.18040 

•3I3 

•3*4 

1.24361 

1.24507 

.360 
.361 

I-3I599 
1.31761 

.407 
.408 

1.30548 
1.39724 

.268 

1.18169 

•3I5 

1.24654 

.362 

1.31923 

.409 

1.39900 

.269 

1.18299 

.316 

1.24801 

.363 

1.32086 

410 

1.40077 

.270 

1.18429 

•3V 

1.24948 

.364 

1.32249 

.411 

1.40254 

.271 

I-l8559 

.318 

1.25095 

•365 

1-32413 

.412 

1.40432 

.272 

1.18689 

•3J9 

1.25243 

.366 

1.32577 

•413 

1.40610 

•273 

1.18820 

.320 

I-2539i 

•367 

1.32741 

.414 

1.40788 

.274 

1.18951 

.321 

1.25540 

.368 

1.32905 

•415 

.40966 

•275 

1.19082 

.322 

1.25689 

•369 

1.33069 

.416 

.4H45 

.276 

1.19214 

•323 

1.25838 

•370 

1.33234 

.417 

.41324 

.277 

1.19346 

•324 

1.25988 

•371 

1-33399 

.418 

.4I503 

.278 

1.19479 

.325 

1.26138 

•372 

I.33564 

.419 

.41682 

.279 

1.19612 

.326 

1.26288 

•373 

1.33730 

.420 

.41861 

.280 

1.19746 

•327 

1.26437 

•374 

1.33896 

I 

.421 

.42041 

.28l 

1.19880 

.328 

1.26588 

•375 

1.34063 

.422 

.42221 

.282 

1.20014 

•329 

1.26740 

•376 

1.34229 

.423 

.42402 

.283 
.284 

1.20149 
1.20284 

•330 

•331 

1.26892 
1.27044 

•377 
•378 

1.34396 
I-34563 

•424 
.425 

•42583 
.42764 

.285 

1.20419 

•332 

1.27196 

•379 

I-34731 

.426 

•42945 

.286 

!-2o555 

•333 

1.27349 

.380 

1.34899 

.427 

•43I27 

.287 

1.20691 

•334 

1.27502 

.381 

1.35068 

.428 

•43309 

.288 

1.20827 

•335 

1.27656 

•382 

I.35237 

.429 

•43491 

.289 

1.20964 

.336 

1.27810 

•383 

1.35406 

•43° 

•43673 

.290 
.291 

I.2I2O2 
I.2I239 

•337 
.338 

1.27864 
1.28118 

•384 
•385 

1-35575 
1-35744 

•431 
•432 

•43856 
.44039 

.292 

I.2I377 

•339 

1.28273 

.386 

I-359I4 

£       O 

•433 

.44222 

•293 

I.2I5I5 

•340 

1.28428 

•387 

o  o 

1.36084 

x- 

•434 

.44405 

.294 

1.21654 

•341 

1.28583 

.388 

o 

1.36254 

£ 

•435 

.44589 

•295 

I.2I794 

•342 

1.28739 

•389 

1.36425 

.436 

•44773 

.296 

I.2I933 

•343 

1.28895 

•390 

1.36596 

•437 

•44957 

.297 

1.22073 

•344 

1.29052 

.391 

1.36767 

.438 

.45142 

.298 

I.222I3 

•345 

1.29209 

.392 

I-36939 

•439 

•45327 

.299 

1.22354 

•346 

1.29366 

•393 

I.37III 

.440 

•45512 

.300 
.301 
.302 

1.22495 
1.22636 
1.22778 

•347 
•348 
•349 

1.29523 
1.29681 
1.29839 

-394 
•395 
•396 

1.37283 

1-37455 
1.37628 

.441 
.442 
•443 

•45697 
1.45883 
1.46069 

•303 

1.22920 

•350 

1.29997 

•397 

o 

1.37801 

•444 

1.46255 

•304 

1.23063 

•351 

1.30156 

•398 

1-37974 

•445 

1.46441 

•305 

1.23206 

•352 

I«3°3i5 

•399 

1.38148 

.446 

1.46628 

.306 

1.23349 

•353 

1.30474 

.400 

1.38322 

•447 

1.46815 

LENGTHS    OF    CIRCULAR  ARCS. 


99 


Height. 

Length. 

Height. 

Length. 

Height. 

Length. 

Height. 

i 

Length. 

.448 

I.47OO2 

.461 

1.49460 

•475 

I.52I52 

.489 

L54893 

•449 

1.47189 

.462 

1.49651 

.476 

1.52346 

•463 

1.49842 

•477 

L5254I 

.490 

I-55o9i 

•45° 

1-47377 

.464 

1.50033 

•478 

1.52736 

.491 

1-55289 

•451 

i-47565 

•465 

I.5O224 

•479 

I.5293I 

.492 

1-55487 

•452 
•453 
•454 
•455 
•456 
•457 
.458 
•459 

1-47753 
1.47942 
1.48131 
1.48320 
1.48509 
1.48699 
1.48889 
1.49079 

.466 
.467 
.468 
.469 

470 

.471 
.472 

•473 

1.50416 
1.50608 
1.50800 
1.50992 

I.5II85 

i-5I378 

i-5I57i 
1.51764 

.480 

.481 
.482 
•483 
.484 
•485 
.486 

•487 

I.53I26 
1.53322 

i-535l8 
i-537i4 
I-539Io 
1.54106 
1.54302 
1-54499 

•493 
•494 

•495 
.496 

•497 
.498 

•499 
.500 

1-55685 
'•55854 
1.56083 
1.56282 
1.56481 
1.56681 
1.56881 
1.57080 

.460 

1.49269 

•474 

i-5!958 

.488 

1.54696 

100 


MATHEMATICAL   TABLES. 


TABLE    No.  VII.— AREAS    OF   CIRCULAR   SEGMENTS,   UP   TO   A 

SEMICIRCLE. 
(DIAMETER  OF  CIRCLE  =i.) 


Height. 

Area. 

Height. 

Area. 

Height. 

Area. 

Height. 

Area. 

.001 

.00004 

.040 

.01054 

.079 

.02889 

.118 

.05209 

.OO2 

.00012 

.041 

.01093 

oRn 

.II9 

.05274 

.003 
.004 
.005 
.OO6 
.007 
.008 
.OOQ 

.OIO 

.00022 
.00034 
.OOO47 
.00062 
.00078 
.OOO95 
.00114 

.00133 

.042 

•043 
.044 

•045 
.046 
.047 
.048 
.049 

•OII33 

•OII73 
.OI2I4 
.01255 
.01297 
.01340 
.01382 
.01425 

•  v/OU 

.081 
.082 
.083 
.084 
.085 
.086 
.087 
.088 

.02943 
.02997 

.03053 
.03108 
.03163 
-03219 

.03275 

-03331 
•03385 

.I2O 

.121 
.122 
.123 
.124 
.125 
.126 
.127 

•05338 
.05404 
.05469 

•05535 
.05600 
.05666 
•05733 
.05799 

.Oil 

.00153 

.050 

.01468 

.089 

.03444 

.128 

.05866 

.012 
.013 
.OI4 
.015 
.Ol6 
.017 
.Ol8 
.OIQ 

.020 

.00175 
.OOI97 
.OO22O 
.00244 
.O0268 
.00294 
.0032O 
.00347 

.00375 

.051 
.052 
•053 
•054 

•055 
.056 

•057 
.058 

•059 

.01512 
•01556 
.01601 
.01646 
.01691 
.01737 
.01783 
.01830 
.01877 

.090 
.091 
.092 

•093 
.094 
.095 

.096 

.097 
.098 

•03501 
•°3538 
.03616 
.03674 
.03732 

.0379° 
.03850 
.03909 
.03968 

.129 

.130 

•131 
.132 

•133 
•134 
•135 
.136 
•137 

•05933 
.O6OOO 
.06067 
•06135 
.06203 
.06271 
.06339 
.06407 
.06476 

.021 

.00403 

.O6O 

.01924 

.099 

.04028 

.138 

•06545 

.022 

.00432 

.O6l 

.01972 

.100 

.04087 

•139 

.06614 

.023 

.00461 

.062 

.0202O 

.101 

.04148 

.I4O 

.06683 

.024 

.00492 

.063 

.02O68 

.102 

.04208 

.141 

•°6753 

.025 
.026 
.027 
.028 
.029 

.00523 

•°°555 
.00587 
.00619 
.00653 

.064 
.065 
.066 
.067 
.068 

.02117 
.O2l66 
.O22I5 
.02265 
•02315 

.103 

.104 

.105 

.106 

.107 

.04269 

•0433° 
.04391 
.04452 
.04514 

.142 

•143 
.144 

•145 
.146 

.O6822 
.06892 
.06963 
.07033 
.07103 

.030 

.031 

.00687 
.00721 

.069 
.070 

.02366 
.02417 

.108 

.109 

.04576 
.04638 

.147 
.148 

.07174 
.07245 

X* 

.032 

.00756 

.071 

.02468 

.no 

.04701 

.149 

07316 

•033 

.00792 

.072 

.02520 

.III 

.04763 

•J50 

.07387 

•034 

.00828 

•073 

•02571 

.112 

.04826 

•151 

•07459 

•035 

.00864 

.074 

.02624 

•113 

.04889 

•152 

•07530 

.036 

.00901 

•075 

.02676 

.114 

•°4953 

•i53 

.07603 

•037 

.00939 

.076 

.02729 

•«5 

.05016 

•154 

.07675- 

.038 

.00977 

.077 

.02782 

.116 

.05080 

•155 

.07747 

•°39 

.01015 

.078 

.02836 

.117 

•05145 

.156 

.07819 

AREAS   OF  CIRCULAR   SEGMENTS. 


101 


Height. 

Area. 

Height. 

Area. 

Height. 

Area. 

Height. 

Area. 

.157 

.07892 

.203 

.11423 

•249 

.15268 

.295     .19360 

.158 
•159 
.160 

.07965 
.08038 

.oSlII 

.204 
.205 
.206 

.11504 
.11584 
.11665 

.250 
.251 

.252 

•r5355 
.15442 

•15528 

.296 
.297 
.298 

•I9451 
•19543 
.19634 

!l62 
.163 

.08185 
.08258 
.08332 

.207 
.208 
.209 

.11746 
.11827 
.11908 

.253 
.254 
.255 

•15615 
.15702 

•15789 

.299 

.300 
.301 

•19725 

.19817 
.19908 

.164 

.08406 

.2IO 

.II99O 

.256 

.15876 

.302 

.20000 

.165 

.08480 

.211 

.12071 

.257 

.15964 

.303 

.20092 

.166 

•08554 

.212 

.12153 

.258 

.16051 

•3°4 

.20184 

.167 

.08629 

.213 

•12235 

•259 

.16139 

•3°5 

.20276 

.168 

.08704 

.214 

.12317 

.260 

.16226 

•3°6 

.20368 

.169 

.08778 

.215 

.12399 

.26l 

.16^14 

.307 

.20460 

.170 

.171 

.08854 
.08929 

.216 
.217 

.12481 
.12563 

.262 
.263 

\J   * 

.16402 
.16490 

.308 
.309 

•20553 
.20645 

.172 

.09004 

.2l8 

.12646 

.264 

.16578 

.310 

.20738 

.173 

.09080 

.219 

.12729 

.265 

.16666 

•311 

.20830 

.174 

•09155 

.220 

.12811 

.266 

«l6755 

.312 

.20923 

.175 

.09231 

.221 

.12894 

.267 

.16843 

•313 

.21015 

.176 

.09307 

.222 

.12977 

.268 

.16932 

•3*4 

.21108 

.177 

.09383 

.223 

.13060 

.269 

.17020 

•315 

.2I2OI 

.178 

.09460 

.224 

.13144 

.270 

.17109 

-316 

.21294 

.179 

•09537 

.225 

.13227 

.271 

.17198 

•SI? 

.21387 

.ISO 

.09613 

.226 

•I33II 

.272 

.17287 

.318 

.21480 

.l8l 

.09690 

.227 

•13395 

•273 

.17376 

•3*9 

•21573 

.182 
.183 

.09767 
•09845 

.228 
.229 

•13478 
.13562 

.274 
.275 

.17465 
•17554 

.320 

.321 

.21667 
.21760 

.184 

.09922 

.230 

.13646 

.276 

.17644 

'O 

.322 

•21853 

.185 
.186 

O  _ 

.O920O 
.10077 

.231 
.232 

•I373I 
•I38I5 

.277 
.278 

•17733 
.17823 

•323 
.324 

.21947 

.2204O 

.187 

•IOI53 

•233 

.13899 

.279 

.17912 

.325 

.22134 

.188 

o 

.10233 

.234 

.13984 

.280 

.18002 

.326 

.22228 

.109 

.10317 

.235 

.14069 

.281 

.18092 

.327 

.22322 

.190 

.10390 

.236 

•I4I54 

.282 

.18182 

.328 

.22415 

.191 

.10469 

•237 

.14239 

•283 

.18272 

.329 

.22509 

.192 

•10547 
.10626 

•239 

.14324 
.14409 

.284 
.285 

.18362 
.18452 

.330 

.•?-?  I 

.22603 
.22607 

.194 
•195 

.196 

.197 

o 

.10705 
.10784 
.10864 
.10943 

.240 

.241 
.242 
.243 

.14494 
.14580 
.14665 
.14752 

.286 
.287 
.288 
.289 

.18542 
.18633 
.18723 
.18814 

«J«J 

.332 

•333 
•334 
•335 

SI 

.22792 
.22886 
.22980 
.23074 

.190 

.11023 

.244 

•14837 

.290 

.18905 

•336 

.23169 

.199 

.IIIO2 

.245 

.14923 

.291 

.18996 

•337 

•23263 

.200 

.IIl82 

.246 

.15009 

.292 

.19086 

.338 

.23358 

.201 

.11262 

.247 

.15096 

.293 

.19177 

•339 

.23453 

.202 

•II343 

.248 

.15182 

.294 

.19268 

•340 

•23547 

IO2 


MATHEMATICAL  TABLES. 


Height. 

Area. 

Height. 

Area. 

Height. 

Area. 

Height. 

Area. 

•34* 

..23642 

•376 

.26998 

.411 

•30417 

•446 

.33880 

•342 

.23737 

•377 

.27095 

.412 

•305*6 

•447 

•33980 

•343 

.23832 

•378 

.27192 

•4*3 

.30614 

•448 

•34079 

•344 

.23927 

•379 

.27289 

•4*4 

.30712 

•449 

•34*79 

•345 
•346 
•347 
•348 
•349 

.24025 
.24117 
.24212 
.24307 
.24403 

.380 

.381 
.382 

•383 
•384 

.27386 
.27483 
.27580 
.27678 

•27775 

•4*5 
.416 

•4*7 
.418 

•4*9 

.30811 
.30910 
.31008 
.31107 

•3*205 

•450 

•453 
•455 
•457 

.34278 
•34378 
•34577 
•34776 
•34975 

•350 

.24498 

•385 

.27872 

.420 

•3*304 

•459 

•35*74 

•35* 

.24593 

.386 

.27969 

.421 

•31403 

.462 

•35*74 

•352 

.24689 

•387 

.28070 

.422 

.31502 

.464 

.35673 

•353 

.24784 

.388 

.28164 

•423 

.31600 

.466 

.35873 

•354 

.24880 

•389 

.28262 

.424 

.31699 

.468 

.36072 

•355 
•356 

.24976 
.25071 

•390 

•28359 

•425 
.426 

•3*798 
•3l897 

.470 

.36272 

•357 
•358 
•359 

.25167 
.25263 
•25359 

•39* 
•392 
•393 
•394 

•28457 
.28554 
.28652 
.28750 

.427 
.428 
•429 

•3*996 
•32095 
•32*94 

.471 

•473 
•475 
•477 

•3637* 
•3657* 
.36771 
.36971 

.360 

•25455 

•395 

.28848 

•430 

.32293 

•479 

.37170 

.361 

•2555* 

•396 

.28945 

•43* 

.32392 

.482 

•37470 

.362 

.25647 

•397 

.29043 

•432 

.32491 

.484 

.37670 

•363 

•25743 

•398 

.29141 

•433 

•32590 

.486 

.37870 

•364 

•25839 

•399 

.29239 

•434 

.32689 

.488 

.38070 

•365 

•25936 

•435 

.32788 

.366 

.26032 

.400 

•29337 

•436 

.32887 

.490 

.38270 

•367 

.26128 

.401 

•29435 

•437 

.32987 

•49* 

.38370 

.368 

.26225 

.402 

•29533 

•438 

.33086 

•492 

•38470 

•369 

.26321 

•403 

.29631 

•439 

•33185 

•493 

•38570 

.404 

.29729 

•494 

.38670 

•370 

.26418 

•405 

.298-27 

.440 

.33284 

•495 

.38770 

•371 

.26514 

.406 

.29926 

•44* 

•33384 

.496 

.38870 

•372 

.266ll 

.407 

.30024 

.442 

•33483 

•497 

•38970 

•373 

.26708 

.408 

.30122 

•443 

.33582 

.498 

.39070 

•374 

.26805 

.409 

.30220 

•444 

.33682 

•499 

.39170 

•375 

.26901 

.410 

•30319 

•445 

•33781 

.500 

.39270   / 

SINES,    COSINES,    &C.   OF  ANGLES. 


I03 


TABLE    No.  VIII.— SINES,    COSINES,   TANGENTS,    COTANGENTS, 
SECANTS,   AND    COSECANTS    OF   ANGLES   FROM  o°  TO  90°. 

ADVANCING  BY  10'  OR  ONE-SIXTH  OF  A  DEGREE.    (RADIUS  =i.) 


Angle. 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

0°  0' 

.OOOOOO 

Infinite. 

.000000 

Infinite. 

.OOOOO 

I.  OOOOOO 

90°  o' 

10 

.002909 

343-775J6 

.002^09 

343-77371 

.OOOOO 

.999996 

5o 

20 

.005818 

171.88831 

.005818 

171.88540 

.OOOO2 

.999983 

40 

30 

.008727 

114.59301 

.008727 

114.58865 

.OOOO4 

.999962 

30 

40 

.011635 

85-9456o9 

.011636 

85.939791 

.OOOO7 

.999932 

20 

5° 

.014544 

68.757360 

.014545 

68.750087 

.OOOII 

.999894 

10 

I   0 

.017452 

57.298688 

•017455 

57.289962 

.00015 

.999848 

89  o 

10 

.020361 

49.114062 

.020365 

49.103881 

.00021 

•999793 

50 

20 

.023269 

42.975713 

.023275 

42.964077 

.OOO27 

.999729 

40 

30 

.026177 

38.201550 

.026l86 

38.188459 

.00034 

•999657 

30 

40 

.029085 

34.382316 

.029097 

34.367771 

.OOO42 

•999577 

20 

5° 

.031992 

3L257577 

.032009 

3I'24f577 

.00051 

.999488 

IO 

2  0 

.034899 

28.653708 

.034921 

28.636253 

.OOO6l 

•999391 

88  o 

10 

.037806 

26.450510 

.037834 

26.431600 

.OOO72 

.999285 

So 

2O 

.040713 

24.562123 

.040747 

24-54I758 

.00083 

.999171 

40 

30 

.043619 

22.925586 

.043661 

22.903766 

.OOO95 

.999048 

3° 

4° 

.046525 

21.493676 

.046576 

21.470401 

.OOIOS 

.998917 

20 

5° 

.049431 

20.230284 

.049491 

20.205553 

I.OOI22 

.998778 

IO 

3  o 

•052336 

19.107323 

.052408 

19.081137 

I.OOI37 

.998630 

87  o 

10 

.055241 

18.102619 

•055325 

18.074977 

I.OOI53 

•998473 

50 

20 

.058145 

17.198434 

.058243 

17.169337 

I.OOl69 

.998308 

40 

30 

.061049 

16.380408 

.061163 

16.349855 

I.OOI87 

•998135 

3° 

40 

.063952 

15-636793 

.064083 

15.604784 

1.00205 

•997357 

20 

50 

.066854 

14.957882 

.067004 

14.924417 

I.OO224 

.997763 

10 

4  o 

.069756 

14.335587 

.069927 

14.300666 

I.OO244 

•997564 

86  o 

10 

.072658 

13-763115 

.072851 

13.726738 

1.00265 

-997357 

5° 

20 

•075559 

13-234717 

.075776 

13.196888 

1.00287 

.997141 

40 

3° 

.078459 

12.745495 

.078702 

12.706205 

I.O03O9 

.996917 

3° 

40 

.081359 

12.291252 

.081629 

12.250505 

1.00333 

.996685 

20 

50 

.084258 

11.86^370 

.084558 

11.826167 

1.00357 

.996444 

10 

5  o 

.087156 

n.473713 

.087489 

11.430052 

1.00382 

•996195 

85  o 

10 

.090053 

11.104549 

.090421 

11.059431 

1.00408 

•995937 

50 

20 

.092950 

10.758488 

•093354 

10.711913 

1.00435 

.995671 

40 

30 

.095846 

io.43343  1 

.096289 

10.385397 

1.00463 

•995396 

3° 

40 

.098741 

10.127522 

.099226 

10.078031 

I.0049I 

•995JI3 

20 

50 

.101635 

9.83912.27 

.102164 

9.7881732 

I.OO52I 

.994822 

10 

Cosine. 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

Angle. 

IO4 


MATHEMATICAL   TABLES. 


Angle. 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

6°  o 

.104528 

9.5667722 

.105104 

9.5143645 

1.00551 

.994522 

84°  o' 

10 

.107421 

9.3091699 

.108046 

9-2553035 

1.00582 

.994214 

5° 

20 

.110313 

9.0651512 

.110990 

9.0098261 

1.00614 

.993897 

40 

30 

.113203 

8.8336715 

.113936 

8.7768874 

1.00647 

.993572 

3° 

40 

.116093 

8.6137901 

.116883 

8.5555468 

1.  00681 

•993238 

20 

50 

.118982 

8.4045586 

•"9833 

8.3449558 

1.00715 

.992896 

IO 

7  o 

.121869 

8.2055090 

.122785 

8.1443464 

I.0075I 

.992546 

83  o 

TO 

.124756 

8.0156450 

•125738 

7.9530224 

1.00787 

.992187 

50 

20 

.127642 

7^344335 

.128694 

7.7703506 

1.00825 

.991820 

40 

30 

.130526 

7.6612976 

•I3l653 

7-5957541 

1.00863 

•991445 

30 

40 

•I334IO 

7.4957100 

•134613 

7.4287064 

I.OO902 

.991061 

20 

50 

.136292 

7.337I909 

.137576 

7-2687255 

1.00942 

.990669 

IO 

\ 

8  o 

•I39I73 

7.1852965 

.140541 

7.H53697 

1.00983 

.990268 

82  o  ' 

10 

.142053 

7.0396220 

.143508 

6.9682335 

I.OI024 

•989859 

50 

20 

.144932 

6.8997942 

.146478 

6.8269437 

1.01067 

.989442 

40 

30 

.147809 

6.7654691 

.149451 

6.6911562 

I.OIIII 

.989016 

30 

40 

.150686 

6.6363293 

.152426 

6.5605538 

I-OII55 

.988582 

20 

50 

•I5356I 

6.5120812 

.155404 

6.4348428 

I.OI2OO 

.988139 

IO 

9  o 

•156434 

6.3924532 

.158384 

6.3I375I5 

I.OI247 

.987688 

81  o 

10 

•159307 

6.2771933 

.161368 

6.1970279 

I.OI294 

.987229 

5° 

20 

.162178 

6.1660674 

.164354 

6.0844381 

I.OI432 

.986762 

40 

30 

.165048 

6.0588980 

•167343 

5.9757644 

I.OI39I 

.986286 

3° 

40 

.167916 

5.9553625 

•170334 

5.8708042 

I.OI440 

.985801 

20 

50 

.170783 

5.855392I 

.173329 

5.7693688 

I.OI49I 

•985309 

IO 

IO  0 

.173648 

5.7587705 

.176327 

5.6712818 

I-OI543 

.984808 

80  o 

10 

.176512 

5-6653331 

.179328 

5-5763786 

I-°i595 

.984298 

5° 

20 

•179375 

5.5749258 

.182332 

5.4845052 

1.01649 

.983781 

40 

30 

.182236 

5.4874043 

.185339 

5.3955I72 

1.01703 

•983255 

30 

40 

.185095 

5.4026333 

.188359 

5.3092793 

1.01758 

.982721 

20 

50 

•J87953 

5.3204860 

.191363 

5.2256647 

1.01815 

.982178 

10 

II   0 

.190809 

5.2408431 

.194380 

5.1445540 

1.01872 

.981627 

79  o 

10 

.193664 

5-I635924 

.197401 

5.0658352 

1.01930 

.981068 

5° 

20 

.196517 

5.0886284 

.200425 

4.9894027 

1.01989 

.980500 

40 

30 

.199368 

5.0I583I7 

.203452 

4-9i5I57o 

1.02049 

•979925 

3° 

40 

.202218 

4.9451687 

.206483 

4.8430045 

I.O2IIO 

•979341 

20 

5° 

.205065 

4.8764907 

.209518 

4-7728568 

I.O2I7I 

.978748 

10 

12  0 

.207912 

4-8097343 

•212557 

4.7046301 

1.02234 

.978148 

78  o 

10 

.210756 

4.7448206 

•215599 

4-6382457 

1.02298 

•977539 

5° 

20 

•2I3599 

4.6816748 

.218645 

4.5736287 

1.02362 

.976921 

40 

30 

.216440 

4.6202263 

.221695 

4.5107085 

1.02428 

.976296 

3° 

40 

.219279 

4.5604080 

.224748 

4.4494181 

1.02494 

.975662 

20 

50 

.222116 

4.5021565 

.227806 

4.3896940 

1.02562 

.975020 

10 

Cosine. 

Secant. 

Cotangent. 

Tangent.     Cosecant. 

Sine. 

Angle. 

SINES,   COSINES,  &C.   OF  ANGLES. 


105 


Angle. 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

13°  o' 

.224951 

4.4454II5 

.230868 

4.3314759 

1.02630 

•974370 

77°  o' 

10 

.227784 

4.3901158 

.233934 

4.2747066 

I.O27OO 

.973712 

So 

20 

.230616 

4.3362150 

.237004 

4.2193318 

I.O277O 

•973045 

40 

30 

•233445 

4.2836576 

.240079 

4.1652998 

1.02842 

•972370 

30 

40 

.236273 

4.2323943 

.243158 

4.1125614 

1.02914 

.971687 

20 

50 

.239098 

4.1823785 

.246241 

4.0610700 

1.02987 

•970995 

10 

14  o 

.241922 

4.1335655 

.249328 

4.0107809 

1.03061 

.970296 

76  o 

10 

•244743 

4.0859130 

.252420 

3.9616518 

1.03137 

.969588 

5° 

20 

.247563 

4.0393804 

•255517 

3.9136420 

1.03213 

.968872 

40 

3° 

.250380 

3.9939292 

.258618 

3.8667131 

1.03290 

.968148 

30 

40 

•253I95 

3.9495224 

.261723 

3.8208281 

1.03363 

.967415 

20 

5° 

.256008 

3.9061250 

.264834 

3-77595!9 

1.03447 

.966675 

IO 

15  o 

.258819 

3-8637033 

.267949 

3.7320508 

1.03528 

.965926 

75  o 

10 

.261628 

3.8222251 

.271069 

3.6890927 

1.03609 

.965169 

5o 

20 

.264434 

3.7816596 

.274195 

3.6470467 

1.03691 

.964404 

40 

30 

.267238 

3-74I9775 

•277325 

3-6058835 

1.03774 

•96363° 

30 

40 

.270040 

3.7031506 

.280460 

3.5655749 

1.03858 

.962849 

20 

50 

.272840 

3.6651518 

.283600 

3.5260938 

1.03944 

.962059 

IO 

16  o 

•275637 

3.6279553 

.286745 

3.4874144 

1.04030 

.961262 

74  o 

10 

.278432 

3.5915363 

.289896 

3.4495120 

I.04II7 

.960456 

5° 

20 

.281225 

3.55587IO 

.293052 

3.4123626 

I.O42O6 

.959642 

40 

30 

.284015 

3.5209365 

.296214 

3-3759434 

1.04295 

.958820 

30 

40 

.286803 

3.4867110 

.299380 

3.3402326 

1.04385 

•957990 

20 

50 

.289589 

3.453I735 

•302553 

3.3052091 

1.04477 

•957-151 

10 

17  o 

.292372 

3.4203036 

.305731 

3.2708526 

1.04569 

•956305 

73  o 

IO 

.295152 

3.3880820 

•308914 

3.2371438 

1.04663 

-955450 

50 

20 

.297930 

3.3564900 

.312104 

3.2040638 

1-04757 

.954588 

40 

30 

.300706 

3.3255095 

•3I5299 

3.1715948 

1.04853 

•953717 

30 

40 

.303479 

3-295I234 

.318500 

3-I397I94 

1.04950 

.952838 

20 

50 

.306249 

3-2653I49 

.321707 

3.1084210 

1.05047 

-95I9S* 

IO 

18  o 

.309017 

3.2360680 

.324920 

3.0776835 

1.05146 

•951057 

72  o 

10 

.311782 

3.2073673 

.328139 

3.0474915 

1.05246 

-95OI54 

50 

20 

.3M545 

3.1791978 

.331364 

3.0178301 

1-05347 

.949243 

40 

30 

•317305 

3-i5I5453 

•334595 

2.9886850 

1.05449 

.948324 

30 

40 

.320062 

3-I243959 

•337833 

2.9600422 

L05552 

•947397 

20 

5° 

.322816 

3.0977363 

.341077 

2.9318885 

I-05657 

.946462 

10 

IQ  0 

•325568 

3.0715535 

.344328 

2.9042109 

1.05762 

•9455*9 

71  o 

10 

.328317 

3.0458352 

.347585 

2.8769970 

1.05869 

.944568 

5° 

20 

•33Io63 

3.0205693 

.350848 

2.8502349 

1.05976 

.943609 

40 

30 

•333807 

2-9957443 

-354II9 

2.8239129 

1.06085 

.942641 

30 

40 

.336547 

2.9713490 

'357396 

2.7980198 

I.06I95 

.941666 

20 

50 

.339285 

2.9473724 

.360680 

2.7725448 

1.06306 

.940684 

10 

Cosine. 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

Angle. 

io6 


MATHEMATICAL   TABLES. 


Angle. 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

20°  0 

.342020 

2.9238044 

.363970 

2.7474774 

1.06418 

.939693 

70°  o' 

10 

•344752 

2.9006346 

.367268 

2.7228076 

1.06531 

.938694 

50 

2O 

.347481 

2.8778532 

.370573 

2.6985254 

.06645 

.937687 

40 

3° 

.350207 

2.8554510 

.373885 

2.6746215 

.06761 

.936672 

30 

40 

.352931 

2.8334185 

•377204 

2.6510867 

.06878 

•935650 

20 

5° 

.355651 

2.8II747I 

.380530 

2.6279I2I 

.06995 

•934619 

10 

21  0 

.358368 

2.7904281 

•383864 

2.6050891 

.07115 

•933580 

69  o 

10 

.361082 

2.7694532 

.387205 

2.5826094 

•07235 

•932534 

50 

20 

•363793 

2.7488144 

.390554 

2.5604649 

•07356 

.931480 

40 

3° 

.366501 

2.7285038 

•3939II 

2.5386479 

.07479 

.930418 

30 

40 

.369206 

2.7085139 

.397275 

2.5171507 

I.O7602 

•929348 

20 

5° 

.371908 

2.6888374 

.400647 

2.4959661 

1.07727 

.928270 

IO 

22  0 

.374607 

2.6694672 

.404026 

2.4750869 

I-07853 

.927184 

68  o 

10 

.377302 

2.6503962 

.407414 

2.4545061 

1.07981 

.926090 

5° 

20 

•379994 

2.6316180 

.410810 

2.4342172 

I.08IO9 

.924989 

40 

3° 

.382683 

2.6131259 

.414214 

2.4142136 

1.08239 

.923880 

30 

40 

•385369 

2.5949137 

.417626 

2.3944889 

1.08370 

.922762 

20 

50 

.388052 

2.5769753 

.421046 

2.3750372 

1.08503 

.921638 

IO 

23  o 

•390731 

2-5593°47 

.424475 

2.3558524 

1.08636 

.920505 

67  o 

10 

•393407 

2.5418961 

.427912 

2.3369287 

1.08771 

.919364 

50 

20 

.396080 

2.5247440 

-43I358 

2.3182606 

1.08907 

.918216 

40 

30 

.398749 

2.5078428 

.434812 

2.2998425 

1.09044 

.917060 

30 

40 

.401415 

2.4911874 

.438276 

2.2816693 

1.09183 

.915896 

20 

50 

.404078 

2.4747726 

.441748 

2.2637357 

1.09323 

.914725 

10 

24  o 

•406737 

2.4585933 

•445229 

2.2460368 

1.09464 

•913545 

66  o 

10 

.409392 

2.4426448 

.448719 

2.2285676 

1.09606 

.912358 

5o 

20 

.412045 

2.4269222 

.452218 

2.2II3234 

1.09750 

.911164 

40 

30 

.414693 

2.4II42IO 

.455726 

2.1942997 

1.09895 

.909961 

3° 

40 

•417338 

2.3961367 

.459244 

2.1774920 

I.IOO4I 

.908751 

20 

5° 

.419980 

2.3810650 

.462771 

2.1608958 

I.IOI89 

•907533 

10 

25  o 

.422618 

2.3662016 

.466308 

2.1445069 

1.10338 

.906308 

65  o 

10 

.425253 

2-35x5424 

.469854 

2.1283213 

I.I0488 

•905075 

5° 

20 

.427884 

2-3370833 

.473410 

2.1123348 

1.10640 

•903834 

40 

30 

.4305  " 

2.3228205 

•476976 

2.0965436 

I.I0793 

.902585 

3° 

40 

•433*35 

2.3087501 

.480551 

2.0809438 

I.I0947 

.901329 

20 

50 

•435755 

2.2948685 

.484137 

2.0655318 

I.III03 

.900065 

10 

26   0 

•438371 

2.2811720 

.487733 

2.0503038 

I.II260 

.898794 

64  o 

10 

.440984 

2.2676571 

.49J339 

2.0352565 

I.H4I9 

.897515 

50 

20 

•443593 

2.2543204 

•494955 

2.0203862 

I.U579 

.896229 

40 

30 

.446198 

2.2411585 

.498582 

2.0056897 

I.II740 

.894934 

30 

40 

.448799 

2.2281681 

.502219 

I.99II637 

I.II903 

.893633 

20 

5° 

•45J397 

2.2153460 

.505867 

1.9768050 

I.I2067 

•892323 

10 

Cosine. 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

Angle. 

SINES,   COSINES,  &C.    OF  ANGLES. 


107 


Angle. 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

27°  o 

•45399° 

2.2026893 

•509525 

1.9626105 

I.I2233 

.891007 

63°  o' 

10 

.456580 

2.1901947 

•5I3I95 

1.9485772 

I.I240O 

.889682 

50 

20 

.459166 

2.1778595 

.516876 

1.9347020 

I.I2568 

•888350 

40 

3° 

.461749 

2.1656806 

.520567 

1.9209821 

I.I2738 

.887011 

3° 

40 

.464327 

2.1536553 

.524270 

1.9074147 

I.I29IO 

.885664 

20 

50 

.466901 

2.1417808 

.527984 

1.8939971 

1.13083 

.884309 

10 

28  0 

.469472 

2.1300545 

.531709 

1.8807265 

LI3257 

.882948 

62  0 

10 

.472038 

2.1184737 

•535547 

1.8676003 

LI3433 

.881578 

5° 

20 

.474600 

2.1070359 

.539195 

1.8546159 

I.I36IO 

.880201 

40 

3° 

•477159 

2.0957385 

•542956 

1.8417409 

I.I3789 

.878817 

30 

40 

•479713 

2.0845792 

.546728 

1.8290628 

I.I3970 

.877425 

20 

50 

.482263 

2.0735556 

•550515 

1.8164892 

I.I4I52 

.876026 

10 

2Q  0 

.484810 

2.0626653 

.554309 

1.8040478 

J-I4335 

.874620 

61  o 

10 

•487352 

2.0519061 

.558118 

1.7917362 

1.14521 

.873206 

50 

20 

.489890 

2.0412757 

•561939 

L7795524 

1.14707 

.871784 

40 

30 

.492424 

2.0307720 

.565773 

1.7674940 

1.14896 

.870356 

3° 

40 

•494953 

2.0203929 

.569619 

L7555590 

1.15085 

.868920 

20 

5° 

•497479 

2.0101362 

.573478 

1-7437453 

i.i5277» 

.867476 

10 

30  o 

.500000 

2.00OOOOO 

.577350 

1.7320508 

1.15470 

.866025 

60  0 

10 

•5°25I7 

1.9899822 

•581235 

1.7204736 

1.15665 

.864567 

5° 

20 

•5°5°30 

1.9800810 

.585134 

1.7090116 

1.15861 

.863102 

40 

30 

•507538 

1.9702944 

•589045 

1.6976631 

1.16059 

.861629 

30 

40 

.510043 

1.9606206 

.592970 

1.6864261 

1.16259 

.860149 

20 

50 

•5*2543 

I-95I°577 

.596908 

1.6752988 

1.16460 

.858662 

10 

31  o 

•515038 

1.9416040 

.600861 

1.6642795 

1.16663 

.857167 

59  o 

IO 

•5^529 

1.9322578 

.604827 

1-6533663 

1.16868 

.855665 

5° 

20 

.520016 

1.9230173 

.608807 

1.6425576 

1.17075 

.854156 

40 

30 

•522499 

1.9138809 

.612801 

1.6318517 

1.17283 

.852640 

30 

40 

•524977 

1.9048469 

.616809 

1.6212469 

I.I7493 

.85IH7 

20 

5° 

•52745° 

1.8959138 

.620832 

1.6107417 

1.17704 

.849586 

IO 

32  o 

.529919 

1.8870799 

.624869 

1.6003345 

1.17918 

.848048 

58  o 

10 

•532384 

1.8783438 

.628921 

1.5900238 

1-18133 

.846503 

50 

20 

.534844 

1.8697040 

.632988 

1.5798079 

1.18350 

.844951 

40 

30 

.5373oo 

1.8611590 

.637079 

1.5696856 

1.18569 

.843391 

30 

40 

•539751 

1.8527073 

.641167 

I-5596552 

1.18790 

.841825 

20 

50 

•542197 

1.8443476 

.645280 

I-5497i55 

1.19012 

.840251 

IO 

33  o 

544639 

1.8360785 

.649408 

1.5398650 

1.19236 

.838671 

57  o 

10 

547076 

1.8278985 

•653531 

1.5301025 

1.19463 

.837083 

5° 

20 

549509 

1.8198065 

.657710 

1.5204261 

1.19691 

.835488 

40 

30 

551937 

i.  8118010 

.661886 

1.5108352 

1.19920 

.833886 

30 

40 

55436o 

1.8038809 

.666077 

1.5013282 

1.20152  1.832277 

20 

5° 

556779 

1.7960449 

.670285 

1.4919039 

1.20386  .830661 

10 

Cosine. 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

Angle. 

io8 


MATHEMATICAL   TABLES. 


Angle. 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

34°  o' 

•559J93 

1.7882916 

.674509 

.4825610 

1.20622 

.829038 

56°  o' 

10 

.561602 

1.7806201 

.678749 

.4732983 

1.20859 

.827407 

50 

20 

.564007 

1.7730290 

.683007 

.4641147 

I.2I099 

•825770 

40 

3° 

.566406 

I-7655I73 

.687281 

.4550090 

I.2I34I 

.824126 

30 

40 

.568801 

1.7580837 

.691573 

.4459801 

1.21584 

•822475 

20 

50 

•57H91 

1.7507273 

.695881 

.4370268 

1.21830 

.820817 

10 

35  o 

•573576 

1.7434468 

.700208 

.4281480 

1.22077 

.819152 

55  o 

10 

•575957 

1.7362413 

.704552 

.4193427 

1.22327 

.817480 

5° 

20 

•578332 

1.7291096 

.708913 

.4106098 

1.22579 

.815801 

40 

30 

.580703 

1.7220508 

•713293 

.4019483 

1.22833 

.814116 

30 

40 

.583069 

1.7150639 

.717691 

•3933571 

1.23089 

.812423 

20 

5° 

•585429 

1.7081478 

.722108 

.3848355 

1-23347 

.810723 

10 

36  o 

•587785 

1.7013016 

•726543 

•3763810 

1.23607 

.809017 

54  o 

IO 

.590136 

1.6945244 

.730996 

•3679959 

1.23869 

.807304 

5o 

20 

.592482 

1.6878151 

•735469 

•3596764 

1.24134 

•805584 

40 

3° 

•594823 

1.6811730 

.739961 

.3514224 

I.2440O 

•803857 

3° 

40 

•597159 

1.6745970 

•744472 

.3432331 

1.24669 

.802123 

20 

So 

•599489 

1.6680864 

.749003 

•335!075 

1.24940 

.800383 

10 

37  o 

.601815 

1.6616401 

•753554 

1.3270448 

I.252I4 

.798636 

53  o 

10 

.604136 

I-6552575 

•758125 

1.3190441 

1.25489 

.796882 

5° 

20 

.606451 

1.6489376 

.762716 

1.3111046 

1.25767 

.795121 

40 

30 

.608761 

1.6426796 

.767627 

1.3032254 

1.26047 

•793353 

30 

40 

.611067 

1.6364828 

•771959 

1.2954057 

1.26330 

•79*579 

20 

5° 

.613367 

1.6303462 

.776612 

1.2876447 

1.26615 

.789798 

IO 

38  o 

.615661 

1.6242692 

.781286 

1.2799416 

1.26902 

.788011 

52  o 

10 

.617951 

1.6182510 

.785981 

1.2722957 

I.27I9I 

.786217 

5° 

20 

.620235 

1.6122908 

.790698 

1.2647062 

1.27483 

.784416 

40 

30 

.622515 

1.6063879 

•795436 

1.2571723 

1.27778 

.782608 

30 

40 

.624789 

1.6005416 

.800196 

1.2496933 

1.28075 

.780794 

20 

50 

.627057 

I-5947511 

.804080 

1.2422685 

1.28374 

.778973 

10 

39  o 

.629320 

1.5890157 

.809784 

1.2348972 

1.28676 

.777146 

51  o 

10 

•631578 

1-5833318 

.814612 

1.2275786 

1.28980 

•775312 

5° 

20 

•633831 

1.5777077 

.819463 

1.2203121 

1.29287 

•773472 

40 

3° 

.636078 

I-572I337 

.824336 

1.2130970 

1.29597 

.771625 

30 

40 

.638320 

1.5666121 

•829234 

1.2059327 

1.29909 

.769771 

20 

50 

•640557 

1.5611424 

.834155 

1.1988184 

1.30223 

.767911 

10 

40  o 

.642788 

I-5557238 

.839100 

1.1917536 

L3054I 

.766044 

50  o 

10 

.645013 

I.5503558 

.844069 

1.1847376 

1.30861 

.764171 

5° 

20 

•647233 

i-5450378 

.849062 

1.1777698 

I.3II83 

.762292 

40 

3° 

.649448 

i-539769o 

.854081 

1.1708496 

L3I509 

.760406 

3° 

40 

•651657 

I-534549I 

•859124 

1.1639763 

1.31837 

•7585J4 

20 

50 

.653861 

I-5293773 

.864193 

1.1571495 

I.32I68 

•756615 

10 

Cosine. 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

Angle. 

SINES,   COSINES,  &C.   OF  ANGLES. 


I09 


Angle. 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

;4i°  o' 

.656059 

I.524253I 

.869287 

1.1503684 

1.32501 

•754710 

49°  o' 

10 

.658252 

I-5I9i759 

.874407 

1.1436326 

1.32838 

.752798 

So 

20 

.660439 

1.5141452 

.879553 

1.1369414 

L33I77 

.750880 

40 

3° 

.662620 

1.5091605 

.884725 

1.1302944 

I-335I9 

.748956 

30 

40 

.664796 

1.5042211 

.889924 

1.1236909 

1.33864 

.747025 

20 

5° 

.666966 

1.4993267 

•^S'SI 

I.II7I305 

1.34212 

.745088 

10 

42  o 

.669131 

1.4944765 

.900404 

I.llo6l25 

I.34563 

.743U5 

48  o 

10 

.671289 

1.4896703 

.905685 

1.1041365 

i.349J7 

•74II95 

50 

20 

•673443 

1.4849073 

.910994 

1.0977020 

i.35274 

•739239 

40 

30 

•675590 

1.4801872 

.916331 

1.0913085 

I.35634 

•737277 

30 

40 

.677732 

I-4755°95 

.921697 

1.0849554 

1-35997 

•735309 

20 

5° 

.679868 

1.4708736 

.927091 

1.0786423 

1.36363 

•733335 

10 

43  o 

.681998 

1.4662792 

•932515 

1.0723687 

I.36733 

•731354 

47  o 

10 

.684123 

1.4617257 

.937968 

1.0661341 

1.37105 

•729367 

So 

20 

.686242 

1.4572127 

•94345  i 

1.0599381 

1.37481 

•727374 

40 

30 

•688355 

I-4527397 

.948965 

1.0537801 

1.37860 

•725374 

30 

40 

.690462 

1.4483063 

•9545°8 

1.0476598 

1.38242 

•723369 

20 

5° 

.692563 

1.4439120 

.960083 

1.0415767 

1.38628 

•721357 

10 

44  o 

.694658 

L4395565 

.965689 

1.0355303 

1.39016 

.719340 

46  o 

10 

.696748 

J-4352393 

.971326 

1.0295203 

1.39409 

.717316 

50 

20 

.698832 

1.4309602 

•976996 

1.0235461 

1.39804 

.715286 

40 

30 

.700909 

1.4267182 

.982697 

1.0176074 

1.40203 

•7I3251 

30 

40 

.702981 

1.4225134 

.988432 

I.OII7088 

1.40606 

.711209 

20 

50 

.705047 

1.4183454 

.994199 

1.0058348 

1.41012 

.709161 

10 

45  o 

.707107 

1.4142136 

I.OOOOOO 

I.OOOOOOO 

1.41421 

.707107 

45  o 

Cosine. 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

Angle. 

no 


MATHEMATICAL   TABLES. 


TABLE    No.    IX.— LOGARITHMIC    SINES,    COSINES,    TANGENTS, 
AND    COTANGENTS    OF   ANGLES   FROM  o°  TO  90°. 

ADVANCING  BY  10',  OR  ONE-SIXTH  OF  A  DEGREE. 


Angle. 

Sine. 

Tangent. 

Cotangent. 

Cosine. 

0° 

O.OOOOOO 

O.OOOOOO 

Infinite. 

IO.OOOOOO 

90° 

10' 

7.463726 

7.463727 

12.536273 

9.999998 

50' 

20 

7.764754 

7.764761 

12.235239 

9-999993 

40 

30 

7.940842 

7.940858 

12.059142 

9.999983 

30 

40 

8.065776 

8.065806 

11.934194 

9.999971 

20 

50 

8.l6268l 

8.162727 

11.837273 

9-999954 

10 

I 

8.241855 

8.241921 

11.758079 

9-999934 

89 

10 

8.308794 

8.308884 

11.691116 

9.999910 

50 

20 

8.366777 

8.366895 

11.633105 

9.999882 

40 

30 

8.417919 

8.418068 

11.581932 

9.999851 

30 

40 

8.463665 

8.463849 

11.536151 

9.999816 

20 

'  50 

8.505045 

8.505267 

n-494733 

9-999778 

IO 

2 

8.542819 

8.543084 

11.456916 

9-999735 

88 

10 

8.577566 

8.577877 

11.422123 

9.999689 

5° 

20 

8.609734 

8.610094 

11.389906 

9.999640 

40 

30 

8.639680 

8.640093 

ii-359907 

9.999586 

3° 

40 

8.667689 

8.668l6o 

11.331840 

9-999529 

20 

50 

8.693998 

8.694529 

11.305471 

9.999469 

10 

3 

8.718800 

8.719396 

11.280604 

9-999404 

87 

10 

8.742259 

8.742922 

11.257078 

9-999336 

5° 

20 

8.764511 

8.765246 

11-234754 

9.999265 

40 

30 

8.785675 

8.786486 

11.213514 

9.999189 

3° 

40 

8.805852 

8.806742 

11.193258 

9.999110 

20 

50 

8.825130 

8.826103 

11.173897 

9.999027 

10 

4 

8.843585 

8.844644 

ii-i55356 

9.998941 

86 

10 

8.861283 

8.862433 

11.137567 

9.998851 

5° 

20 

8.878285 

8.879529 

11.120471 

9-998757 

40 

30 

8.894643 

8.895984 

11.104016 

9.998659 

30 

40 

8.910404 

8.911846 

11.088154 

9-998558 

20 

5° 

8.925609 

8.927156 

11.072844 

9.998453 

10 

5 

8.940296 

8.941952 

11.058048 

9.998344 

85 

10 

8-954499 

8.956267 

n-043733 

9.998232 

50 

20 

8.968249 

8.970133 

11.029867 

9.998116 

40 

30 

8.98i573 

8.983577 

11.016423 

9.997996 

30 

40 

8.994497 

8.996624 

11.003376 

9.997872 

20 

5° 

9.007044 

9.009298 

10.990702 

9-997745 

10 

Cosine. 

Cotangent. 

Tangent. 

Sine. 

Angle. 

LOGARITHMIC   SINES,   TANGENTS,  &c. 


Ill 


Angle. 

Sine. 

Tangent. 

Cotangent. 

Cosine. 

6° 

9.019235 

9.021620 

10.978380 

9.997614 

84° 

10' 

9.031089 

9.033609 

10.966391 

9.997480 

50' 

20 

9.042625 

9.045284 

10.954716 

9.997341 

40 

30 

9.053859 

9.056659 

10.943341 

9.997199 

30 

40 

9.064806 

9.067752 

10.932248 

9.997053 

20 

5° 

9.075480 

9.078576 

10.921424 

9.996904 

10 

7 

9.085894 

9.089144 

10.910856 

9.996751 

83 

10 

9.096062 

9.099468 

10.900532 

9.996594 

50 

20 

9.105992 

9«I09559 

10.890441 

9.996433 

40 

30 

9.115698 

9.119429 

10.880571 

9.996269 

30 

40 

9.125187 

9.129087 

10.870913 

9.996100 

2O 

5° 

9.134470 

9.138542 

10.861458 

9.995928 

10 

8 

9-143555 

9.147803 

10.852197 

9-995753 

82 

10 

9-I5245I 

9.156877 

10.843123 

9-995573 

50 

20 

9.161164 

9-!65774 

10.834226 

9-995390 

40 

3° 

9.169702 

9.174499 

10.825501 

9-995203 

3° 

40 

9.178072 

9.183059 

10.816941 

9-995OI3 

20 

5o 

9.186280 

9.191462 

10.808538 

9.994818 

10 

9 

9.194332 

9.199713 

10.800287 

9.994620 

81 

10 

9.202234 

9.207817 

10.792183 

9.994418 

5° 

20 

9.209992 

9.215780 

10.784220 

9.994212 

40 

30 

9.217609 

9.223607 

10.776393 

9.994003 

30 

40 

9.225092 

9.231302 

10.768698 

9-993789 

20 

50 

9.232444 

9.238872 

10.761128 

9-993572 

IO 

10 

9.239670 

9.246319 

10.753681 

9-993351 

80 

IO 

9.246775 

9.253648 

10.746352 

9.993127 

5° 

20 

9.253761 

9.260863 

10.739137 

9.992898 

40 

30 

9.260633 

9.267967 

10.732033 

9.992666 

30 

40 

9-267395 

9.274964 

10.725036 

9.992430 

20 

5° 

9.274049 

9.281858 

10.718142 

9.992190 

IO 

II 

9.280599 

9.288652 

10.711348 

9.991947 

79 

IO 

9.287048 

9-295349 

10.704651 

9.991699 

50 

20 

9.293399 

9.3oi95i 

10.698049 

9.991448 

40 

30 

9.299655 

9.308463 

10.691537 

9.991193 

30 

40 

9.305819 

9.314885 

10.685115 

9.990934 

20 

5° 

9.3ii893 

9.321222 

10.678778 

9.990671 

IO 

12 

9.317879 

9.327475 

10.672525 

9.990404 

78 

10 

9.323780 

9.333646 

10.666354 

9.990134 

5° 

20 

9.329599 

9-339739 

IO.66026I 

9.989860 

40 

30 

9-335337 

9-345755 

10.654245 

9.989582 

30 

40 

9.340996 

9-35l697 

10.648303 

9.989300 

20 

5° 

9.346779 

9.357566 

10.642434 

9.989014 

IO 

Cosine. 

Cotangent. 

Tangent. 

Sine. 

Angle. 

112 


MATHEMATICAL  TABLES. 


Angle. 

Sine. 

Tangent. 

Cotangent. 

Cosine. 

13° 

9.352088 

9.363364 

10.636636 

9.988724 

77° 

10' 

9-357524 

9.369094 

10.630906 

9.988430 

20 

9.362889 

9.374756 

10.625244 

9.988133 

40 

30 

9.368185 

9.380354 

10.619646 

9.987832 

30 

40 

9-3734I4 

9.385888 

I0.6l4II2 

9.987526 

20 

5° 

9-378577 

9.391360 

10.608640 

9.987217 

IO 

14 

9»383675 

9.396771 

10.603229 

9.986904 

76 

10 

9-3887II 

9.402124 

10.597876 

9.986587 

50 

20 

9.393685 

9.407419 

10.592581 

9.986266 

40 

30 

9.398600 

9.412658 

10.587342 

9.985942 

30 

40 

9-403455 

9.417842 

10.582158 

9.985613 

20 

5° 

9.408254 

9.422974 

10.577026 

9.985280 

IO 

15 

9.412996 

9.428052 

10.571948 

9.984944 

75 

IO 

9.417684 

9.433080 

10.566920 

9.984603 

50 

20 

9.422318 

9.438059 

10.561941 

9.984259 

40 

30 

9.426899 

9.442988 

10.557012 

9.983911 

30 

40 

9.431429 

9.447870 

10.552130 

9.983558 

20 

5° 

9.435908 

9.452706 

10.547294 

9.983202 

IO 

16 

9-440338 

9.457496 

10.542504 

9.982842 

74 

IO 

9.444720 

9.462242 

10.537758 

9.982477 

5° 

20 

9.449054 

9.466945 

10.533055 

9.982109 

40 

30 

9-453342 

9.471605 

10.528395 

9.981737 

30 

40 

9.457584 

9.476223 

10.523777 

9.981361 

20 

5° 

9.461782 

9.480801 

10.519199 

9.980981 

IO 

17 

9.465935 

9.485339 

10.514661 

9.980596 

73 

10 

9.470046 

9.489838 

10.510162 

9.980208 

50 

20 

9.474II5 

9.494299 

10.505701 

9.979816 

40 

30 

9.478142 

9.498722 

10.501278 

9.979420 

30 

40 

9.482128 

9.503109 

10.496891 

9.979019 

20 

50 

9.486075 

9.507460 

10.492540 

9.978615 

10 

18 

9.489982 

9.5II776 

10.488224 

9.978206 

72 

IO 

9.493851 

9.516057 

16.483943 

9-977794 

5° 

20 

9.497682 

9-520305 

10.479695 

9-977377 

40 

3° 

9.501476 

9.524520 

10.475480 

9-976957 

30 

40 

9.505234 

9.528702 

10.471298 

20 

50 

9.508956 

9.532853 

10.467147 

9.976103 

IO 

19 

9.512642 

9.536972 

10.463028 

9.975670 

71 

10 

9.516294 

9.541061 

I0.458939 

9.975233 

50 

20 

9.5I99II 

9-545II9 

10.454881 

9.974792 

40 

30 

9.523495 

9.549M9 

10.450851 

9-974347 

30 

40 

9.527046 

9.553I49 

10.446851 

9.973897 

20 

50 

9.530565 

10.442879 

9-973444 

10 

Cosine. 

Cotangent. 

Tangent. 

Sine. 

Angle. 

LOGARITHMIC   SINES,   TANGENTS,  &C. 


Angle. 

Sine. 

Tangent. 

Cotangent. 

Cosine. 

20° 

9-534052 

9.561066 

10.438934 

9.972986 

70° 

10' 

9-537507 

9.564983 

10.435017 

9.972524 

5O/ 

20 

9-54093I 

9.568873 

10.431127 

9.972058 

40 

30 

9-544325 

9.572738 

10.427262 

9.971588 

30 

40 

9.547689 

9.576576 

10.423424 

9.97III3 

20 

50 

9.551024 

9.580389 

10.419611 

9.970635 

10 

21 

9-554329 

9.584177 

10.415823 

9.970152 

69 

10 

9.557606 

9.587941 

10.412059 

9.969665 

5° 

2O 

9-560855 

9.591681 

10.408319 

9.969173 

40 

30 

9-564075 

9-595398 

10.404602 

9.968678 

30 

40 

9.567269 

9.599091 

10.400909 

9.968178 

20 

50 

9-570435 

9.602761 

10.397239 

9.967674 

1C 

22 

9-573575 

9.606410 

I0-39359o 

9.967166 

68 

IO 

9-576689 

9.610036 

10.389964 

9.966653 

5° 

2O 

9-579777 

9.613641 

10.386359 

9.966136 

40 

3° 

9.582840 

9.617224 

10.382776 

9.965615 

30 

40 

9.585877 

9.620787 

10.379213 

9.965090 

20 

50 

9.588890 

9.624330 

10.375670 

9.964560 

10 

23 

9.591878 

9.627852 

10.372148 

9.964026 

67 

IO 

9.594842 

9-63I355 

10.368645 

9.963488 

5° 

20 

9.597783 

9.634838 

10.365162 

9.962945 

40 

30 

9.600700 

9.638302 

10.361698 

9.962398 

30 

40 

9-603594 

9.641747 

10.358253 

9.961846 

20 

50 

9.606465 

9.645174 

10.354826 

9.961290 

IO 

24 

9.609313 

9.648583 

10.351417 

9.960730 

66 

10 

9.612140 

9.651974 

10.348026 

9.960165 

5° 

20 

9.614944 

9.655348 

10.344652 

9-959596 

40 

30 

9.617727 

9.658704 

10.341296 

9-959023 

30 

40 

9.620488 

9.662043 

10-337957 

9-958445 

20 

50 

9.623229 

9-665366 

10.334634 

9.957863 

10 

25 

9.625948 

9.668673 

10.331328 

9.957276 

65 

10 

9.628647 

9.671963 

10.328037 

9.956684 

5° 

20 

9.631326 

9.675237 

10.324763 

9.956089 

40 

30 

9-633984 

9.678496 

10.321504 

9-955488 

3° 

40 

9.636623 

9.681740 

10.318260 

9.954883 

20 

50 

9.639242 

9.684968 

10.315032 

9.954274 

10 

26 

9.641842 

9.688l82 

10.311818 

9.953660 

64 

10 

9-644423 

9.691381 

10.308619 

9.953042 

5° 

20 

9.646984 

9.694566 

10.305434 

9.952419 

40 

30 

9.649527 

9.697736 

10.302264 

9-95I79I 

3° 

40 

9.652052 

9.700893 

10.299107 

9.95JI59 

20 

50 

9-654558 

9.704036 

10.295964 

9.950522 

10 

Cosine. 

Cotangent. 

Tangent. 

Sine. 

Angle. 

MATHEMATICAL   TABLES. 


Angle. 

Sine. 

Tangent. 

Cotangent. 

Cosine. 

27° 

9.657047 

9.707166 

10.292834 

9.949881 

63° 

10' 

9-65951? 

9.710282 

10.289718 

9-949235 

50/ 

20 

9.661970 

9.713386 

I0.2866I4 

9.948584 

40 

30 

9.664406 

9.716477 

10.283523 

9.947929 

30 

40 

9.666824 

9-7I9555 

10.280445 

9.947269 

20 

5° 

9.669225 

9.722621 

10.277379 

9.946604 

10 

28 

9.671609 

9.725674 

10.274326 

9-945935 

62 

10 

9.673977 

9.728716 

10.271284 

9.945261 

5° 

20 

9.676328 

9.731746 

10.268254 

9.944582 

40 

30 

9.678663 

9-734764 

10.265236 

9.943899 

3° 

40 

9.680982 

9-737771 

IO.262229 

9.943210 

20 

50 

9.683284 

9.740767 

10.259233 

9.942517 

10 

29 

9-685571 

9-743752 

10.256248 

9.941819 

61 

10 

9.687843 

9.746726 

10.253274 

9.941117 

5o 

20 

9.690098 

9.749689 

IO.2503II 

9.940409 

40 

3° 

9.692339 

9.752642 

10.247358 

9.939697 

3° 

40 

9.694564 

9-755585 

10.244415 

9.938980 

20 

5° 

9.696775 

9-7585I7 

10.241483 

9.938258 

10 

30 

9.698970 

9-76I439 

10.238561 

9-937531 

60 

10 

9.701151 

9-764352 

10.235648 

9.936799 

50 

20 

9-7033I7 

9-767255 

10.232745 

9.936062 

40 

30 

9.705469 

9.770148 

10.229852 

9-935320 

30 

40 

9.707606 

9-773033 

10.226967 

9-934574 

20 

5° 

9.709730 

9.775908 

IO.224092 

9.933822 

10 

31 

9.711839 

9.778774 

IO.22I226 

9.933066 

59 

10 

9-7I3935 

9.781631 

10.218369 

9.932304 

5° 

20 

9.716017 

9.784479 

10.215521 

9-93I537 

40 

3° 

9.718085 

9.787319 

I0.2I268l 

9.930766 

30 

40 

9.720140 

9.790151 

10.209849 

9.929989 

20 

5° 

9.722181 

9.792974 

I0.2O7026 

9.929207 

10 

32 

9.724210 

9-7957^9 

10.204211 

9.928420 

58 

10 

9.726225 

9.798596 

IO.20I404 

9.927629 

5° 

20 

9.728227 

9.801396 

10.198604 

9.926831 

40 

30 

9.730217 

9.804187 

10.195813 

9.926029 

3° 

40 

9-732I93 

9.806971 

10.193029 

9.925222 

20 

50 

9-734I57 

9.809748 

10.190252 

9.924409 

10 

33 

9.736109 

9.812517 

10.187483 

9-923591 

57 

10 

9.738048 

9.815280 

10.184720 

9.922768 

50 

20 

9-739975 

9.818035 

10.181965 

9.921940 

40 

30 

9.741889 

9.820783 

10.179217 

9.921107 

30 

40 

9-743792 

9.823524 

10.176476 

9.920268 

20 

50 

9-745683 

9.826259 

10.173741 

9.919424 

10 

Cosine. 

Cotangent. 

Tangent. 

Sine. 

Angle. 

LOGARITHMIC   SINES,   TANGENTS,  &C. 


Angle. 

Sine. 

Tangent. 

Cotangent. 

Cosine. 

34° 

9.747562 

9.828987 

IO.I7IOI3 

9.918574 

56° 

10' 

9.749429 

9.831709 

10.168291 

9.917719 

5o' 

20 

9.751284 

9-834425 

10.165575 

9.916859 

40 

3° 

9-753128 

9-837I34 

10.162866 

9-9I5994 

30 

40 

9.754960 

9.839838 

I0.l6oi62 

9-9I5I23 

20 

50 

9.756782 

9-842535 

10.157465 

9.914246 

10 

35 

9-75859I 

9.845227 

10.154773 

9-9I3365 

55 

10 

9.760390 

9.847913 

10.152087 

9.912477 

5° 

20 

9.762177 

9-850593 

10.149407 

9.911584 

40 

3° 

9-763954 

9.853268 

10.146732 

9.910686 

30 

40 

9.765720 

9-855938 

10.144062 

9.909782 

20 

5° 

9-767475 

9.858602 

10.141398 

9.908873 

10 

36 

9.769219 

9.86l26l 

10.138739 

9.907958 

54 

10 

9.770952 

9.863915 

10.136085 

9.907037 

5° 

20 

9.772675 

9.866564 

10.133436 

9.906111 

40 

30 

9.774388 

9.869209 

10.130791 

9.905179 

30 

40 

9.776090 

9.871849 

IO.I28I5I 

9.904241 

20 

50 

9.777781 

9.874484 

10.125516 

9.903298 

IO 

37 

9-779463 

9.877114 

10.122886 

9.902349 

53 

10 

9.781134 

9.879741 

10.120259 

9.901394 

5° 

20 

9.782796 

9-882363 

10.117637 

9-900433 

40 

30 

9.784447 

9.884980 

10.115020 

9-899467 

30 

40 

9.786089 

9.887594 

IO.II24O6 

9.898494 

20 

50 

9.787720 

9.890204 

10.109796 

9.897516 

10 

38 

9.789342 

9.892810 

10.107190 

9.896532 

52 

10 

9-79°954 

9.895412 

10.104588 

9-895542 

5° 

20 

9-792557 

9.898010 

10.101990 

9.894546 

40 

30 

9.794150 

9.900605 

10.099395 

9-893344 

30 

40 

9-795733 

9.903197 

10.096803 

9-892536 

20 

50 

9.797307 

9-905785 

10.094215 

9.891523 

10 

39 

9.798872 

9.908369 

10.091631 

9.890503 

51 

10 

9.800427 

9.910951 

10.089049 

9-889477 

50 

20 

9.801973 

9-9!3529 

10.086471 

9.888444 

40 

3° 

9.803511 

9.916104 

10.083896 

9.887406 

30 

40 

9.805039 

9.918677 

10.081323 

9.886362 

20 

50 

9.806557 

9.921247 

10.078753 

9.885311 

IO   1 

40 

9.808067 

9.923814 

10.076186 

9.884254 

50 

IO 

9.809569 

9.926378 

10.073622 

9.883191 

5° 

20 

9.811061 

9.928940 

IO.07I060 

9.882I2I 

40 

30 

9.812544 

9.931499   10.068501 

9.881046 

30 

40 

9.814019 

9.934056   10.065944 

9.879963 

20 

50 

9.815485 

9.936611    10.063389 

9.878875 

IO 

Cosine. 

Cotangent.       Tangent. 

Sine. 

Angle. 

u6 


MATHEMATICAL   TABLES. 


Angle. 

Sine. 

Tangent. 

Cotangent. 

Cosine. 

41° 

9.816943 

9-939l63 

10.060837 

9.877780 

49° 

10' 

9.818392 

9-94I7I3 

10.058287 

9.876678 

5o' 

20 

9.819832 

9.944262 

10.055738 

9-87557l 

40 

30 

9.821265 

9.946808 

10.053192 

9.874456 

30 

40 

9.822688 

9-949353 

10.050647 

9-873335 

20 

50 

9.824104 

9.951896 

10.048104 

9.872208 

10 

42 

9.825511 

9-954437 

10.045563 

9.871073 

48 

IO 

9.826910 

9-956977 

10.043023 

9-869933 

5° 

20 

9.828301 

9-959516 

10.040484 

9-868785 

40 

3° 

9.829683 

9.962052 

10.037948 

9.867631 

3° 

40 

9.831058 

9.964588 

10.035412 

9.866470 

20 

50 

9.832425 

9.967123 

10.032877 

9.865302 

10 

43 

9-833783 

9.969656 

10.030344 

9.864127 

47 

IO 

9-835*34 

9.972188 

10.027812 

9.862946 

5o 

20 

9.836477 

9.974720 

10.025280 

9-861758 

40 

30 

9.837812 

9.977250 

10.022750 

9.860562 

30 

40 

9.839140 

9.979780 

IO.O2O22O 

9.859360 

20 

50 

9.840459 

9.982309 

10.017691 

9.858I5I 

IO 

44 

9.841771 

9.984837 

10.015163 

9-856934 

46 

IO 

9.843076 

9-987365 

10.012635 

9-8557II 

5° 

20 

9.844372 

9.989893 

10.010107 

9.854480 

40 

30 

9.845662 

9.992420 

10.007580 

9.853242 

30 

40 

9.846944 

9-994947 

10.005053 

9.851997 

20 

5° 

9.848218 

9-997473 

10.002527 

9-850745 

10 

45 

9.849485 

10.000000 

IO.OOOOOO 

9.849485 

45 

Cosine. 

Cotangent. 

Tangent. 

Sine. 

Angle. 

RHUMBS,   OR   POINTS  OF   THE  COMPASS. 


117 


TABLE    No.   X.— RHUMBS,  OR   POINTS   OF  THE    COMPASS. 


Points. 

Angles. 

NORTH. 

NORTH. 

SOUTH. 

SOUTH. 

I/ 

2°  48'  45" 

N^E 

N  34  W 

S^E 

S  "*•/  W 

YZ 

5  37  3o 

N  Yz  E 

N  YZ  w 

S  Yi  E 

s  -^  w 

3A 

8  26  15 

N  ^>A  E 

N  24  w 

s  24  E 

s  24  w 

i 

ii  15    o 

N  by  E 

N  by  w 

s  by  E 

s  by  w 

1*4 

i4     3  45 

N  by  E  ^4  E 

N  by  w  Y(.  w 

s  by  E  %  E 

s  by  w  ^  w 

i^ 

16  52  30 

N  by  E  YZ  E 

N  by  w  YZ  w 

s  by  E  YZ  E 

s  by  w  YZ  w 

I3A 

19  4i   15 

N  by  E  24  E 

N  by  w  24  w 

s  by  E  24  E 

s  by  w  24  w 

2 

22    30      0 

NNE 

NNW 

SSE 

ssw 

2/4 

25   18  45 

NNE  14  E 

NNW  14  W 

SSE  14  E 

ssw  14  w 

2/2 

28     7  30 

NNE  %  E 

NNW  YZ  w 

SSE  YZ  E 

ssw  %  w 

23/4 

30  56  15 

NNE  24  E 

NNW  24  W 

SSE  24  E 

ssw  24  w 

3 

33  45     o 

NE  by  N 

NW  by  N 

SE  by  s 

sw  by  s 

3% 

36  33  45 

NE^N 

NW  24  N 

SE  24  s 

sw24s 

$% 

39  22  30 

NE  YZ  N 

NW^N 

SE  %  S 

sw  Y^  s 

&A 

42   ii   15 

NE  ^4  N 

NW  %  N 

SE  Y*.  S 

sw  Yt  s 

4 

45     o     o 

NE 

NW 

SE 

sw 

4^4 

47  48  45 

NE  14  E 

NW  14  W 

SE  ^4  E 

sw  14  w 

45* 

5°  37  30 

NE  YZ  E 

NW  %  w 

SE  YZ  E 

sw  YZ  w 

4^4 

53  26  15 

NE  24  E 

NW  24  w 

SE24E 

sw24w 

5 

56  15     o 

NE  by  E 

NW  by  w 

SE  by  E 

sw  by  w 

5^4 

59     3  45 

ENE  24  N 

WNW  24  N 

ESE  24  S 

wsw  24  s 

5/^ 

61  52  30 

ENE  YZ  N 

WNW  %  N 

ESE  YZ  S 

wsw  YZ  s 

5^4 

64  41   15 

ENE  14  N 

WNW  ^4  N 

ESE  ^4  s 

wsw  ^4  s 

6 

67  30    o 

ENE 

WNW 

ESE 

wsw 

6^4 

70  18  45 

ENE  ^4  E 

WNW   %   W 

ESE  ^4  E 

wsw  14  w 

6^j 

73     7  30 

ENE  YZ  E 

WNW  YZ  w 

ESE  YZ  E 

wsw  YZ  w 

624 

75  56  15 

ENE  24  E 

WNW  24  w 

ESE  24  E 

wsw  24  w 

7 

78  45     o 

E  by  N 

w  by  N 

E  by  s 

w  by  s 

7^4 

8  1  33  45 

E24N 

w  24  N 

E  24  s 

w  24  s 

7^2 

84    22    30 

W  YZ  N 

E  YZ  s 

w  YZ  s 

7^C 

87     II     15 

E  14  N 

W  I^  N 

E  14  S 

w^  s 

8 

90    o    o 

EAST. 

WEST. 

EAST. 

WEST. 

u8 


MATHEMATICAL   TABLES. 


TABLE   No.  XL— RECIPROCALS   OF   NUMBERS 

FROM    I    TO    IOOO. 


No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

I 

I.OOOOOO 

40 

.O2500O 

79 

.012658 

118 

.008475 

2 

.500000 

41 

.024390 

80 

119 

.008403 

3 

4 

6 

7 
8 

9 
10 

•333333 

.250000 
.200000 
.166667 
.142857 
.125000 
.iimi 

.100000 

42 

43 

44 

45 
46 

47 
48 
49 

.023810 
.023256 
.022727 
.022222 
.021739 
.021277 
.020833 
.O2O408 

\J\J 

81 
82 

83 

84 

85 
86 

87 
88 

.012346 
.012195 
.012048 
.011905 
.011765 
.011628 
.011494 
.011364 

120 

121 
122 
123 
124 

I25 
126 
127 

•008333 
.008264 
.008197 
.008130 
.008065 
.OO8000 
.007937 
.007874 

ii 

.090909 

50 

.O2OOOO 

89 

.011236 

128 

.007813 

12 

•083333 

51 

.019608 

go 

.OIIIII 

129 

.007752 

13 
14 
15 

16 

0 

.076923 
.071429 

.066667 
.062500 

.058824 

52 
53 
54 
55 
56 

.019231 
.018868 
.018519 
.Ol8l82 
.017857 

91 
92 
93 
94 
95 

.010989 
.010870 
.010753 
.010638 
.010526 

130 

132 

J34 

.007692 
•007634 
.007576 
.007519 
.007463 

10 

.052632 

57 
58 

.017544 
.017241 

96 
97 

.010417 
.010309 

136 

.007407 
•007353 

20 

.050000 

59 

.016949 

98 

.OIO2O4 

137 

.007299 

21 

.047619 

60 

.016667 

99 

.010101 

138 

.007246 

22 

•045455 

61 

.016393 

100 

.010000 

139 

.007194 

23 

.043478 

62 

.Ol6l29 

101 

.009901 

140 

.007143 

24 

.041667 

63 

•OI5873 

102 

.009804 

141 

.007092 

25 

.040000 

64 

.015625 

103 

.009709 

142 

.007042 

26 

.038462 

65 

•015385 

I04 

.009615 

143 

.006993 

27 

•037037 

66 

.015152 

I05 

.009524 

144 

.006944 

28 

•035714 

67 

•014925 

106 

.009434 

J45 

.006897 

29 

•034483 

68 

.014706 

107 

.009346 

146 

.006849 

30 

31 
32 

•033333 

.032258 

.031250 

69 
70 

.014493 

.014286 
.014085 

108 
109 

no 

.009259 
.009174 

.009091 

147 

148 
149 

.006803 
.006757 
.006711 

33 

.030303 

72 

.013889 

in 

.009009 

150 

.006667 

34 

.029412 

73 

.013699 

112 

.008929 

151 

.006623 

35 

.028571 

74 

•OI35I4 

113 

.008850 

152 

.006579 

36 

.027778 

75 

•OI3333 

114 

.008772 

153 

.006536 

37 

.027027 

76 

.013158 

115 

.008696 

154 

.006494 

38 

.026316 

77 

.012987 

116 

.008021 

155 

.006452 

39 

.025641 

78 

.012821 

117 

•008547 

156 

.006410 

RECIPROCALS   OF   NUMBERS. 


No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

157 

.006369 

2O2 

.004950 

247 

.004049 

292 

•003425 

158 

.006329 

203 

.004926 

248 

.004032 

293 

.003413 

159 

.006289 

2O4 

.004902 

249 

.OO40l6 

294 

.003401 

1  60 

161 

162 

i63 

164 

.006250 
.OO62II 
.006173 
.006135 
.006098 

205 
2O6 
207 
208 
209 

.004878 
.004854 
.004831 
.004808 
.004785 

250 

251 

252 

253 
254 

.OO4OOO 
.003984 
.003968 
•003953 
.003937 

295 
296 
297 
298 
299 

.003390 
.003378 
.003367 
•003356 
•003344 

165 

.006061 

2IO 

.004762 

255 

.003922 

300 

•0°3333 

1  66 

.006024 

211 

.004739 

256 

.003906 

3OI 

.003322 

167 

.005988 

212 

.004717 

257 

.003891 

302 

.003311 

168 

.005952 

2I3 

.004695 

258 

.003876 

303 

•003301 

169 

.005917 

214 

.004673 

259 

.003861 

3°4 

.003289 

170 

171 
172 

173 

174 

.005882 
.005848 
.005814 
.005780 
.005747 

215 
216 
217 
218 
219 

.004651 
.004630 
.004608 
.004587 
.004566 

26O 
26l 
262 
263 

264 

.003846 
.003831 
.003817 
.003802 
.003788 

305 
306 

307 
308 

309 

.003279 
.003268 
.003257 
•003247 
.003236 

J75 

.005714 

220 

•004545 

265 

.003774 

310 

.003226 

176 

.005682 

221 

.004525 

266 

.003759 

311 

.003215 

177 

.005650 

222 

.004505 

267 

•003745 

3I2 

.003205 

178 

.005618 

223 

.004484 

268 

.003731 

3*3 

.003195 

179 

.005587 

224 

.004464 

269 

.003717 

3J4 

•003185 

180 
181 
182 

183 

184 

•005556 
.005225 
•005495 
.005464 
•005435 

225 
226 
227 
228 
229 

.004444 
.004425 
.004405 
.004386 
.004367 

27O 
271 

272 

273 
274 

.003704 
.003690 
.003676 
.003663 
.003650 

3i5 
316 

3i7 
3i8 
3i9 

.003175 
.003165 
•003155 
.003145 
•003135 

185 
1  86 
187 
188 
189 

.005405 
•005376 
.005348 
.005319 
.005291 

230 
231 
232 
233 

234 

.004348 
.004329 
.004310 
.004292 
.004274 

275 
276 
277 
278 
279 

.003636 
.003623 
.003610 
.003597 
.003584 

320 
321 

322 
323 
324 

.003125 
.003115 
.003106 
.003096 
.003086 

igo 

.005263 

235 

.004255 

280 

•003571 

325 

.003077 

191 

.005236 

236 

.004237 

28l 

•003559 

326 

.003067 

/-> 

192 

.005208 

237 

.004219 

282 

.003546 

327 

.003058 

*93 

.005181 

238 

.OO42O2 

283 

•003534 

328 

.003049 

194 

.005155 

239 

.004184 

284 

.003522 

329 

.003040 

*95 

196 
197 
198 
199 

.005128 
.005102 
.005076 
.005051 
.005025 

24O 

241 
242 

243 
244 

.004167 
.004149 
.004132 
.004115 
.004098 

285 
286 
287 
288 
289 

.003509 
.003497 
.003484 
.003472 
.003460 

330 
331 
332 

333 
334 

.003030 
.O03O2I 
.003012 
.003003 
.002994 

200 

.OO50OO 

245 

.004082 

2QO 

.003448 

335 

.002985 

201 

.004975 

246 

.004065 

29I 

.003436 

336 

.002976 

I2O 


MATHEMATICAL  TABLES. 


No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

337 

.002967 

382 

.0026l8 

427 

.002342 

472 

.002119 

338 

.002959 

383 

.002611 

428 

.002336 

473 

.OO2II4 

339 

.002950 

384 

.002604 

429 

.002331 

474 

.002IIO 

340 

.002941 

385 
386 

.002597 
.002591 

430 

.002326 

475 
476 

.002105 
.002IOI 

34i 

342 
343 

.002933 
.002924 
.002915 

387 
388 

•389 

.002584 
.002577 
.002571 

431 

432 

433 

.O0232O 
.002315 
.002309 

477 
478 
479 

.002096 
.002092 
.002088 

344 

.002907 

434 

.002304 

345 

.002899 

390 

.002564 

435 

.002299 

480 

.002083 

346 

.002890 

391 

.002558 

436 

.O02294 

481 

.OO2079 

347 

.002882 

392 

.002551 

437 

.002288 

482 

.OO2O75 

348 

.002874 

393 

.002545 

438 

.002283 

483 

.002070 

349 

.002865 

394 

.002538 

439 

.OO2278 

484 

.002066 

350 
35' 
352 
353 
354 

.002857 
.002849 
.002841 
.002833 
.002825 

395 
396 
397 
398 
399 

.002532 
.002525 
.002519 
.002513 
.002506 

440 

441 
442 
443 
444 

.OO2273 
.OO2268 
.OO2262 
.OO2257 
.O02252 

485 
486 
487 
488 
489 

.002062 
.002058 
.002053 
.OO2049 
.OO2045 

355 
356 

.002817 
.002809 

400 

401 

.OO25OO 
.002494 

445 
446 

.OO2247 
.OO2242 

490 

49  1 

.002041 
.002037 

357 

.OO28OI 

402 

.002488  !  447 

.002237 

492 

.002033 

358 

.002793 

403 

.002481 

44« 

.OO2232 

493 

.OO2O28 

359 

.002786 

404 

.002475 

449 

.002227 

494 

.OO2O24 

360 
361 

.002778 
.002770 

405 
406 

.002469 
.002463 

450 

45  1 

.OO2222 
.OO22I7 

495 
496 

.00202O 
.002016 

362 
363 

.002762 

.002755 

407 
408 

.002457 
.002451 

452 

453 

.OO22I2 
.OO2208 

497 
498 

.002012 
.OO2008 

364 

.002747 

409 

.002445 

454 

.O022O3 

499 

.002OO4 

365 
366 

367 
368 

369 

.002740 
.002732 
.002725 
.002717 
.002710 

410 

411 
412 

4i3 
414 

.002439 

.002433 
.002427 
.002421 
.002415 

455 
456 
457 
458 
459 

.002198 
.002193 
.002188 
.002183 
.002179 

500 

501 

502 

503 
5°4 

.00200O 
.001996 
.001992 
.001988 
.001984 

370 

.002703 

415 

.002410 

460 

.OO2I74 

5°5 

.001980 

37i 

.002695 

416 

.002407 

461 

.002169 

506 

.001976 

372 

.002688 

4i7 

.002398 

462 

.002165 

507 

.001972 

373 

.O0268l 

418 

.002392 

463 

.OO2l6o 

508 

.001969 

374 

.002674 

419 

.002387 

464 

.OO2I55 

509 

.001965 

375 
376 

.002667 
.00266O 

420 

.002381 

465 
466 

.002151 
.002146 

5Jo 

.001961 

377 

.002653 

421 

.002375 

467 

.002141 

5n 

.001957 

378 
379 

.002646 
.002639 

422 
423 
424 

.002370 
.002364 
.002358 

468 
469 

.002137 
.OO2I32 

5" 
513 

5*4 

.001953 
.001949 
.001946 

380 

.002632 

425 

•002353 

470 

.002128 

5i5 

.OOI942 

38i 

.002625 

426 

.002347 

471 

.002123 

5i6 

.001938 

RECIPROCALS   OF   NUMBERS. 


121 


No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

5*7 

.001934 

562 

.001779 

607 

.001647 

652 

.001534 

518 

.001931 

563 

.001776 

608 

.001645 

653 

.001531 

519 

.001927 

564 

.001773 

609 

.001642 

654 

.001529 

520 

521 

522 

523 
524 

.001923 
.001919 
.001916 
.OOI9I2 
.001908 

565 
566 

567 
568 

569 

.001770 
.001767 
.001764 
.001761 

.001757 

610 
6n 
612 

613 
614 

.001639 
.001637 
.001634 
.001631 
.001629 

655 
656 

657 
658 

659 

.001527 
.001524 
.OOI522 
.OOI52O 
.001517 

525 

.001905 

570 

.001754 

615 

.OOl626 

660 

.001515 

526 

.OOI90I 

571 

.001751 

616 

.001623 

661 

.001513 

527 

.001898 

572 

.001748 

617 

.OOl62I 

662 

.OOI5II 

528 

.001894 

573 

.001745 

618 

.00l6l8 

663 

.001508 

529 

.001890 

574 

.OOI742 

619 

.OOl6l6 

664 

.001506 

530 

.001883 

575 
576 

.001739 
.001736 

620 

621 

.OOl6l3 
.OOl6lO 

665 
666 

.001504 
.001502 

532 

\J 

.OOl88o 

577 

.001733 

622 

.OOl6o8 

667 

.001499 

533 
534 

.001876 
.001873 

578 
579 

.001730 
.001727 

623 

624 

.001605 
.001603 

668 
669 

.OOI497 
.001495 

535 
536 

.001869 
.001866 

580 

.001724 
.001721 

625 
626 

.OOI6OO 
.001597 

670 
671 

.001493 
.001490 

537 

r> 

.001862 

582 

.OOI7l8 

627 

.001595 

672 

.001488 

538 

.001859 

f\ 

583 

.001715 

628 

.001592 

673 

.001486 

539 

.001855 

584 

.001712 

629 

.001590 

674 

.001484 

540 

542 
543 
544 

.001852 
.001848 
.001845 
.001842 
.001838 

585 
586 

587 
588 

589 

.001709 
.001706 
.001704 
.OOI7OI 
.001698 

630 
631 
632 
633 
634 

.001587 
.001585 
.001582 
.001580 
.001577 

675 
676 
677 
678 
679 

.001481 
.001479 
.001477 
.001475 
.001473 

545 
546 
547 
548 
549 

.001835 
.001832 
.001828 
.001825 
.001821 

590 

592 
593 
594 

.001695 
.001692 
.001689 
.OOI686 
.001684 

635 
636 

637 
638 

639 

•001575 
.OOI572 
.OOI57O 
.001567 
.001565 

680 

681 
682 
683 
684 

.OOI47I 
.001468 
.001466 
.001464 
.001462 

55° 

.00l8l8 

595 

.OOl68l 

640 

.00156^ 

685 

.001460 

55i 

.00l8l5 

596 

.001678 

641 

<J    \J 

.001560 

686 

.001458 

.00l8l2 

597 

.001675 

642 

.001558 

687 

.001456 

553 

.00l8o8 

598 

.001672 

643 

•OOI555 

688 

.001453 

554 

.001805 

599 

.001669 

644 

•001553 

689 

.OOI45I 

555 
556 
557 
558 
559 

.001802 
.001799 
.001795 
.001792 
.001789 

600 
601 
602 
603 
604 

.001667 
.001664 
.OOl66l 
.001658 
.001656 

645 
646 
647 
648 
649 

.001550 
.001548 
.001546 
.001543 
.001541 

690 

691 
692 
693 
694 

.001449 
.001447 
.OOI445 
.001443 
.OOI44I 

560 

.001786 

605 

.001653 

650 

.001538 

695 

.001439 

561 

.001783 

606 

.001650 

651 

.001536 

696   .001437 

122 


MATHEMATICAL   TABLES. 


No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

697 

•001435 

742 

.001348 

787 

.OOI27I 

832 

.OOI2O2 

698 

.001433 

743 

.001346 

788 

.001269 

833 

.OOI20O 

699 

.001431 

744 

.001344 

789 

.001267 

834 

.001199 

70O 

701 
702 

703 
704 

.001429 
.001427 
.001425 
.OOI422 
.OOI42O 

745 
746 

747 
748 
749 

.001342 
.OOI34O 
.001339 
.001337 
.001335 

7QO 
79I 
792 

793 
794 

.OOI266 
.OOI264 
.001263 
.OOI26l 
.OOI259 

835 
836 

837 
838 

839 

.001196 
.001195 
.001193 
.OOII92 

705 

.OOI4l8 

75° 

•001333 

795 

.001258 

840 

.001190 

706 

.OOI4l6 

751 

.001332 

796 

.OOI256 

841 

.001189 

707 

.001414 

752 

.001330 

797 

.OOI255 

842 

.001188 

708 

.OOI4I2 

753 

.001328 

798 

.001253 

843 

.OOIl86 

709 

.OOI4IO 

754 

.001326 

799 

.001251 

844 

.001185 

7IO 

711 
712 

713 

714 

.OOI4O8 
.001406 
.001404 
.001403 
.OOI4OI 

755 
756 
757 
758 
759 

.001325 
.001323 
.OOI321 
.001319 
.001318 

800 

80  1 

802 
803 
804 

.OOI250 
.001248 
.001247 
.001245 
.OOI244 

845 
846 

847 
848 

849 

.001183 
.OOIl82 
.OOIlSl 
.001179 
.001178 

715 

•001399 

760 

.001316 

805 

o   /* 

.OOI242 

850 

.001176 

7l6 

.001397 

761 

.001314 

806 

.OOI24I 

851 

.001175 

717 

7l8 

.001395 
.001393 

762 

763 

.OOI3I2 
.001311 

807 
808 

.001239 
.001238 

852 
853 

.001174 
.001172 

719 

.001391 

764 

.001309 

809 

.001236 

854 

.OOII7I 

72O 

721 
722 

723 
724 

.001389 
.001387 
.001385 
.001383 
.001381 

765 
766 
767 
768 
769 

.001307 
.001305 
.001304 
.001302 
.001300 

810 

811 
812 

813 

814 

.001235 
.OOI233 
.OOI232 
.OOI230 
.001229 

855 
856 

857 
858 

859 

.OOII70 
.OOIl68 
.OOIl67 
.OOll66 
.001164 

725 
726 

727 
728 

729 

.001379 
.001377 
.001376 
.001374 
.001372 

770 

771 

772 
773 
774 

.OOI299 
.001297 
.001295 
.OOI294 
.001292 

816 

8i7 
818 
819 

.OOI227 
.OOI225 
.OOI224 
.OOI222 
.OOI22I 

860 

86  1 
862 
863 
864 

.001163 
.OOIl6l 
.OOIl6o 
.001159 
.001157 

730 

.001370 

775 

.001290 

820 

.OOI220 

865 

.OOII56 

73  1 

.001368 

776 

.001289 

821 

.001218 

866 

.001155 

732 

.001366 

777 

.001287 

822 

.001217 

867 

.001153 

733 

.001364 

778 

.001285 

823 

.OOI2I5 

868 

.001152 

734 

.001362 

779 

.001284 

824 

.OOI2I4 

869 

.OOII5I 

735 
736 
737 
738 
739 

.001361 
.001359 
•001357 
•001355 
•001353 

780 
781 

782 
783 
784 

.001282 
.OOI28O 
.001279 
.OOI277 
.00-1276 

825 
826 
827 
828 
829 

.OOI2I2 
.OOI2II 
.OOI2O9 
.OOI2O8 
.OOI206 

870 

871 

872 

873 
874 

.OOII49 
.OOII48 
.OOII47 
.001145 
.OOII44 

740 

.001351 

785 

.001274 

830 

.OOI2O5 

875 

.001143 

741 

.001350 

786 

.001272 

831 

.001203 

876 

.001142 

RECIPROCALS   OF   NUMBERS. 


No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

Reciprocal. 

No. 

xjtr 

Reciprocal^ 

877 

.OOII4O 

908 

.OOIIOI 

939 

.OOIO65 

970 

.OOIO3I 

878 
879 

.001139 
.001138 

909 
QIO 

.001100 

.001099 

940 
941 

.001064 
.001063 

97I 

972 

Q7-J 

.OOI030 
.OOI029 
.OOIO28 

880 

881 

.001136 
.OOII35 

911 
9I2 

.001098 
.001096 

942 
943 

.OOI062 
.001060 

7  /  o 

974 
975 

.OOI027 
.OOIO26 

882 
883 

.001134 
.OOII33 

913 
914 

.001095 
.001094 

944 
945 

.OOI059 
.001058 

976 
977 

.OOIO25 
.OOIO24 

884 

.OOII3I 

915 

.001093 

946 

.OOI057 

078 

.OOIO22 

885 

.OOII3O 

916 

.001092 

947 

.001056 

7  1  w 

979 

.OOI02I 

886 

.OOII29 

917 

.001091 

948 

.OOI055 

887 

.OOII27 

918 

.001089 

949 

.OOI054 

980 

.OOIO20 

888 

.OOII26 

919 

.001088 

981 

.OOIOI9 

889 
890 

.001125 
.OOII24 

920 

92I 

.001087 
.001086 

95° 

95i 
952 

.001053 
.OOI052 
.OOI050 

982 

983 
984 

.OOIOlS 
.001017 
.001016 

891 
892 

893 
894 

'  .001122 
.OOII2I 
.OOII20 
.001119 

922 

923 
924 

925 

.001085 
.001083 
.001082 
.001081 

953 
954 
955 
956 

.OOI049 
.001048 
.OOIO47 
.001046 

985 
986 

987 
988 

.OOIOI5 
.OOIOI4 
.OOIOI3 
.OOIOI  2 

895 

.OOIIlS 

926 

.001080 

957 

.OOIO45 

080 

.OOIOI  I 

896 

.OOIIl6 

927 

.001079 

958 

.OOI044 

ytjy 

897 

.001115 

928 

.001078 

959 

.OOIO43 

990 

.OOIOIO 

898 

.OOIII4 

929 

.001076 

r\f\f\ 

991 

.001009 

899 

.001112 

900 

.OOI042 

992 

.001008 

930 

.001075 

961 

.OOIO4I 

993 

.001007 

goo 

.OOIIII 

931 

.001074 

962 

.OOIO40 

QQ4. 

.001006 

901 

.001110 

932 

.001073 

963 

.001038 

77^ 

995 

.001005 

902 

.001109 

933 

.001072 

964 

.OOI037 

006 

.001004 

903 
904 

.OOII07 
.001106 

934 
935 

.001071 
.001070 

965 
966 

.001036 
.001035 

yyw 

997 
008 

.001003 

.001002 

9°5 

.OOIIO5 

936 

.001068 

967 

.OOI034 

yy" 

QQQ 

.OOIOOI 

906 

.OOIIO4 

937 

.001067 

968 

.001033 

y  vy 

907 

.OOIIO3 

938 

.001066 

969 

.OOI032 

IOOO 

.00  1000 

c% 


WEIGHTS    AND    MEASURES. 


WATER  AND  AIR  AS  STANDARDS  FOR  WEICzHT  AND  MEASURE. 


WATER  AS  A  STANDARD. 

There  are  four  notable  temperatures  for  water,  namely, 

32°  F.,  or      o°  C.  =  the  freezing  point,  under  one  atmosphere. 

3  9°.  i     or      4°  =  the  point  of  maximum  density. 

62°        or    1 6°. 66  =  the  British  standard  temperature. 

212°        or  1 00°  =  the  boiling  point,  under  one  atmosphere. 

The  temperature  62°  F.  is  the  temperature  of  water  used  in  calculating 
the  specific  gravity  of  bodies,  with  respect  to  the  gravity  or  density  of 
water  as  a  basis,  or  as  unity.  In  France,  the  temperature  of  maximum 
density,  39°.  i  F.,  or  4°  C.,  is  used  for  this  purpose,  for  solids. 

Weight  of  one  cubic  foot  of  Pure  Water. 

At  32°  F.  —  62.418  pounds. 

At  39°. i  =  62.425-^    „ 

At  62°       (Standard  temperature)  =  62.355       » 
At  212°  =  59.640 

The  weight  of  a  cubic  foot  of  water  is,  it  may  be  added,  about  1000 
ounces  (exactly  998.8  ounces),  at  the  temperature  of  maximum  density. 

The  weight  of  water  is  usually  taken  in  round  numbers,  for  ordinary 
calculations,  at  62.4  Ibs.  per  cubic  foot,  which  is  the  weight  at  52°.3  F. ;  or 
it  is  taken  at  62^  Ibs.  per  cubic  foot,  where  precision  is  not  required,  equal 
to  1f^°  Ibs. 

The  weight  of  a  cylindrical  foot  of  water  at  62°  F.  is  48.973  pounds. 

Weight  of  one  cubic  inch  of  Pure  Water. 

At  32°  F.  =  .03612  pound,  or  0.5779  ounce. 

At  39°.  i     -.036125     „         ,,0.5780      „ 

At  62°       =  .03608       „         „  0.5773      „       or  252.595  grains. 

At  212°         -    .03451  „  „    0.5522         „ 

The  weight  of  one  cylindrical  inch  of  pure  water  at  62°  F.  is  .02833 
pound,  or  0.4533  ounce. 


WATER  AND  AIR  AS   STANDARDS.  125 

Volume  of  one  pound  of  Pure  Water. 

At  32°  F.  =  .016021  cubic  foot,  or  27.684  cubic  inches. 

At39°.i  -.016019         „  ,,27.680  „ 

At  62°  =  .016037          „  „  27.712  „ 

At  212°  =  .016770         „  „  28.978 

The  volume  of  one  ounce  of  pure  water  at  62°  F.  is  1.732  cubic  inches. 

The  Gallon. 

The  weight  of  one  gallon  of  water  at  the  standard  temperature,  62°  F., 
is  10  pounds,  and  the  correct  volume  is  0.160372  cubic  foot,  or  277.123 
cubic  inches.  But  in  an  Act  of  Parliament,  which  came  into  force  in  1825, 
the  volume  of  one  gallon  is  stated  to  be  277.274  cubic  inches;  this  is  the 
commonly  accepted  volume.  See  page  339. 

The  volume  of  10  pounds  of  water  at  62°  F.  is,  therefore,  to  the  volume 
of  the  imperial  gallon,  as  i  to  1.000545. 

And,  the  weight  of  an  imperial  gallon  of  water  at  62*  F.  is  10.00545 
pounds  avoirdupois;  or  10  pounds,  38.15  grains. 

One  cubic  foot  of  water  contains  6.2355  gallons  of  277.123  cubic  inches, 
or  6.23208  gallons  of  277.275  cubic  inches,  or  approximately  6J^  gallons. 
One  gallon  is  equal  to  .1604  cubic  foot. 

The  volume  of  water  at  62°  F.,  in  cubic  inches,  multiplied  by  '.00036, 
gives  the  capacity  in  gallons. 

The  capacity  of  one  gallon  is  equal  to  one  square  foot,  two  inches  deep 
nearly  (exactly  1.924  inches);  or  to  one  circular  foot,  2^/2,  inches  deep 
nearly  (exactly  2.45  inches). 

One  ton  of  water  at  62°  F.  contains  224  gallons. 

Other  Measures  of  Water. 

Volume  of  given  weights  of  water,  at  62.4  pounds  per  cubic  foot: — 

i  ton 35-90  cubic  feet. 

i  cwt 1.795 

i  quarter 449 

i  pound /      -016  cubic  foot,  or 

{  27.692  cubic  inches, 
i  ounce 1.731 

i  tonne,  at  39°.!  F 35-3*56  cubic  feet. 

i  kilogramme,  at  39°.  i  F. . .  L  '°353  c^ic.  f°?t'  °r 

(  61.025  cubic  inches. 

i  tonne,  at  5  2°. 3  F.                     )  ,  .    f    . 

(62.4  pounds  per  cubic  foot)  } 35-33°  cublc  feet. 

Thirty-six  cubic  feet,  or  iy§  cubic  yards,  of  water,  at  62.4  pounds  per 
cubic  foot,  being  at  the  temperature  5  2°.  3  F.,  weigh  about  one  ton  (exactly 
6.4  pounds  more). 

One  cubic  yard,  or  twenty-seven  cubic  feet,  of  water  weighs  about 
15  cwt,  or  y^  ton  (exactly  4.8  pounds  more).  It  is  equal  to  168.36  gallons. 

One  cubic  metre  of  water  is  equal  in  volume  to  35.3156  cubic  feet, 
or  1.308  cubic  yards,  or  220.09  gallons;  and  at  62.4  pounds  per  cubic  foot, 
it  weighs  i  ton  nearly  (exactly  36.3  pounds  less).  It  is  nearly  equivalent 


126  WEIGHTS   AND   MEASURES. 

to  the  old  English  tun  of  4  hogsheads — 210  imperial  gallons,  and  is  a 
better  unit  for  measuring  sewage  or  water-supply  than  the  gallon. 

The  cubic  metre  is  generally  used  on  the  Continent  for  such  measurements. 

A  pipe  one  yard  long  holds  about  as  many  pounds  of  water  as  the  square 
of  its  diameter  in  inches  (exactly  2  per  cent.  more). 

Pressure  of  Water. 

A  pressure  of  one  Ib.  per  square  inch  is  exerted  by  a  column  of  water 
2.3093  feet,  or  27.71  inches  high,  at  62°  F. ;  and  a  pressure  of  one  atmos- 
phere, or  14.7  Ibs.  per  s'quare  inch,  is  exerted  by  a  column  of  water 
33.947  feet  high,  or  10.347  metres,  at  62°  F. 

A  column  of  water  at  62°  F.,  one  foot  high,  presses  on  the  base  with  a 
force  of  0.433  Ib.,  or  6.928  ounces  per  square  inch.  A  column  100  feet 
high  presses  with  a  force  of  43^  Ibs.  per  square  inch.  A  column  one 
metre  high  presses  with  a  force  of  1.422  Ibs.  per  square  inch. 

A  column  of  water  one  inch  high,  presses  on  the  base  with  a  force  of 
0.5773  ounce  per  square  inch,  or  5.196  Ibs.  per  square  foot. 

A  column  of  water  one  mile  deep,  weighing  62.4  pounds  per  cubic  foot, 
presses  on  the  base  with  a  force  of  about  one  ton  per  square  inch  (fresh 
water  exactly  48  Ibs.  more;  sea- water  exactly  107.5  ^s.  more). 

Water  is  hardly  compressible  under  pressure.  Experiment  appears  to 
show  that  for  each  atmosphere  of  pressure  it  is  condensed  47^  million ths 
of  its  bulk. 

Sea-water. 

One  cubic  foot  of  average  sea-water,  at  62°  F.,  weighs  64  pounds,  and 
the  weight  of  fresh  water  is  to  that  of  sea-water  as  39  to  40,  or  as  i  to  1.026. 
Thirty-five  cubic  feet  of  sea-water  weighs  one  ton. 
One  cubic  yard  of  sea-water  weighs  15}^  cwt.  nearly  (8  Ibs.  less). 
One  cubic  metre  of  sea-water  weighs  fully  one  ton  (20  Ibs.  more). 
Average  sea-water  is  composed  as  follows : — 

Per  100  parts.       Per  100  parts. 

Chloride  of  sodium  (common  salt), 2.50 

Sulphuret  of  magnesium, 0.53 

Chloride  of  magnesium, 0.33 

Carbonate  of  lime,  ) 

Carbonate  of  magnesia,   J  ' 

Sulphate  of  lime, o.oi 

Solid  matter,  say, 3.40 

Water, 96.60 


showing  that  sea-water  contains  ^jtn  Part  °f  ^ts  weight  of  solid  matter  in 
solution. 

According  to  Reclus,  the  mean  specific  gravity  of  sea-water  is  1.028.  In 
the  Mediterranean  Sea,  it  is  1.029;  in  the  Black  Sea,  1.016.  The  mean 
quantity  of  salts,  or  solid  matter,  in  solution,  is  3.44  per  cent,  three-fourths 
of  which  is  common  salt.  In  the  Red  Sea,  the  water  contains  4.3  per  cent; 
in  the  Baltic  Sea,  5  per  cent. ;  and  at  Cronstadt,  2  per  cent. 


WATER  AND  AIR  AS   STANDARDS.  I2/ 

Ice  and  Snow. 

One  cubic  foot  of  ice  at  32°  F.  weighs  57.50  Ibs. 

One  pound  of  ice  at  32°  F.  has  a  volume  of  .0174  cubic  foot,  or  30.067 
cubic  inches. 

The  volume  of  water  at  32°  F.  is  to  that  of  ice  at  32°  F.,  as  i.ooo  to 
1.0855;  the  expansion  in  passing  into  the  solid  state  being  above  8*^  per 
cent,  of  the  volume  of  water. 

The  specific  density  of  ice  is  0.922,  that  of  water  at  62°  F.  being  =  i. 

The  melting  point  of  ice  is  32°  F.,  or  o°  C.,  under  the  ordinary  atmos- 
pheric pressure,  of  14.7  Ibs.  per  square  inch.  Under  greater  pressure  the 
melting  point  is  lower,  being  at  the  rate  of  .0133°  F.  for  each  additional 
atmosphere  of  pressure. 

The  specific  heat  of  ice  is  .504,  that  of  water  being  =  i. 

One  cubic  foot  of  fresh  snow  weighs  5.20  Ibs.  Snow  has  12  times  the 
bulk  of  water,  and  its  specific  gravity  is  .0833. 

French  and  English  Measures  of  Water. 

One  litre  of  water  is  equal  to  0.2201  gallon,  or  1.761  pints:  about 
i^  pints.  One  gallon  is  equal  to  4.544  litres,  and  one  pint  is  .568  litre. 

One  litre  of  water  at  39°. i  F.,  or  4°  C.,  the  temperature  of  maximum 
density,  weighs  one  kilogramme,  or  2.2046  Ibs.;  at  the  temperature  62°  F., 
or  1 6°. 7  C.,  it  weighs  2.202  Ibs. 

1000  litres  =  one  cubic  metre,  equal  to  35.3156  cubic  feet;  and,  at 
39°.  i  F.,  or  4°  C.,  weigh  1000  kilogrammes,  or  one  ton  nearly  (35.4  Ibs.  less). 

AIR   AS   A   STANDARD. 

The  mean  pressure  of  the  atmosphere  at  the  level  of  the  sea,  is  equal 
to  14.7  Ibs.  per  square  inch,  or  2116.4  Ibs.  per  square  foot;  or  to  1.0335 
kilogrammes  per  square  centimetre.  This  is  called  one  atmosphere  of 
pressure.  The  following  are  measures  of  pressures  (see  also  pages  145, 158): — 

One  atmosphere  of  pressure : — (i.)  A  column  of  air  at  32°  F.,  27,801  feet, 
or  about  5  ^  miles  high,  of  uniform  density  equal  to  that  of  air  at  the  level 
of  the  sea.  (2.)  A  column  of  mercury  at  32°  F.,  29.922  inches  or  76  centi- 
metres high;  nearly  30  inches.  At  62°  F.,  the  height  is  30  inches.  (3.)  A 
column  of  water  at  62°  F.,  33.947  feet  or  10.347  metres  high;  nearly  34  feet. 

A  pressure  of  i  Ib.  per  square  inch: — (i.)  A  column  of  air  at  32°  F., 
1891  feet  high,  of  uniform  density  as  above.  (2.)  A  column  of  mercury  at 
32°  F.,  2.035  inches  or  51.7  millimetres  high.  At  62°  F.,  the  height  is  2.04 
inches.  (3.)  A  column  of  water  at  62°  F.,  2.31  feet  or  27.72  inches  high. 

A  pressure  of  i  Ib.  per  square  foot: — (i.)  A  column  of  air  at  32°  F.,  13.13 
feet  high,  of  uniform  density  as  above.  (2.)  A  column  of  mercury  at  32°  F., 
.0141  inch  or  .359  millimetre  high.  At  62°  F.,  the  height  is  .01417  inch. 
(3.)  A  column  of  water  at  62°  F.,  .1925  inch  high. 

The  density,  or  weight  of  one  cubic  foot  of  pure  air,  under  a  pressure 
of  one  atmosphere,  or  14.7  Ibs.  per  square  inch,  is 

At  32°  F.,      =      .080728  pound,  or  1.29  ounce,  or  565.1  grains. 
At62°F.,      =      .076097       „       „    1.217     »      »    532-7      » 

The  weight  of  a  litre  of  pure  air,  under  one  atmosphere,  at  32°  F.,  is 
1.293  grammes,  or  19.955  grains. 


128  WEIGHTS  AND   MEASURES. 

The  weight  of  air,  compared  with  that  of  water  at  three  notable  tempera- 
tures, and  at  5 2°. 3,  under  one  atmosphere,  is  as  follows: — 

Weight  of  water  at32°F.,         773.2  times  the  weight  of  air  at  32°  F. 

•>•>  >5         39  •*}  773-27         ?>  j>  jj 

,»  »         62°,  772.4  „  „  „ 

62°,  819.4  „  „  62°. 

52°-3,  82° 

The  volume  of  one  pound  of  air  at  32°  F.,  and  under  one  atmosphere  of 
pressure,  is  12.387  cubic  feet.  The  volume  at  62°  F.,  is  13.141  cubic  feet. 

The  specific  heat  of  air  at  constant  pressure  is  .2377,  and  at  constant 
volume  .1688,  that  of  water  being  *  i. 


GREAT   BRITAIN    AND    IRELAND.— IMPERIAL   WEIGHTS 
AND    MEASURES. 

The  origin  of  English  measures  is  the  grain  of  corn.  Thirty-two  grains 
of  wheat,  dried  and  gathered  from  the  middle  of  the  ear,  weighed  what  was 
called  one  pennyweight;  20  pennyweights  were  called  one  ounce,  and 
20  ounces  one  pound.  Subsequently,  the  pennyweight  was  divided  into 
24  grains.  Troy  weight  was  afterwards  introduced  by  William  the  Conqueror, 
from  Troyes,  in  France ;  but  it  gave  dissatisfaction,  as  the  troy  pound  did 
not  weigh  so  much  as  the  pound  then  in  use;  consequently,  a  mean  weight 
was  established,  making  16  ounces  equal  to  one  pound,  and  called  avoir- 
dupois (avoir  du  poids). 

Three  grains  of  barleycorn,  well-dried,  placed  end  to  end,  made  an  inch 
— the  basis  of  length.  The  length  of  the  arm  of  King  Henry  I.  was  made 
the  length  of  the  ulna,  or  ell,  which  answers  to  the  modern  yard.  The 
imperial  standard  yard  is  a  solid  square  bar  of  gun-metal,  kept  in  the 
office  of  the  Exchequer  at  Westminster,  38  inches  in  length,  i  inch  square, 
at  the  temperature  62°  F.,  composed  of  copper  16  ounces,  tin  2^  ounces, 
and  zinc  i  ounce.  Two  cylindrical  holes  are  drilled  half  through  the  bar, 
one  near  each  end,  and  the  centres  of  these  holes  are  36  inches,  or  3  feet, 
apart — the  length  of  the  imperial  standard  yard.  Compared  with  a  pendu- 
lum vibrating  seconds  of  mean  time,  at  the  level  of  the  sea,  in  the  latitude 
of  London,  in  a  vacuum,  the  yard  is  as  36  inches  in  length  to  39.1393 
inches,  the  length  of  the  pendulum. 

Measures  of  capacity  were  based  on  troy  weight;  it  was  enacted  that 
8  pounds  troy  of  wheat,  from  the  middle  of  the  ear,  well  dried,  should 
make  i  gallon  of  wine  measure,  and  that  8  such  gallons  should  make 
i  bushel. 

The  imperial  gallon  is  now  the  only  standard  measure  of  capacity,  and  it 
contains  277.274  cubic  inches.  It  is  said  to  be  the  volume  of  10  pounds 
avoirdupois  of  distilled  water,  weighed  in  air,  at  62°  F. 

Note. — The  exact  volume  of  10  pounds  of  distilled  water  at  62°  F.  is 
277.123  cubic  inches. 


GREAT   BRITAIN   AND   IRELAND. — LENGTH.  1 29 

Tables  of  weights  and  measures  are  conveniently  classified  thus — 
i.  Length;  2.  Surface;  3.  Volume;  4.  Capacity;  5.  Weight. 

The  following  are  some  of  the  principal  units  of  measurement : — 

The  acre,  for  land  measure. 

The  mile,  for  itinerary  measure.  ' 

The  yard,  for  measure  of  drapery,  &c. 

The  coomb,  for  capacity  of  corn,  &c. 

The  gallon,  for  capacity  of  liquids. 

The  grain,  for  chemical  analysis. 

Impound,  for  grocers'  ware,  &c. 

The  stone  of  8  pounds,  for  butchers'  meat. 

The  stone  of  14  pounds,  for  flour,  oatmeal,  &c. 

I.    MEASURES  OF  LENGTH. — Tables  No.  12. 

Lineal  Measure. 

3  barleycorns,  or 
12  lines,  or  f 

72  points,  or          > Ilnch- 

1000  mils 


3  inches 

4  inches 
9  inches 

12  inches 

1 8  inches, 

3  feet... 

'Yz  feet ... 

5  feet ... 
2  yards.. 

yards 


palm. 

hand. 

span. 

foot. 

cubit. 

yard. 

military  pace. 

geometrical  pace. 

fathom. 

rod,  pole,  or  perch. 


40  poles,  or  )  r    i 

220  yards       /  '"'  l  furlon& 

8  furlongs,  or  } 

1 760  yards,  or       > i  mile. 

5280  feet  j 

3  miles i  league. 

2240  yards,  or  )  ... 

1.272  miles        }  '  Insh  mlle' 


The  inch  is  also  divided  into  halves,  quarters,  eighths,  and  sixteenths; 
sometimes  into  tenths. 

The  hand  is  used  as  a  measure  of  the  height  of  horses. 

The  military  pace  is  the  length  of  the  ordinary  step  of  a  man. 

The  geometrical  pace  is  the  length  of  two  steps.     A  thousand  of  such 
paces  were  reckoned  to  a  mile. 

The  fathom  is  used  in  soundings  to  ascertain  depths,  and  for  measuring 
cordage  and  chains. 

9 


130  WEIGHTS  AND  MEASURES. 

Land  Measure. 

7.92  inches  i  link. 

100  links,  or    \ 
66  feet,  or      (  ,    . 

22  yards,  or  f I  cham' 

4  poles 

10  chains '. i  furlong. 

80  chains,  or  )  ., 

8  furlongs     } '  mile- 

They*?*,  or  woodland  pole  or  perch,  is  18  feet. 
The  for tst  pole  is  21  feet. 

Nautical  Measure. 

6086.44  feet?  or  j 

1000  fathoms,  or     (  f   i  nautical  mile, 

10  cables,  or        [  "  1          or  knot. 

1.1528  statute  miles  ) 

3  nautical  miles i  league. 

60  nautical  miles,  or  \ 

69.168  statute  miles  or    >  i  degree. 

20  leagues  j 

(  Circumference 

360  degrees <  of  the  earth  at 

(  the  equator. 

The  above  value  of  the  nautical  mile  is  that  which  is  commonly  taken, 
and  is  the  length  of  a  minute  of  longitude  at  the  equator.  The  mean 
length  of  a  minute  of  latitude  at  the  mean  level  of  the  sea  is  nearly  6076 
feet,  or  1.1508  statute  miles. 

The  nautical  fathom  is  the  thousandth  part  of  a  nautical  mile,  and  is,  on 
an  average,  about  ^-th  longer  than  the  common  fathom. 

Cloth  Measure. 

2%  inches i  nail. 

2  nails i  finger-length. 

4  nails,  or  9  inches  i  quarter. 

4  quarters i  yard. 

5  quarters i  ell. 

WIRE-GAUGES. 

The  "  Birmingham  Wire-Gauge  "  is  a  scale  of  notches  in  the  edge  of  a 
plate,  of  successively  increasing  or  decreasing  widths,  to  designate  a  set  of 
arbitrary  sizes  or  diameters  of  wire,  ranging  from  about  half  an  inch  down  to 
the  smallest  size  easily  drawn,  say,  four-thousands  of  an  inch.  The  practical 
utility  of  such  a  gauge  is  obvious,  when  it  is  considered  how  far  beyond  the 
means  supplied  by  the  graduations  of  an  ordinary  scale  of  feet  and  inches 
is  the  measurement  of  the  gradations  of  the  wire-gauge.  But  the  "Birming- 
ham Wire-Gauge"  is  a  variable  measure.  The  principle,  if  there  was  any, 
on  which  it  was  originally  constructed,  is  not  known.  Mr.  Latimer  Clark 
states  that,  when  plotted,  the  widths  of  the  gauge  range  in  a  curve  approxi- 


GREAT  BRITAIN   AND   IRELAND.— WIRE-GAUGES. 


mating  to  a  logarithmic  curve,  such  as  would  be  found  by  the  successive 
addition  of  10  or  12  per  cent,  to  the  width  of  the  notches  of  the  gauge. 
However  that  may  be,  there  are  many  varieties  of  the  wire-gauge  in  existence. 
The  oldest  and  best-known  gauge  is  that  of  which  the  numbers  were  care- 
fully measured  by  Mr.  Holtzapffel,  and  published  by  him  in  1847.  It  has 
been,  and  still  is,  widely  followed  in  the  manufacture  of  wire;  and  also  of 
tubes  in  respect  of  their  thickness.  It  gives  40  measurements  ranging  from 
.454  inch  to  .004  inch,  and  is  contained  in  Table  No.  13.  Although 
there  are  only  40  marks  in  the  table,  there  are  60  different  sizes  of  wire 
made,  for  which  intermediate  sizes  have  been  added  to  the  gauge.  This 
table  has  also  been  used  in  rolling  sheet  iron,  sheet  steel,  and  other 
materials,  and  for  joiners'  screws;  but  it  appears  to  be  falling  into  disuse 
for  these  purposes. 

BIRMINGHAM  WIRE-GAUGE  (HoltzapffeFs). — Table  No.  13. 
For  Wire  and  Tubes  chiefly;  and  for  Sheet  Iron  and  Steel  formerly. 


Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

OOOO 

•454 

7 

.180 

17 

.058 

27 

.Ol6 

OOO 

•425 

8 

.165 

18 

.049 

28 

.014 

00 

.380 

9 

.148 

J9 

.042 

29 

.013 

O 

•340 

10 

•134 

20 

•035 

30 

.012 

I 

.300 

ii 

.120 

21 

.032 

31 

.OIO 

2 

.284 

12 

.ICQ 

22 

.028 

32 

.009 

3 

•259 

13 

•095 

23 

.025 

33 

.008 

4 

.238 

14 

.083 

24 

.022 

34 

.007 

5 

.220 

15 

.072 

25 

.020 

35 

.005 

6 

.203 

16 

.065 

26 

.018 

36 

.004 

BIRMINGHAM  METAL-GAUGE,  or  PLATE-GAUGE  (HottzapffeTs). — 
Table  No.  14. 

For  Sheet  Metals,  Brass,  Gold,  Silver,  &c. 


Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

No. 

Inch. 

No. 

Inch. 

No. 

.Inch. 

No. 

Inch. 

I 

.004 

IO 

.024 

19 

.064 

28 

.120 

2 

.005 

II 

.029 

20 

.067 

29 

.124 

3 

.008 

12 

•034 

21 

.072 

30 

.126 

4 

.OIO 

J3 

.036 

22 

.074 

31 

•133 

5 

.OI2 

14 

.041 

23 

.077 

32 

•143 

6 

.013 

15 

.047 

24 

.082 

33 

•145 

7 

.015 

16 

.051 

25 

•°95 

34 

.148 

8 

.Ol6 

17 

•057 

26 

.103 

35 

.158 

9 

.019 

18 

.O6l 

27 

•113 

36 

.167 

Another  of  Holtzapffel's  tables,  No.  14,  the  Plate-Gauge,  has  been,  and 
may  now,  to  some  extent,  be,  employed  for  most  of  the  sheet  metals,  except- 


132 


WEIGHTS   AND   MEASURES. 


LANCASHIRE  GAUGE  (Holtzapffel' s). — Table  No.  15. 
For  Round  Steel  Wire,  and  for  Pinion  Wire. 


Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

80 

.013 

57 

.042 

34 

.109 

II 

.189 

M 

•295 

79 

.014 

56 

.044 

33 

.III 

IO 

.I9O 

N 

.302 

78 

.015 

55 

.050 

32 

•H5 

9 

.191 

0 

.316 

77 

.Ol6 

54 

•055 

3i 

.118 

8 

.192 

P 

•323 

76 

.018 

53 

.058 

30 

.125 

7 

•195 

Q 

•332 

75 

.019 

52 

.060 

29 

•134 

6 

.198 

R 

•339 

74 

.022 

51 

.064 

28 

•  138 

5 

.2OI 

S 

.348 

73 

.023 

5o 

.067 

27 

.141 

4 

.204 

T 

.358 

72 

.024 

49 

.070 

26 

•143 

3 

.209 

U 

.368 

7i 

.026 

48 

•073 

25 

.146 

2 

.2I9 

V 

•377 

70 

.027 

47 

.076 

24 

.148 

I 

.227 

w 

.386 

69 

.029 

46 

.078 

23 

.150 

A 

•234 

X 

•397 

68 

.030 

45 

.080 

22 

•152 

B 

.238 

Y 

.404 

67 

.031 

44 

.084 

21 

•157 

C 

.242 

Z 

•413 

66 

.032 

43 

.086 

20 

.160 

D 

.246 

Ai 

.420 

65 

•033 

42 

.091 

19 

.164 

E 

.250 

Bi 

•431 

64 

•034 

4i 

•095 

18 

.167 

F 

•257 

Ci 

•443 

63 

•035 

40 

.096 

17 

.169 

G 

.261 

Di 

•452 

62 

.036 

39 

.098 

16 

.174 

H 

.266 

E! 

.462 

61 

.038 

38 

.IOO 

15 

•175 

I 

.272 

Fi 

•475 

60 

•°39 

37 

.102 

14 

.177 

J 

.277 

Gi 

.484 

59 

.040 

36 

.105 

13 

.180 

K 

.281 

Hi 

•494 

58 

.041 

35 

.107 

12 

.185 

L 

.290 

ing  iron  and  steel :  as  copper,  brass,  gilding-metal,  gold,  silver,  and  platinum. 
The  intervals  are  closer  or  smaller  than  those  of  the  wire-gauge,  and  the 
maximum  size,  for  No.  36,  is  J/6  inch.  When  thicker  sheets  are  wanted, 
their  measures  are  sought  in  the  Birmingham  wire-gauge. 

The  last  table,  No.  15,  by  HoltzaprTel,  the  Lancashire  Gauge,  is  employed 
exclusively  for  the  bright  steel  wire  prepared  in  Lancashire,  and  the  steel 
pinion-wire  for  watch  and  clock  makers.  The  larger  sizes  are  marked  by 
capital  letters,  to  distinguish  them  from  the  others.  This,  the  second  part 
of  the  table,  is  known  as  the  Letter-Gauge. 

Needle- Gauge,  for  needle  wire.  The  sizes  correspond  with  some  of  those 
of  the  Holtzapffel  wire-gauge.  The  following  are  the  relative  marks  for 
equal  sizes  on  the  two  gauges : — 

Needle  wire -gauge — Nos.  i,        2,    2^,    3,    4,    5,  thence  to  21, 
corresponding  to  B.  W.-G. — 18^,  19,  19^2,  20,  21,  22,  thence  to  38. 

Music  Wire-gauge,  for  the  strings  of  pianofortes.  The  marks  used  are 
Nos.  6  to  20.  The  following  are  the  relative  marks  for  equal  sizes  with  the 
Holtzapffel  wire-gauge : — 

Music  wire-gauge — Nos.  6,  7,  8,  9,  10,  n,  12,  14,  16,  18,  20, 
corresponding  to  B.  W.-G.— 26,  25%,  25,  24^,  24,  23^,  23,  22,  21,  20,  19. 
No.  6,  the  thinnest  wire  now  used,  measures  about  one  fifty- fifth  of  an  inch 
in  diameter,  and  No.  20  about  one  twenty-fifth  of  an  inch. 


GREAT   BRITAIN   AND   IRELAND. — WIRE-GAUGES. 


133 


The  preceding  Tables  of  Gauges  have  been  extracted  from  Holtzapffel's 
estimable  work  on  Turning  and  Mechanical  Manipulation,  1847. 

Messrs.  Rylands  Brothers,  of  Warrington,  manufacture  iron  wire  accord- 
ing to  the  gauge  in  Table  No.  16. 

WARRINGTON  WIRE-GAUGE  (Rylands  Brothers). — Table  No.  16. 


Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

7/0 

1/2 

0 

.326 

8 

•159 

15 

.069 

6/0 

15/32 

I 

.300 

9 

.146 

16 

.0625,  or  J/i6 

5/o 

7/16 

2 

.274 

TO 

•133 

i7 

•053 

4/0 

13/32 

3 

•25,  or  y± 

io# 

.125,  or  ^6 

18 

.047 

3/o 

3/8 

4 

.229 

II 

.117 

*9 

.041 

2/O 

1  1/32 

5 

.209 

12 

.10,  or  Vio 

20 

.036 

6 

.191 

13 

.090 

21 

.0315,  or  V32 

7 

.174 

14 

.079 

22 

.028 

For  sheets,  the  wire-gauge  that  seems  to  be  adhered  to  by  the  iron-sheet 
rollers  of  South  Staffordshire,  is  a  scale  comprising  32  measurements,  ranging 
from  .3125  inch  to  .0125  inch,  contained  in  Table  No.  17. 

BIRMINGHAM  WIRE-GAUGE. — Table  No.  17. 
For  Iron  Sheets  chiefly. 


No. 

Size. 

No. 

Size. 

No. 

Size. 

No. 

Size. 

Inch. 

Inch. 

Inch. 

Inch. 

I 

.3125  (s/l6) 

9 

.15625(5/32) 

17 

.05625 

25 

•02344 

2 

.28125 

10 

.140625 

18 

•05        ('/») 

26 

.021875 

3 

•25          (X) 

ii 

•I25       '(#) 

19 

•04375 

27 

.020312 

4 

•234375 

12 

.1125 

20 

•0375 

28 

.01875 

5 

.21875 

13 

.10        (r/io) 

21 

•034375 

29 

.01719 

6 

.203125 

14 

.0875 

22 

•031250/32) 

30 

.015625 

7 

.1875  (3/x6) 

15 

•075 

23 

.028125 

31 

.01406 

8 

.171875 

16 

•0625  0/l6) 

24 

•025      (V4o) 

32 

•OI25  (x/8o) 

Sir  Joseph  Whitworth,  in  1857,  introduced  his  Standard  Wire-Gauge, 
ranging  from  a  half  inch  to  a  thousandth  of  an  inch,  and  comprising  62 
measurements,  as  given  in  Table  No.  18.  It  commences  with  the 
smallest  size,  and  increases  by  thousandths  of  an  inch  up  to  half  an  inch. 
The  smallest  size,  Vioooth  of  an  inch,  is  No.  i ;  No.  2  is  2/ioooths  of  an  inch, 
and  so  on,  increasing  up  to  No.  20  by  intervals  of  Vioooth  of  an  inch;  from 
No.  20  to  No.  40  by  2/ioooths;  frOm  No.  40  to  No.  100  by  s/I000ths  of  an 
inch.  The  sizes  are  designated  or  marked  by  their  respective  values  in 
thousandths  of  an  inch. 

The  Standard  Imperial  Wire  Gauge  came  into  force  on  the  ist  March, 
1884.  It  supersedes  other  gauges,  which  are  rendered  illegal. 


134  WEIGHTS  AND   MEASURES. 

SIR  JOSEPH  WHITWORTH  &  Co.'s  STANDARD  WIRE-GAUGE. — Table  No.  18. 


Mark. 

Size. 

Mark. 

Size. 

Mark, 

Size. 

Mark. 

Size. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

No. 

Inch. 

I 

.001 

17 

.017 

55 

•°55 

200 

.200 

2 

.OO2 

18 

.018 

60 

.060 

220 

.220 

3 

.003 

!9 

.019 

65 

.065 

240 

.240 

4 

.004 

20 

.O2O 

70 

.070 

260 

.260 

5 

.005 

22 

.022 

75 

•075 

280 

.280 

6 

.006 

24 

.024 

80 

.080 

300 

.300 

7 

.007 

26 

.026 

85 

.085 

325 

.325 

8 

.008 

28 

.028 

90 

.090 

350 

•350 

9 

.OO9 

3° 

.030 

95 

•095 

375 

•375 

10 

.OIO 

32 

.032 

IOO 

.IOO 

400 

.400 

ii 

.Oil 

34 

•034 

no 

.no 

425 

•425 

12 

.OI2 

36 

.036 

I2O 

.120 

45° 

•45° 

13 

.013 

38 

.038 

135 

•135 

475 

•475 

14 

.014 

40 

.040 

150 

.150 

500 

.500 

15 

.015 

45 

•045 

165 

.165 

16 

.Ol6 

5° 

.050 

1  80 

.180 

STANDARD  IMPERIAL  WIRE-GAUGE. 
Table  No.  19. 


Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

7/0 

.500 

16 

.064 

38 

.0060 

«/o 

.464 

i7 

.056 

39 

.0052 

5/o 

•432 

18 

.048 

40 

.0048 

4/0 

.400 

i9 

.040 

4i 

.0044 

'/o 

•372 

20 

,036 

42 

.0040 

•/• 

•348 

21 

.032 

43 

.0036 

o 

.324 

22 

.028 

44 

.0032 

I 

.300 

23 

.024 

45 

.OO28 

2 

.276 

24 

.022 

46 

.0024 

3 

.252 

25 

.O2O 

47 

.OO2O 

4 

.232 

26 

.018 

48 

.O0l6 

5 

.212 

27 

.0164 

49 

.OOI2 

6 

.192 

28 

.0148 

50 

.OOIO 

7 

.176 

29 

.0136 

8 

.l6o 

3° 

.OI24 

9 

.144 

3i 

.OIl6 

10 

.128 

32 

.OI08 

ii 

.116 

33 

.OIOO 

12 

.IO4 

34 

.0092 

13 

.092 

35 

.0084 

14 

.080 

36 

.0076 

15 

.072 

37 

0068 

GREAT  BRITAIN   AND  IRELAND. — FRACTIONS  OF   INCH.      135 


INCHES  AND  THEIR  EQUIVALENT  DECIMAL  VALUES  IN  PARTS  OF  A  FOOT. 

—Table  No.  20. 


Inches. 

Fraction  of  foot. 

Foot. 

I  

«/„  

-083? 

2 

'/6 

..  XA  . 

.1667 
25 

4 
e... 

g 

•3333 
.4167 

J" 

6 

7 

? 

•5 
.5873 

8 
9  

10 

1  1 

2^ 

3/4  
M6 

."/_, 

.6667 
75 
.8333 
.0167 

12 

I 

I.O 

FRACTIONAL  PARTS  OF  AN  INCH,  AND  THEIR  DECIMAL  EQUIVALENTS. 

Tables  No.  21. 

Eighths. 


Eighths. 

Fractions. 

Inch. 

I 

I/O       ., 

.12$ 

2 

3  
4 
5" 

1  

% 

5/8 

•25 

375 
.621; 

6 
7... 

S 

"•0 

•75 
.  .875 

8 

I 

I.O 

Twelfths. 


Twelfths. 

Fractions. 

Inch. 

I  

.-   X/M. 

.    .OS'?'?'? 

'# 

2  
3 

4  
6 

"/• 

•/«  

•A 

£  

i/ 

.125 
16667 
•25 

33333 
.41667 

c 

8  

9 
10  
ii 

12  

/2     

y» 

::!:: 

*& 

i       .... 

.58333 

66666 

•75 
83333 
.91667 
i.o 

136 


Sixteenths 


WEIGHTS  AND   MEASURES. 
and  Thirty-seconds. — Tables  No.  21  (continued). 


Thirty- 
Seconds. 

Sixteenths. 

Fractions. 

Inch. 

I 

i/,2 

O3.I  2  1 

2 
-2 

I 

/32 

Vi6 
3/_  a 

•WOX  ^  J 
.0625 

OQ'?7i; 

4 

$ 

2 

/  32   •  • 

V. 

5/,_   . 

•^VO  /  0 
.125 
I  ^62^ 

6 

7 
8 

3 

4 

'32 
V.6 

Vs.  

v,     '/4 

•*3**^j 

•1875 
.21875 

•25 
28125 

10 

1  1 

5 

/  32       

V,6 

II  / 

•3125 

•3/1  -?7  t 

12 
17 

6 

/  32    

'/. 

'3/,2 

•JT-O  /  J 

•375 
4062^ 

14 

1C 

7 

/32 
7A6 

15A2  .. 

•4375 
4.6871; 

16 
17 

8 

•A 

'7A2     - 

•5 

C-2I2C 

18 

10 

9 

/32 
9/,6 
«9/M 

•  jo^-^o 

•5625 

CQ^7C 

20 

21 

10 

/  32   

5/8 

2IA2  . 

OVo  /  D 
•625 
6^62^ 

22 
2^ 

ii 

/32   

ii/  , 
/i6 

23A2  :, 

.v^^^-j 
.6875 
7l87S 

24 

25 

12 

/32 

3/4 

2s/,2  .. 

•  /  ^^/O 

•75 
.7812^ 

26 

27 

13 

732            ...... 

13/i6 

27A2 

.8125 
8/1771; 

28 
20 

14 

/  32    

V. 

29/,2   . 

•"T-O  /  D 
.875 

0062  1; 

30 
31 

15 

732     • 
I5A6 

3IA2  .. 

•yw'-'^o 

•9375 
06871; 

32 

16 

/  32 

I 

•y»-M~'  /  o 
1.0 

II.   MEASURES  OF  SURFACE.  —  Tables  No.  22. 

Superficial  Measure. 

144  square  inches,  or   7  r 

183.35  circular  inches  )  ...........................  i  square  foot. 

9  square  feet  ......................................  i  square  yard. 

100  square  feet  ......................................  i  square. 

272^  square  feet,  or  )  . 

30^  square  yards     }  ' 


The  square  is  used  in  measuring  flooring  and  roofing. 
The  rod  is  used  in  measuring  brick-work. 


GREAT  BRITAIN   AND   IRELAND.  —  SURFACE,   VOLUME.         137 

Builders'  Measurement. 

i  superficial  part  ........................  i  square  inch. 

12  parts  .....................................  "i  inch"  (12  square  inches). 

12  "inches"  ...............................  i  square  foot. 

This  table  is  employed  in  the  superficial  or  flat  measure  of  boards,  glass, 
stone,  artificers'  work,  &c. 

Land  Measure. 
9  square  feet  .................  .  ..................  i  square  yard. 


1  6  square  poles  .............................  .  .  .  .  i  square  chain. 

40  square  poles,  or  )  , 

1  2  10  square  yards       J 
4  roods,  or 
10  square  chains,  or 


1 60  square  poles,  or 


i  acre.* 


4,840  square  yards,  or 
43,560  square  feet 

640  acres,  or          )  ., 

3,097,600  square  yards  \   i  square  mile. 

30  acres i  yard  of  land. 

100  acres i  hide  of  land. 

40  hides i  barony. 

*  The  side  of  a  square  having  an  area  of  one  acre  is  equal  to  69.57  lineal  yards. 

III.  MEASURES  OF  VOLUME. — Tables  No.  24. 

Solid  or  Cubic  Measure. 

1728  cubic  inches          ^ 

2200.15  cylindrical  inches  (  i  cubic  foot. 

3300.23  spherical  inches     f 
6600.45  conical  inches       ) 

2  7  cubic  feet i  cubic  yard,  or  load. 

35.3156  cubic  feet  or  1  ; 

1.308    cubic  yards     J 

Note. — The  numbers  of  cylindrical,  spherical,  and  conical  inches  in  a  cubic  foot,  are 
as  I,  1.5,  3. 

Builders'  Measurement. 


i  solid  part  12  cubic  inches. 

1 2  solid  parts i  "inch"  (144  cubic  inches). 

12  "inches" i  cubic  foot. 

This  table  is  used  in  measuring  square-sided  timber,  stone,  &c. 


138 


WEIGHTS  AND   MEASURES. 


Note. — The  cubic  contents  of  a  piece, 

6  inches  square  and  4  feet  long  is  i  cubic  foot. 


12 

17 
24 


DECIMAL  PARTS  OF  A  SQUARE  FOOT,  IN  SQUARE  INCHES. — Table  No.  23. 


Hundredth 
Parts. 

Square 
Inches. 

Hundredth 
Parts. 

Square 
Inches. 

Hundredth 
Parts. 

Square 
Inches. 

Hundredth 
Parts. 

Square 
Inches. 

I 

1.44 

26 

37-4 

51 

73-4 

76 

109.4 

2 

2.88 

27 

38.9 

52 

74-9 

77 

II0.9 

3 

4-32 

28 

40-3 

53 

76.3 

78 

II2.3 

4 

5.76 

29 

41.8 

54 

77.8 

79 

II3.8 

5 

7.20 

30 

43-2 

55 

79.2 

80 

II5.2 

6 

8.64 

31 

44.6 

56 

80.6 

81 

116.6 

7 

10.  1 

32 

46.1 

57 

82.1 

82 

118.1 

8 

"•5 

33 

47-5 

58 

83-5 

83 

II9-5 

9 

13.0 

34 

49.0 

59 

85.0 

84 

121.  0 

10 

14.4 

35 

5°-4 

60 

86.4 

85 

122.4 

ii 

15-8 

36 

51-8 

61 

87.8 

86 

123.8 

12 

17-3 

37 

53-3 

62 

89-3 

87 

125.3 

13 

18.7 

38 

54-7 

63 

90.7 

88 

126.7 

14 

20.  2 

39 

56-2 

64 

92.2 

89 

128.2 

15 

21.6 

40 

57-6 

65 

93-6 

90 

129.6 

16 

23.0 

4i 

58.0 

66 

95-° 

9i 

I3I.O 

i7 

24.5 

42 

60.5 

67 

96-5 

92 

132.5 

18 

25-9 

43 

61.9 

68 

97-9 

93 

133.9 

i9 

27.4 

44 

63-4 

69 

99.4 

94 

135.4 

20 

28.8 

45 

64.8 

70 

100.8 

95 

136.8 

21 

30.2 

46 

66.2 

7i 

IO2.2 

96 

138.2 

22 

31.7 

47 

67.7 

72 

103.7 

97 

139.7 

23 

33.1 

48 

69.1 

73 

IO5.I 

98 

I4I.I 

24 

34-6 

49 

70.6 

74 

106.6 

99 

142.6 

25 

36.0 

50 

72.0 

75 

108.0 

IOO 

144.0 

IV.  MEASURES  OF  CAPACITY. — Tables  No.  25. 

Liquid  Measure. 

8.665  cubic  inches i  gill  or  quartern. 

4  gills  (34.659  cubic  inches) i  pint. 

2  pints i  quart. 

2  quarts i  pottle. 

4  quarts,  or  8  pints  (277.274  cubic  inches) i  gallon. 


6.2355  gallons i  cubic  foot. 

The  barn-gallon,  for  milk,  is  equal  to  2  imperial  gallons. 


GREAT  BRITAIN   AND  IRELAND.—  CAPACITY.  139 

Dry  Measure. 

2  pints  ................................................  i  quart. 

4  quarts  ................................................  i  gallon. 

2  gallons  ..............................................  i  peck. 

8  gallons01}  (T'28366  cubic  feet)  ................  i  bushel. 

2  bushels  ..............................................  i  strike. 

4  bushels  .......................................  .  .....  i  coomb. 

5  bushels  ..............................................  i  sack. 

8  bushels  .......  .......................................  i  quarter. 

4  quarters  (41.077  cubic  feet)  .....................  i  chaldron. 

5  quarters  .............................................  i  wey  or  load. 

2  loads  ................................................  i  last. 

In  the  Weights  and  Measures  Act  of  1878,  it  is  only  declared  that  the 
Imperial  Standard  Gallon  contains  10  pounds  of  water  at  62°  F.,  and  that 
8  gallons  shall  be  a  bushel.  Assuming  that  i  cubic  inch  of  water  weighs 
252.458  grains,  the  Imperial  Standard  Gallon  has  a  capacity  of  277.27384 
cubic  inches,  or,  say,  277.274  cubic  inches,  as  before  announced,  page  125. 

The  Imperial  Standard  bushel,  which  is  equal  to  8  gallons,  has  a  capacity 
of  2218.19072  cubic  inches.  The  internal  diameter  of  the  standard  bushel, 
1  7.8  inches,  is  double  its  internal  depth,  8.9  inches.  Heaped  measure,  which 
was  used  for  such  goods  as  could  not  be  stricken  —  as  coals,  potatoes,  fruit, 
is  now  legally  abandoned.  Coals  are  sold  by  weight;  and  for  other  round 
goods,  the  measure  is  filled  level  with  the  brim  as  nearly  as  is  practicable. 
The  Market  Garden  bushel  is  made  large  enough  to  hold  as  much  fruit  as 
the  heaped  bushel  held,  filled  level,  so  as  to  pack  one  on  another. 

Coal  and  Coke  Measure. 

3  bushels  (heaped)  ...........................   i  sack. 

9  bushels  ..............................  .  ........   i  vat. 

36  bushels,  or  12  sacks  (58.66  cubic  feet)  i  chaldron. 
5^  chaldrons  ....................................   i  room. 

2  1  chaldrons  ....................................   i  score. 

Old  Wine  and  Spirit  Measure. 
4  gills  or  quarterns  .......................  ...  i  pint.  canons. 

2  pints  .........................................  i  quart. 

4  quarts  (231  cubic  inches)  .........  .  .....  i  gallon  =       -8333 

10  gallons  ......................................  i  anker  =     8.333 

1  8  gallons  ......................................  i  runlet  =    15. 

3  1  ^  gallons  .  .  .  ...................................  i  barrel  =    26.250 

42  gallons  ......................................  i  tierce  =   35. 


..............................  ,  puncheon    =    7o. 

126  gallons,  or       } 

2  hogsheads,  or  V  ...........................  i  pipeorbutt=  105. 

i^4  puncheons       j 

2  pipes  or     )  ^  Itun  =2IO< 

3  puncheons  J 


140  WEIGHTS  AND  MEASURES. 

By  this  measure  wines,  spirits,  cider,  perry,  mead,  vinegar,  oil,  &c.,  are 
measured;  but  the  contents  of  every  cask  are  reckoned  in  imperial  gallons 
when  sold.  The  imperial  gallon  is  one-fifth  larger  than  the  old  wine 
gallon. 

Old  Ale  and  Beer  Measure. 


*  pnts  ............  .....  ..................  i  quart. 

4  quarts  (282  cubic  inches)  ............  i  gallon  =  1.017 

9  gallons  ..................................  i  firkin  =  9.  1  53 

2  firkins,  or  1  8  gallons  ..................  i  kilderkin  =  18.306 


-     54.9*8 
3  barrels,  or  , 

108  gallons       }  ...........................  '  butt  '09.836 

The  imperial  gallon  is  one-sixtieth  smaller  than  the  old  beer  gallon. 

Apothecaries'  Fluid  Measure. 

60  minims  (rn,)  ...........................  i  fluid  drachm  (/  5). 

8   drachms   (water,    1.732    cubic  )       n  -j 

inches,  437^  grains)  }'  ""'d  ounce    (/|). 

20  ounces  .....................  .  ...........  i  pint  (  °  ). 

8  pints  (water,  70,000  grains)  ........  i  gallon  (gall.). 


1  drop 

60  drops 

4  drachms 

2  ounces  (water,  875  grains) 

3  ounces 


gram. 

drachm. 

tablespoonful. 

wineglassful. 

teacupful. 


V.  MEASURES  OF  WEIGHT. — Tables  No.  26. 

Avoirdupois  Weight. 
ms,  or    { 
437^2  grains 


1 6  drachms,  or    I  ,     . 

> i  ounce  (oz.). 


16  ounces,  or  )  .  ,.         .  1V 

}  .................................  '  Pound  (™P«i»l) 


7000  grans 

8  pounds  ..................................  i  stone  (London  meat  market). 

14  pounds  .........................................  i  stone. 

28  pounds,  or    ) 
,  stones  |  ...............................  '  Barter 

4  quarters,  or    \ 

8  stones,  or       >  ...............................  i  hundredweight  (cwt). 

112  pounds  ' 

20  hundredweights  ...............................  i  ton. 

The  grain  above  noted,  of  which  there  are  7000  to  the  pound  avoirdupois, 
is  the  same  as  the  troy  grain,  of  which  there  are  5760  to  the  troy  pound. 

Hence  the  troy  pound  is  to  the  avoirdupois  pound  as  i  to  1.215,  or  as 
14  to  17. 


GREAT   BRITAIN   AND   IRELAND.  —  WEIGHTS.  141 

The  troy  ounce  is  to  the  avoirdupois  ounce  as  480  grains,  the  weight  of 
the  former,  to  4S7/4  grains,  the  weight  of  the  latter;  or,  as  i  to  .9115. 
In  Wales,  the  iron  ton  is  20  cwt.  of  120  Ibs.  each. 

Troy  Weight. 
24  grains  ........................................  i  pennyweight  (dwt.}. 


25  pounds  ......  .  ................................  i  quarter. 

4  quarters,  or  i  oo  pounds  ..................  i  hundredweight. 

By  troy  weight  are  weighed  gold,  silver,  jewels,  and  such  liquors  as  are 
sold  by  weight. 

Diamond  Weight. 
i  diamond  grain  .............................  0.8  troy  grain. 

i  carat  .........................................  4  diamond  grains. 

iS/4  carats  .....................................  i  troy  ounce. 

Apothecaries'  Weight. 

The  revised  table  of  weights  of  the  British  Pharmacopeia  is  as  follows: 
it  is  according  to  the  avoirdupois  scale  :  — 

437  YZ  grains  ..................................  .  .  i  ounce. 

1  6  ounces  ...................................  i  pound. 

In  the  old  table  of  Apothecaries'  Weight,  superseded  by  the  preceding 
table,  the  troy  scale  was  followed,  thus  :  — 

Old  Apothecaries'  Weight. 
20  grains  ....................................  i  scruple  (  9). 

,8  56}.'  ..drachm  (3). 


576 


1  2  ounces,  or  )  j 

60  grains  •  '  POund 


Weights  of  Current  Coins. 


i  farthing,        .  8  inch  diameter,  .............    x/10  ounce. 

i  halfpenny,  i.o  „  .............    Vs        » 

i  penny,         1.2  „  .............    «/3 

i  threepenny  piece  ............................    I/20      „ 

i  fourpenny  piece  ............................  ..    */I5       „ 

i  sixpence  .......................................    '/IQ      „ 

i  shilling  ..........................................    Vs        » 

i  florin  ...........................................    2/s        » 

i  half-crown  .....................................    T/2        „ 

5  shillings  or  10  sixpences  ............  %  ......  i  „ 

i  sovereign  ......................................    x/4  ounce,  fully. 

For  the  exact  weight  in  grains  of  these  coins,  see  Table  of  British  Money. 


142  WEIGHTS  AND   MEASURES. 

Coal  Weight. 

14  pounds i  stone. 

28  pounds i  quarter  hundredweight. 

56  pounds i  half  hundredweight. 

88  pounds i  bushel.* 

i  sack,  of  n  2  pounds i  hundredweight. 

i  double  sack,  of  224  pounds...  2  hundredweights. 

20  hundredweights,  or  I 
10  double  sacks  J  " 

26^/2  hundredweights i  chaldron  (London). 

53  hundredweights i  chaldron  (Newcastle). 

7  tons T  room. 

21  tons  4  cwt i  barge  or  keel. 

*  Sundry  Btishel  Measures. 

i  Cornish  bushel  of  coal  is  90  or  94  pounds;  heaped,  101  pounds. 
i  Welsh  bushel,  average  weight  93  pounds. 

I  Newcastle  bushel  is  80  or  84  pounds.     Bradley  Main,  92^  pounds. 
I  London  bushel,  80  or  84  pounds. 

t  In  Wales  the  miners'  coal-ton  is  21  cwt.  of  120  Ibs.  each. 


Wool  Weight. 


7  pounds 

2  cloves,  or  14  pounds 


2  stones 


y2  tods 

2  weys 

12  sacks,  or  39  hundredweight. 
12  score,  or  240  pounds 


clove. 

stone. 

tod. 

wey. 

sack. 

last. 

pack. 


Hay  and  Straw  Weight. 

truss  of  straw 36  pounds. 

load  of  straw 1 1  hundredweights,  64  pounds. 

truss  of  old  hay 56  pounds. 

load  of  old  hay 18  hundredweight. 

cubic  yard  of  old  hay 15  stone. 

truss  of  new  hay 60  pounds. 

load  of  new  hay 19  hundredweights,  32  pounds. 

cubic  yard  of  new  hay 6  stone. 

Corn  and  Flour  Weight. 

1  peck,  or  stone  of  flour 14  pounds. 

i  o  pecks i  boll  =140       „ 

2  bolls i  sack  =280       „ 

14  pecks i  barrel  =196      „ 

i  bushel  of  wheat 60       „ 

i  bushel  of  barley 47       „ 

i  bushel  of  oats 40       „ 

Six  bushels  of  wheat  should  yield  one  sack  of  flour;  i  last  of  corn  is  80  bushels. 


GREAT  BRITAIN   AND   IRELAND. — MISCELLANEOUS.  143 

MISCELLANEOUS  TABLES. — No.  27. 

Whatmaris  Drawing  Papers. — Sizes  of  Sheets. 

Antiquarian 53  inches  long,  3 1  inches  wide. 

Double-elephant 40  „  27  „ 

Atlas 34  „  26  „ 

Colombier 34  „  23  „ 

Imperial 30  „  22  „ 

Elephant 28  „  23  „ 

Super-royal 27  „  19  „ 

Royal 23  „  19 

Medium 22  „  17  „ 

Demy 20  „  15  „ 

Commercial  Numbers  and  Stationery. 

1 2  articles i  dozen. 

13  articles i  long  dozen. 

1 2  dozen i  gross. 

20  articles i  score. 

5  score i  common  hundred. 

6  score i  great  hundred. 


30  deals 

4  quarters 

24  sheets  of  paper 

20  quires 

2 1  ^  quires 

5  dozen  skins  of  parchment. 


quarter. 

hundred. 

quire. 

ream. 

printers'  ream. 

roll. 


Measures  relating  to  Building. 
Load  of  timber,  unhewn  or  rough 40  cubic  feet. 

Load,  hewn  or  squared (      5°  cubic  feet,  reckoned 

(  to  weigh  20  cwt. 

Stack  of  wood 108  cubic  feet. 

Cord  of  wood 128         „ 

(In  dockyards,  40  cubic  feet  of  hewn  timber  are  reckoned  to  weigh 
20  cwt. ;  50  cubic  feet  is  a  load.) 

i  oo  superficial  feet i  square. 

Hundred  of  deals 120  deals. 

Load  of  i-inch  plank 600  square  feet. 

(Load  of  plank  more  than  i-inch  thick  =  600  -f-  thickness  in  inches. 

Planks,  section 1 1  by  3  inches. 

Deals,  section 9  by  3      „ 

Battens,  section 7  by  2^  „ 

A  reduced  deal  is  i  %  inches  thick,  1 1  inches  wide,  and  1 2  feet  long. 

Bundle  of  4  feet  oak-heart  laths 120  laths. 

Load  of  „  „        37^  bundles. 

Bundle  of  5  feet  oak-heart  laths 100  laths. 

Load  of  „  „         30  bundles. 


144  WEIGHTS  AND   MEASURES. 

Measures  relating  to  Building  (continued.} 

Load  of  statute  bricks 500. 

Load  of  plain  tiles 1000. 

Load  of  lime 32  bushels. 

Load  of  sand 36       „ 

Hundred  of  lime 35       „ 

Hundred  of  nails,  or  tacks 1 20. 

Thousand  of  nails,  or  tacks 1 200. 

Fodder  of  lead I9//2  cwt. 

Sheet  lead 6  to  10  pounds  per  sq.  ft. 

Hundred  of  lead 112  pounds. 

Table  of  glass 5  feet. 

Case  of  glass 45  tables. 

Case  of  glass {  (Newcastle  and  Normandy 

(          glass,  25  tables). 

Stone  of  glass 5  pounds. 

Seam  of  glass 24  stone. 

Sundry  Commercial  Measures. 

Dicker  of  hides 10  skins. 

Last  of  hides 20  dickers. 

Weigh  of  cheese 256  pounds. 

Barrel  of  herrings 26  2/3  gallons. 

Cran  of  herrings 37^        „ 

Pocket  of  hops i  y2  to  2  cwt. 

Bag  of  hops 3^  cwt,  nearly. 

Last  of  potash,  cod-fish,  white  her-  ) 

rings,  meal,  pitch,  tar. }      '  2  barrels' 

Barrel  of  tar 26^  gallons. 

Barrel  of  anchovies 30  pounds. 

Barrel  of  butter 224       „ 

Barrel  of  candles 120       „ 

Barrel  of  turpentine 2  to  2  y2  cwt. 

Barrel  of  gunpowder 100  pounds. 

Last  of  gunpowder 24  barrels. 

Measures  for  Ships. 

i  ton,  displacement  of  a  ship, 35  cubic  feet. 

i  ton,  registered  internal  capacity  of  do., 100       do. 

i  ton,  shipbuilders'  old  measurement, 94        do. 

COMPARISON  OF  COMPOUND  UNITS. — Tables  No.  28. 

Measures  of  Velocity. 

j    1.467  feet  per  second, 
unite  per  hour |  88.0  feet  per  minute. 

i  knot  per  hour 1.688  feet  per  second. 

i  foot  per  second .682  mile  per  hour. 

i  foot  per  minute .01136  mile  per  hour. 


GREAT  BRITAIN   AND   IRELAND. — COMPOUND   UNITS.          145 
Measures  of  Volume  and  Time. 

i  cubic  foot  per  second..  I      2'222  cub!c  yards  Per  minute- 

I  * 3 3- 333  CUD1C  yards  per  hour. 

i  cubic  foot  per  minute 2.222  cubic  yards  per  hour. 

i  cubic  yard  per  hour .45  cubic  foot  per  minute. 

i  cubic  inch  per  second . .  j      2>o83  cubic  foot  Per  hour- 

(     12.984  gallons  per  hour. 

i  gallon  per  second 569. 1 24  cubic  feet  per  hour. 

i  gallon  per  min  ute 9.485  cubic  feet  per  hour. 

Measures  of  Pressure  and  Weight.     (See  also  page  127.) 

(.  144  Ibs.  per  square  foot, 
i  Ib.  per  square  inch <j  1296  Ibs.  per  square  yard. 

(  .5786  ton  per  square  yard. 

i  atmosphere  (14.7  Ibs.  per  {  Q                                        , 

square  inch)' . . . . . .  j  8'5°3  l°n  P<*  square  yard. 

("  .00694  Ib.  per  square  inch, 

i  Ib.  per  square  foot <  .1111  ounce  per  square  inch. 

(  .0804  cwt.  per  square  yard, 

i  Ib.  per  square  inch  ..         .  (  2-°355  inches  of  mercury  at  32°  F. 

(  2.308  feet  of  water  at  52°.3  F. 

i  inch  of  mercury  at  32°  F.     j          '^l  J.b'  PeJ  S(luare  inch' 

(        1.133  feet  of  water  at  52^3  F. 

(          -4333  Ib.  per  square  inch, 
i  foot  of  water,  at  52°.3  F. ..  <      62.4  Ibs.  per  square  foot. 

(          .8823  inch  of  mercury  at  32°  F. 

Measures  of  Weight  and  Volume. 

{405.1  grains  per  cubic  inch. 
.926  ounce  per  cubic  inch. 
24.107  cwt.  per  cubic  yard. 
1.205  tons  per  cubic  yard. 

i  grain  per  cubic  inch ..  /    3-95°  ounces  per  cubic  foot. 

(      .247  pounds  per  cubic  foot. 

i  ounce  per  cubic  inch 1 08  pounds  per  cubic  foot. 

i  cwt.  per  cubic  yard 4. 1 48  pounds  per  cubic  foot. 

i  ton  per  cubic  yard 82.963  pounds  per  cubic  foot. 

'  i  pound  for  1122  cubic  feet 


i  grain  per  gallon  (i  in  70,000 
parts  by  weight,  of  water) 


i  pound  for  41.5  cubic  yards. 
i  pound  for  31.8  cubic  metres. 


220  grains  for  i  cubic  metre. 
.503  ounce  for  i  cubic  metre. 

Measures  of  Power. 

i  Ib.  of  fuel  per  H.P.  (  ^980,000  foot-pounds  per  Ib.  of  fuel, 
per  hour  221.76  million  foot-pounds  per  cwt.  of  fuel. 

"  (         2,565  heat-units  per  Ib.  of  fuel, 
i, 000,000 foot-pounds  )  0  rr    , 

per  Ib.  of  fuel J  x'98  pounds  of  fuel  per  H.P.  per  hour. 

10 


146  WEIGHTS  AND   MEASURES. 


FRANCE.— THE    METRIC    STANDARDS   OF  WEIGHTS 
AND   MEASURES. 

The  primary  metric  standards  are: — the  metre,  the  unit  of  length;  and 
the  kilogramme,  the  unit  of  weight,  derived  from  the  metre :  being  the  two 
platinum  standards  deposited  at  the  Palais  des  Archives  at  Paris. 

The  standard  metre  is  defined  to  be  equal  to  one  ten-millionth  part  of 
the  quadrant  of  the  terrestrial  meridian,  that  is  to  say,  the  distance  from 
the  equator  to  the  pole,  passing  through  Paris,  which,  by  the  latest  and 
most  authoritative  measurement,  is  39.3762  inches,  in  terms  of  the  Imperial 
standard  at  62°  F.  By  the  latest  and  most  accurate  measurement,  the 
actual  standard  metre  at  32°  F.  is,  in  terms  of  the  Imperial  standard  at  62°  F., 
39.37043  inches;  and  its  legal  equivalent,  declared  in  the  Metric  Act  of 
1864,  is  39.3708  inches,  being  the  same  as  that  adopted  in  France. 

The  standard  kilogramme  (1000  grammes)  is  defined  to  be  the  weight  of 
a  cubic  decimetre  of  distilled  water  at  its  maximum  density,  at  4°.o  C. 
or  39°.  i  F.  This  is  legally  taken  to  be 

2.20462125  Ibs.,  or 

2  Ibs.,  3  oz.,  4.383  drachms,  or 

15,432.34874  grains. 

There  is  in  the  Standard  Department  at  Westminster  a  newly-constructed 
subdivided  standard  yard,  laid  down  upon  a  bar  of  Baily's  metal,  upon 
which  a  subdivided  metre  has  also  been  laid  down. 

The  metric  unit  of  capacity  is  the  litre,  denned  to  be  equal  to  a  cubic 
decimetre.  Its  Imperial  equivalent  is  0.22009  gallon. 

There  is  no  other  official  standard  of  weight  and  measure  in  France 
than  the  metre  and  the  kilogramme;  there  is  no  standard  litre  or  unit  of 
capacity. 

The  metric  system  is  not  really  founded  on  the  length  of  a  quadrant  of  the 
meridian,  and  although  it  is  described  as  a  scientific  system,  because  of  the 
simple  and  definite  relation  between  the  metre,  which  is  its  basis  and  unit  of 
length,  and  the  kilogramme  and  litre,  which  are  the  units  of  weight  and 
capacity,  it  is  admitted  that  it  has  been  found  impossible  practically  to 
carry  it  out  with  scientific  accuracy.  The  standard  kilogramme  is  admitted 
not  to  be  actually  the  weight  of  a  cubic  decimetre  of  pure  water  at  the 
specified  temperature,  nor  the  litre  a  measure  of  capacity  holding  a  cubic 
decimetre  of  pure  water.  The  real  standard  unit  of  weight  is  declared,  even 
by  men  of  science  in  France,  to  be  merely  the  platinum  kilogramme-weight 
deposited  at  the  Palais  des  Archives,  as  the  real  standard  unit  and  basis  of 
the  metric  system  is  the  platinum  metre,  also  deposited  there.  It  is  an 
accomplished  fact,  however,  that  all  civilized  nations  have  tacitly  agreed  to 
recognize  the  metric  system  as  affording  for  the  future  the  advantages  of  a 
universal  system  of  weights  and  measures,  and  to  adopt  the  standards 
deposited  at  the  Palais  des  Archives  as  the  primary  units  of  the  system. 

The  French  metric  system  has  been  adopted,  and  its  use  made  compul- 
sory by  the  following  States: — France  and  Belgium,  in  1801;  Holland,  in 
1819;  Greece,  in  1836;  Italy  and  Spain,  in  1859;  Portugal,  in  1860-68; 
the  German  Empire,  in  1872;  Colombia,  Venezuela,  in  1872;  Ecuador, 


FRANCE. — THE   METRIC   STANDARDS — LENGTH.  147 

Brazil,  Peru,  and  Chili,  in  1860;  also  by  the  Argentine  Confederation,  and 
Uruguay. 

Great  Britain  and  Ireland,  in  1864,  adopted  the  metric  system,  so  far  as  to 
render  contracts  in  terms  of  the  French  metric  system  permissive. 

The  United  States  of  North  America,  in  1866,  legalized  the  French  metric 
system  concurrently  with  the  old  system;  it  was  also  legalized  in  British 
North  America. 

Switzerland,  in  1856,  legalized  the  foot  of  three  decimetres  as  the  unit  of 
length,  with  a  decimal  scale;  the  unit  of  weight  being  the  pound  of  500 
grammes,  or  half  a  kilogramme,  with  two  distinct  scales  of  multiples  and 
parts,  one  decimal,  the  other  according  to  the  old  custom. 

Sweden,  in  1855,  by  a  law  made  compulsory  in  1858,  adopted  a  decimal 
system  of  weights  and  measures,  having  for  the  unit  of  length  a  foot  of  0.297 
metre,  and  the  unit  of  weight  a  pound  of  0.42  kilogramme: — being  the 
original  units  decimally  treated. 

Denmark  adopted  the  metric  system  so  far  as  the  pound  of  500  grammes. 
The  pound  is  decimally  treated,  and  since  1863  the  use  of  the  greatest 
parts  of  the  multiples  of  the  pound  not  conformable  to  decimal  sub- 
division has  been  prohibited. 

Austria,  in  1853,  adopted  a  pound  of  500  grammes,  with  decimal  divisions, 
for  customs  and  fiscal  purposes. 

Russia  awaits  the  example  of  those  countries  with  which  she  has 
commercial  relations,  especially  of  England. 

In  Morocco  and  Tunis,  the  weights  and  measures  have  no  relation  with 
the  metric  system. 

On  the  2oth  May,  1875,  the  international  convention  for  the  adoption 
of  the  French  metrical  system  of  weights  and  measures  was  signed  at  Paris 
by  the  plenipotentiaries  of  France,  Austria,  Germany,  Italy,  Russia,  Spain, 
Portugal,  Turkey,  Switzerland,  Belgium,  Sweden,  Denmark,  the  United 
States,  the  Argentine  Republic,  Peru,  and  Brazil.  A  special  clause 
reserves  to  States  not  included  in  the  above  list  the  right  of  eventually 
adhering  to  the  convention. 

I.  FRENCH  MEASURES  OF  LENGTH. — Table  No.  29. 

i  o  millimetres i  centimetre. 

10  centimetres i  decimetre. 

10  decimetres,  or  "j 

100  centimetres,  or  > i  METRE. 

1000  millimetres        ) 

10  metres i  decametre. 

i  o  decametres i  hectometre. 

10  hectometres,  or  1000  metres i  KILOMETRE  (kilo.) 

10  kilometres i  myriametre. 


i  toise  (old  measure) =1.949  metres. 

1000  toises i  mille  =  1.949  kilometres. 

2000  toises i  itinerary  league       =3.898          „ 

2280.329  toises i  terrestrial  league     =  4. 444          „ 

2850.411  toises i  nautical  league        =5-555          » 

i  noeud (British  nautical  mile)  =  1.855          „ 


148 


WEIGHTS  AND   MEASURES. 


FRENCH  WIRE-GAUGES  (Jauges  de  Fits  de  JFer). 

The  French  wire-gauge,  like  the  English,  has  been  subject  to  variation. 
Table  No.  30  contains  the  values  of  the  "points,"  or  numbers,  of  the 
Limoges  gauge;  table  No.  31  gives  the  values  of  a  wire-gauge  used  in  the 
manufacture  of  galvanized  iron;  and  table  No.  32  the  values  of  a  gauge 
which  comprises  wire  and  bars  up  to  a  decimetre  in  diameter. 

FRENCH  WIRE-GAUGE  (Jauge  de  Limoges]. — Table  No.  30. 


Number. 

Diameter. 

Number. 

Diameter. 

Number. 

Diameter. 

Millimetre. 

Inch. 

Millimetre. 

Inch. 

Millimetre. 

Inch. 

0 

•39 

.0154 

9 

'•35 

•0532 

18 

3-40 

•134 

I 

•45 

.0177 

10 

1.46 

•°575 

i9 

3-95 

.156 

2 

.56 

.0221 

ii 

1.68 

.O66l 

20 

4-50 

.177 

3 

.67 

.0264 

12 

i.  80 

.0706 

21 

5.10 

.201 

4 

•79 

.O3II 

13 

1.91 

.0752 

22 

5-65 

.222 

5 

.90 

•0354 

14 

2.02 

•0795 

23 

6.20 

.244 

6 

1.  01 

.0398 

15 

2.14 

.0843 

24 

6.80 

.268 

7 

1.  12 

.0441 

16 

2.25 

.0886 

8 

1.24 

.0488 

17 

2.84 

.112 

FRENCH  WIRE-GAUGE  FOR  GALVANIZED  IRON  WIRE. — Table  No.  31, 


Number. 

Diameter. 

Number. 

Diameter. 

Number. 

Diameter. 

M'metre. 

Inch. 

M'metre. 

Inch. 

M'metre. 

Inch. 

I 

.6 

.0236 

9 

1.4 

•Q551 

17 

3-o 

.118 

2 

•  7 

.0276 

10 

i-5 

.0591 

18 

3-4 

•134 

3 

.8 

•03^ 

ii 

1.6 

.0630 

*9 

3-9 

•154 

4 

•9 

•°354 

12 

1.8 

.0709 

20 

4.4 

•173 

5 

I.O 

•0394 

J3 

2.0 

.0787 

21 

4.9 

•193 

6 

i.i 

•0433 

14 

2.2 

.0866 

22 

5-4 

.213 

7 

1.2 

•0473 

15 

2.4 

•0945 

23 

5-9 

.232 

8 

'•3 

.0512 

16 

2.7 

.I06 

FRENCH  WIRE- AND  BAR -GAUGE. — Table  No.  32. 


Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

Mark. 

Size. 

Millimetre. 

Millimetre. 

Millimetre. 

Millimetre. 

P 

5 

8 

13 

16 

27 

24 

64 

I 

6 

9 

14 

i7 

30 

25 

70 

2 

7 

10 

15 

18 

34 

26 

76 

3 

8 

II 

16 

I9 

39 

27 

82 

4 

9 

12 

18 

20 

44 

28 

88 

5 

10 

13 

20 

21 

49 

29 

94 

6 

ii 

!4 

22 

22 

54 

30 

IOO 

7 

12 

15 

24 

23 

59 

FRANCE.  —  THE   METRIC   STANDARDS.  149 

II.  FRENCH  MEASURES  OF  SURFACE.  —  Table  No.  33. 

i  oo  square  millimetres  .............   i  square  centimetre. 

i  oo  square  centimetres  .............   i  square  decimetre. 


'square  metre,  or 

ioo  square  metres,  or  centiares...   i  square  decametre,  or  ARE. 
100  square  decametres,  or  ares...   i  square  hectometre,  or  HECTARE. 
ioo  square  hectometres,  or  hectares  i  square  myriametre. 

Land  is  measured  in  terms  of  the  centiare,  the  are,  and  the  hectare  or 
arpent  metrique  (metric  acre).     There  is  also  the  decare,  of  10  ares. 

III.  FRENCH  MEASURES  OF  VOLUME.  —  Tables  No.  34. 

Cubic  Measure. 

1000  cubic  millimetres  ................   i  cubic  centimetre. 

i  ooo  cubic  centimetres  ...............   i  cubic  decimetre. 

i  ooo  cubic  decimetres  ................   i  cubic  metre. 

Wood  Measure. 

10  decisteres  ...........................  i  stere*  (i  cubic  metre). 

i  voie  (Paris)  .........................  2  steres. 

i  voie  de  charbon  (charcoal)  ......  0.2  stere  (x/s  cubic  metre). 

i  corde  .................................  4  steres. 

*  The  stere  measures  1.  14  metres  x  0.88  metre  x  i  metre,  the  billets  of  wood  being 
x.  14  metres  in  length. 

IV    FRENCH  MEASURES  OF  CAPACITY.  —  Tables  No.  35. 

Liquid  Measure. 

i  o  cubic  centimetres  ....................  i  centilitre. 

10  centilitres  ..............................  i  decilitre. 

i  o  decilitres  ..............................  i  LITRE. 

10  litres  ....................................  i  decalitre. 

Dry  Measure. 

10  litres.  .......................................   i  decalitre. 

10  decalitres,  or 
ioo  litres 


1000 


\  i  hectolitre. 

10  hectolitres,  or )  .  ..  ..       ,        ,  .  . 

oo  litres  / i  kilolitre  (i  cubic  metre). 


The  use  of  measures  equal  to  a  double-litre,  a  half -litre,  a  double-decilitre,  a 
half-decilitre,  is  sanctioned  by  law. 


ISO 


WEIGHTS  AND   MEASURES. 


V.    FRENCH  MEASURES  OF  WEIGHT.  —  Table  No.  36. 


i  o  milligrammes  ..................  i  centigramme. 

10  centigrammes  ..................  i  decigramme. 

10  decigrammes  ..................  i  GRAMME. 

10  grammes  ........................  i  decagramme. 

10  decagrammes  ..................  i  hectogramme. 

10  hectogrammes,  or  I    .........  %  KILOGRAMME 

1000  grammes  j 

i  o  kilogrammes  ...................  i  myriagramme. 


M    } 


!es'or      i  quintal  metrique. 

ioo  kilogrammes  j 

10  quintaux,  or  )  f         i  millier,  tonneau  de  mer,  or  tonne 

j  "  "  (  (weight  of  i  cubic  metre  of  water  at  39°.  i). 


1000  kilogrammes 


EQUIVALENTS  OF  BRITISH   IMPERIAL  AND  FRENCH   METRIC 
WEIGHTS   AND    MEASURES. 

I.  MEASURES  OF  LENGTH. — Tables  No.  37. 

A   DECIMETRE   DIVIDED   INTO   CENTIMETRES   AND   MILLIMETRES. 


INCHES   AND   TENTHS. 


METRIC  DENOMINATIONS 
AND  VALUES. 

EQUIVALENTS  IN  IMPERIAL  DENOMINATIONS. 

Metres. 

Inches. 

Feet. 

Yards. 

Miles. 

•r 

i  millimetre 

i/ 

/  IOOO 

;    0.03937 







i  centimetre 

I/ 

/  IOO 

-    0.39370 







i  decimetre 

'Ao 

*     3-93704 







I  METRE    .... 

I 

-39-37043 

-     3.28087 

1.09362 



I'dekametre 

IO 

=  32.80869 

10.93623 



i  hectometre 

IOO 

— 



109.36231 



I  KILOMETRE 

1,000 

— 

=  3280.87 

7f    1,093.6231 

-0.62138 

i  mynametre 

10,000 

— 



=  10,936.231 

-6.21377 

IMPERIAL  AND   METRIC   EQUIVALENTS. 


Tables  No.  37  (continued). 


IMPERIAL  DENOMINATIONS. 

\  - 

EQUIVALENTS  IN  METRIC  DENOMINATIONS,  ^x 

Centimetres. 

Metres. 

Kilometres. 

inch  (25  4  millimetres)      .    ... 

=  2.53995 

0.30480 
=            0.91439 
1.82878 
5.02915 
=         2O.II662 
=      2OI.l662 
=  1,609.3296 

=  0.20117 
-  1.60933 

foot  or  1  2  inches       

yard,  or  3  feet,  or  36  inches.... 
fathom,  or  2  yards,  or  6  feet.... 
pole  or  5  /^  yards 

chain,  or  4  poles,  or  22  yards... 
furlong,  40  poles,  or  220  yards 
i  mile,  8  furlongs,  or  1760  yards 

EQUIVALENT  VALUES  OF  MILLIMETRES  AND  INCHES. — Tables  No.  38. 

MILLIMETRES  =  INCHES. 


Millimetres. 

Inches. 

Millimetres. 

Inches. 

Millimetres. 

Inches. 

Millimetres. 

Inches. 

I 

•0394 

27 

1.0630 

53 

2.0866 

79 

3-II03 

2 

.0787 

28 

I.I024 

54 

2.I26O 

80 

3.1496 

3 

.Il8l 

29 

I.I4I7 

55 

2.1654 

81 

3.1890 

4 

•1575 

30 

1.1811 

56 

2.2047 

82 

3.2284 

5 

.1968 

31 

1.2205 

57 

2.2441 

83 

3.2677 

6 

.2362 

32 

1.2598 

58 

2.2835 

84 

3.3071 

7 

.2756 

33 

1.2992 

59 

2.3228 

85 

3-3465 

8 

•3I50 

34 

1.3386 

60 

2.3622 

86 

3-3859 

9 

•3543 

35 

1.3780 

61 

2.4016 

87 

3-4252 

10 

•3937 

36 

1-4173 

62 

2.4410 

88 

3.4646 

ii 

•4331 

37 

1.4567 

63 

2.4803 

89 

3-5040 

12 

.4724 

38 

1.4961 

64 

2.5197 

90 

3-5433 

13 

.5118 

39 

I-5354 

65 

2.5591 

9i 

3.5827 

14 

•5512 

40 

1.5748 

66 

2.5984 

92 

3.6221 

15 

.5906 

4i 

1.6142 

67 

2.6378 

93 

3.6614 

16 

.6299 

42 

1-6536 

68 

2.6772 

94 

3.7008 

17 

.6693 

43 

1.6929 

69 

2.7166 

95 

3.7402 

18 

.7087 

44 

1.7323 

70 

2.7559 

96 

3.7796 

*$ 

.7480 

45 

1.7717 

7i 

2-7953 

97 

3.8189 

20 

.7874 

46 

1.8110 

72 

2.8347 

98 

3-8583 

21 

.8268 

47 

1.8504 

73 

2.8740 

99 

3.8977 

22 

.8661 

48 

1.8898 

74 

2.9134 

100 

3-9370 

23 

•9055 

49 

1.9291 

75 

2.9528 

=  i  decimetre. 

24 

.9449 

50 

1.9685 

76 

2.9922 

25 

.9843 

51 

2.0079 

77 

3-°3I5 

26 

1.0236 

52 

2.0473 

78 

3.0709 

152 


WEIGHTS  AND   MEASURES. 


Tables  No.  38  (continued}. 

INCHES  DECIMALLY  =  MILLIMETRES. 


Inches. 

Millimetres. 

Inches. 

Millimetres. 

Inches. 

Millimetres. 

Inches. 

Millimetres. 

.OI 

•25 

.26 

6.60 

.60 

15.2 

.94 

23.9 

.02 

•51 

.28 

7.II 

.62 

15-7 

.96 

24.4 

•03 

.76 

•30 

7.62 

.64 

I6.3 

.98 

24.9 

.04 

i.  02 

•32 

8.13 

.66 

16.8 

I.OO 

25-4 

•°5 

1.27 

•34 

8.64 

.68 

17-3 

2.0O 

50.8 

.06 

1-52 

•36 

9.14 

.70 

17.8 

3.00 

76.2 

.07 

1.78 

.38 

9-65 

.72 

18.3 

4.OO 

101.6 

.08 

2.03 

.40 

10.2 

•74 

18.8 

5.OO 

127.0 

.09 

2.29 

.42 

10.7 

.76 

19-3 

6.00 

152.4 

.IO 

2.54 

.44 

II.  2 

.78 

19.8 

7.00 

177.8 

.12 

3-05 

.46 

II.7 

.80 

20.3 

8.00 

203.2 

.14 

3.56 

.48 

12.2 

.82 

20.8 

9.00 

228.6 

.16 

4.06 

•5° 

12.7 

.84 

21.3 

10.00 

254.0 

.18 

4-57 

•S2 

13.2 

.86 

21.8 

11.00 

279.4 

.20 

5.08 

•54 

13-7 

.88 

22.4 

12.00 

304.8 

.22 

5-59 

.56 

14.2 

.90 

22.9 

=  i  foot. 

.24 

6.10 

•58 

14.7 

.92 

23-4 

INCHES  IN  FRACTIONS  =  MILLIMETRES. 


Eighths. 

Sixteenths. 

Thirty-seconds. 

Millimetres. 

Eighths. 

Sixteenths. 

Thirty-seconds. 

Millimetres. 

I 

•79 

17 

13-5 

I 

2 

1-59 

9 

18 

14-3 

3 

2.38 

I9 

I 

2 

4 

3-17 

5 

IO 

20 

15-9 

5 

3-97 

21 

16.7 

3 

6 

4.76 

ii 

22 

17-5 

7 

5.56 

23 

18.3 

2 

4 

8 

6-35 

6 

12 

24 

19.0 

9 

7.14 

25 

19.8 

5 

IO 

7-94 

13 

26 

2O.6 

ii 

8.73 

27 

21.4 

3 

6 

12 

9-52 

7 

14 

28 

22.2 

X3 

10.32 

29 

23.0 

7 

14 

ii.  ii 

15 

30 

23.8 

15 

11.91 

31 

24.6 

4 

8 

16 

12.7 

8 

16 

32 

25-4 

By  means  of  the  preceding  tables  of  equivalent  values  of  inches  and 
millimetres,  the  equivalent  values  of  inches  in  centimetres  and  decimetres, 
and  even  in  metres,  may  be  found  by  simply  altering  the  position  of  the 
decimal  point.  This  method  naturally  follows  from  the  decimal  subdivisions 
of  French  measure. 

Take,  for  example,  the   tabular  value  of  i    millimetre,  and   shift  the 


IMPERIAL  AND   METRIC   EQUIVALENTS. 


153 


decimal  point  successively,  by  one  digit,  towards  the  right-hand  side;  the 
values  of  a  centimetre,  a  decimetre,  and  a  metre  are  thereby  expressed  in 
inches,  as  follows : — 

i  millimetre °394  inches. 

i  centimetre Q-394        » 

i  decimetre 3.94          „ 

i  metre 39.4 

At  the  same  time,  it  appears  that,  by  selecting   the  tabular  value  of 
10  millimetres,  the  value  of  its  multiples  are  given  more  accurately,  thus, — 

10  millimetres,  or  i  centimetre °-3937  inches. 

i  decimetre 3.937         „ 

i  metre 39.37  „ 

Again : — 

100  millimetres,  or  i  decimetre  =     3-937  inches. 

i  metre =39-37      » 

Similarly,  for  example : — 

.32  inch  =         8.13  millimetres. 
3.2      „     =       81.3          „ 

=  ( 813.0  „  or 

"  \       .813  metre. 


32.0 


II."  SQUARE  MEASURES,  OR  MEASURES  OF  SURFACE.— Tables  No.  39. 


METRIC 

i  square  millimetre 

i  square  centimetre 

i  square  decimetre  


i  square  metre,  or  centiare 

i  ARE,    or    square    dekametre,    or    100 
square  metres 

i  hectare,  or  metrical  acre,  or  100  ares, 
or  10,000  square  metres 


•j 


IMPERIAL  SQUARE  MEASURES. 

.00155  square  inch. 
.155  square  inch. 
15.5003  square  inches. 
10.7641  square  feet,  or 

1.1960  square  yards. 
1076.41  square  feet,  or 

119.60  square  yards. 
11,960.11  square  yards,  or 

2.4711  acres,  or 
2  acres  and  2280.1240  square 
yards. 


IMPERIAL 


METRIC  SQUARE  MEASURES. 


Imperial  Measures. 

Square 
Centimetres. 

Square  Metres. 

Ares. 

Hectares. 

i  square  inch  

—  6.4SIA8 

i  square  ft.,  or  144  sq.  inches 
i  square  yard,  or  9  square  ) 
feet,  or  1296  sq.  inches  ( 
i    perch  or  rod,  or   30^ 
square  yards 

=         0.092901 
=         0.836112 

=       25.292 

— 

— 

i  rood,  or  40  perches,  or' 
1210  square  yards  
i  acre,  or  4  roods,  or  4840 
SQua.re  yards 

— 

=  1011.696 
=  4046.782 

=  10.11696 
=  40.4678 

=      0.40468 

i  square  mile,  or  640  acres 

— 

— 

— 

=  258.98944 

154 


WEIGHTS   AND   MEASURES. 


III.   CUBIC  MEASURES.  —  Tables  No.  40. 


METRIC 


IMPERIAL  CUBIC  MEASURES. 


i  cubic  centimetre =        0.061025  cubic  inch. 

,  .     ,     .  f  61.021524  cubic  inches,  or 

i  cubic  decimetre -  {    0^3*56  cubic  foot 

/  35-3*56  cubic  feet,  or 
i  cubic  metre =  |    1.308  cubic  yards. 

IMPERIAL  =  METRIC  CUBIC  MEASURES. 

i  cubic  inch =      16.387  cubic  centimetres. 

,  •    f  (  28.3153  cubic  decimetres,  or 

icubicfoot =  1    0.028315  cubic  metre. 

i  cubic  yard =       0.764513  cubic  metre. 

WOOD  MEASURE. 

i  stere,  or  cubic  metre 

i  decistere 3.53 16  cubic  feet. 

.     ,        .    /        ,v  ,v        T»    •    (  70.6312  cubic  feet,  or 

i  voie  de  bois  (wood),  or  2  steres,  Pans  <  \.-  ^^  cubic    ards 

i  voie  de  charbon  (charcoal)  =  i  sack   f  5  ^  bushels,  or 

=  '/j  stere t      7.063  cubic  feet. 

i  corde  of  wood  =  4  cubic  metres 141.26  cubic  feet. 

IV.   MEASURES  OF  CAPACITY. — Tables  No.  41. 

METRAN?VNA°LMu!ESATIONS  EQUIVALENTS  IN  IMPERIAL  DENOMINATIONS. 

Litres.        Gills.  Pints.          Quarts.  Gallons.  Bushels.      Quarters. 

Centilitre X/JOO    0.0704   0.0176 

Decilitre J/io     0-7043    0.1761 

Dekalitre 10       —  2.2009     °-275I 

Hectolitre 100       —  22.009        2.7511    0.344 

Kilolitre 1000       —  220.09        27-5IZ      3-439 

EQUIVALENTS  IN  METRIC  DENOMINATIONS. 
IMPERIAL  DENOMINATIONS. 

Litres.  Dekalitres.  Hectolitres. 

gill =  0.1420 

pint,  or  4  gills =  0.5679 

quart,  or  2  pints =  1-1359 

gallon,  or  4  quarts =  4-5435 

peck,  or  2  gallons =  9.0869       =   0.9087 

bushel,  or  8  gallons =  36.3477       =    3.6348 

quarter,  or  8  bushels  =290.7816       =29.0782        =2.9078 


IMPERIAL   AND   METRIC   EQUIVALENTS. 


155 


V.  MEASURES  OF  WEIGHT. — Tables  No.  42. 

METRIC  WEIGHTS    =    IMPERIAL  AVOIRDUPOIS  WEIGHTS. 

i  kilogramme  —  2  Ibs.  3  oz.  4  drachms,  10.473 7 4  grains. 


METRIC  WEIGHTS. 

EQUIVALENTS  IN  IMPERIAL  DENOMINATIONS. 

Grammes. 

Grains. 

Ounces. 

Pounds. 

Hundred- 
weights. 

Tons. 

Milligramme  
Centigramme  

x/iooo 
Vioo 

'/xo 
I 
10 
100 
1,000 
10,000 
100,000 

1,000,000 

0.0154 
0.1543 
1-5432 
I5-4323 
154.3235 
I543-2349 
15432.3487 

0.3527 
3.5274 
35.2739 

2.2046 
22.0462 
220.462  1 
2204.6212 

1.9684 
19.6841 

0.9842 

Decigramme 

GRAMME    

Dekagramme  
Hectogramme  .  . 

KILOGRAMME  
Myriagramme  
Quintal,  or  100  kilog. 
Millier,  or  metric  ton 

IMPERIAL  AVOIRDUPOIS 


METRIC  WEIGHTS. 


IMPERIAL  AVOIRDUPOIS 
WEIGHTS. 

Grammes. 

Decigrammes. 

Kilogrammes. 

Millier,  or 
Metric  Ton. 

i  drachm  

=        1.77184 

i  ounce,  or  1  6  drams 

=     28.34954 

=     2.83495 





i  pound,  or  1  6  ounces 

=  453.59265 

-45.35926 

0-45359 



i  hundredweight,     ) 
or  112  pounds  J 

=     50.80237 

— 

i  ton,  or  20  hun-  ) 
dredweights       J 



=  1016.04754 

=  I.OI6O4 

METRIC  WEIGHTS    =    IMPERIAL  TROY  WEIGHTS. 

i  kilogramme  =  2  troy  Ibs.  8  oz.  3  dwts.,  .34874  grain. 


METRIC  WEIGHTS. 

Grains. 

Pennyweights. 

Ounces. 

Troy  Pound. 

Milligramme  .  .  . 
Centigramme  ... 
Decigramme  .  .  . 
GRAMME  

0.01543 
O.I5432 
I-54323 
—              jc  A727A 







Dekagramme... 
Hectogramme.. 
KILOGRAMME... 

A  j'^rj^o^r 
154.32349 
-       1543.23487 
=  15,432.34874 

-0.64301 
=  6.43014 

=     0.32I5I 
=     3-2I507 
-32.15073 

=  2.67922 

iS6 


WEIGHTS   AND   MEASURES. 

IMPERIAL  TROY    =     METRIC  WEIGHTS. 


IMPERIAL  TROY  WEIGHTS. 

EQUIVALENTS  IN  METRIC  DENOMINATIONS. 

Milligramme. 

Gramme. 

Dekagramme. 

Hecto- 
gramme. 

Kilo- 
gramme. 

i  troy  grain  

64.79895 

0.06480 

J-SSS1? 
3I-I°349 
373-24i95 

3-II035 
37-324I9 

3-73242 

0-37324 

i     „    dwt,  or  24  gr. 
i     „    oz.,  or  480   „ 
i     „    Ib.,or5,y6o  „ 

APPROXIMATE  EQUIVALENTS  OF  ENGLISH  AND 
FRENCH  MEASURES. 

The  following  are  approximately  equal  English  and  French  measures  of 
length : — 

i  pole,  or  peflch  (5^  yards)...       5  metres  (exactly  5.029  metres). 

i  chain  (22  yards) 20  metres  (exactly  20.1166  metres). 

i  furlong  (220  yards) 200  metres  (exactly  201.166  metres). 

5  furlongs i  kilometre  (exactly  1.0058  kilometres). 

f      3  decimetres  (exactly  3. 048  decimetres),  or 
"  jj    30  centimetres. 

One  metre  =  3. 28  feet  =  3  feet  3  inches  and  3  eighths  all  but  l/5I2  inch; 
=  40  inches  nearly  (  T/64th  or  1.6  per  cent.  less). 

.100  metre  (i  decimetre)  =  4  inches  nearly  (exactly  3I5/i6  inches). 
.010  metre  (i  centimetre)  =    .4  inch,  or  */10ths  inch,  nearly. 
.001  metre  (i  millimetre)  =    .04  inch,  or  Viooths  inch,  or  two-thirds 

of  I/\6  inch,  or  x/25  inch,  nearly. 

One  inch  is  about  2^  centimetres  (exactly  2.54). 

One  inch  is  about  25  millimetres  (exactly  25.4). 

One  yard  is  "/i2ths  of  a  metre,     n  metres  are  equal  to  12  yards. 

Approximate  rule  for  converting  metres,  or  parts  of  metres,  into  yards : — 
Add  Vnth  (i£  per  cent.  less). 

For  converting  metres  into  inches: — Multiply  by  40;  and  to  convert 
inches  into  metres,  or  parts  of  metres,  divide  by  40. 

One  kilometre  is  about  ^6  mile  (it  is  0.6  per  cent.  less). 

One  mile  is  about  1.6  or  i  s/s  kilometres  (it  is  0.6  per  cent.  Iess)  =  i6io 
metres,  about. 

With  respect  to  superficial  measures: — 
One  square  centimetre  is  about  ye.  5  part  of  a  square  inch. 
One  square  inch  is  equal  to  about  6.5  square  centimetres. 
One  square  metre  contains  fully  10^  square  feet,  or  nearly  iI/5  square  yards. 
One  square  yard  is  nearly  6/7  ths  of  a  square  metre. 
One  acre  is  over  4000  square  metres  (about  1.2  per  cent.  more). 
One  square  mile  is  nearly  260  hectares  (about  0.4  per  cent.  less). 


FRENCH   AND   ENGLISH   COMPOUND   UNITS.  157 

With  respect  to  cubic  measures,  and  to  capacity : — 
One  cubic  yard  is  about  ^  cubic  metre  (it  is  2  per  cent.  more). 
One  cubic  metre  is  nearly  ij^  cubic  yard  (it  is  i^i  per  cent.  less). 
One  cubic  metre  is  nearly  35  x/3  cubic  feet  (it  is  .05  per  cent.  less). 
One  litre  is  over  i^  pints  (it  is  0.57  per  cent.  more). 
One  gallon  contains  above  4^  litres  (it  holds  about  i  per  cent.  more). 
One  kilolitre  (a  cubic  metre)  holds  nearly  i  ton  of  water  at  62°  F.  (i^ 
per  cent,  less),  or  220^  gallons. — One  cubic  foot  contains  28.3  litres. 

With  respect  to  weights: — The  ton  and  the  gramme  stand   at  nearly 
equal  distances  above  and  below  the  kilogramme,  thus : — 

i  ton  is 1,016,047.5  grammes, 

i  kilogramme  is 1,000.0  grammes, 

i  gramme i.o  gramme, 

in  the  ratio  of  about  1,000,000  :  1,000  :  i. 

One  gramme  is  nearly  15^  grains  (about  ^  per  cent.  less). 
One  kilogramme  is  about  2  I/5  pounds  avoirdupois  (about  J/4  per  cent, 
more). 

A  thousand  kilogrammes,  or  a  metric  ton,  is  nearly  one  English  ton 
(about  i  ]/?,  per  cent.  less). 

One  hundredweight  is  nearly  5 1  kilogrammes  (  2/5  per  cent.  less). 


EQUIVALENTS   OF   FRENCH   AND    ENGLISH   COMPOUND 
UNITS    OF    MEASUREMENT. 

Weight,  Pressure,  and  Measure. 

i  kilogramme  per  metre j          '6^  Poun<J  Per  foot', 

1         2.016  pounds  per  yard. 

i  pound  per  foot 1.488  kilogrammes  per  metre. 

i  pound  per  yard .496  kilogramme  per  metre. 

1000  kilogrammes  per  metre .300  ton  per  foot. 

i  ton  per  foot 3333-333  kilogrammes  per  metre. 


1000  tons    er  mile' 


i  ton  per  mile  ...........................  63-1.0  kilogrammes  per  kilometre. 

,  .,  •„•  f  1422.^2  pounds  per  square  inch. 

i  kilogramme  per  square  millimetre  \  j^^     P  ^  jnch 

,000  pounds  per  square  inch  .........  {  ™°"  ^nZetre^  ^ 

•     ,                            (  i.STS    kilogrammes   per    square 

i  ton  per  square  inch  ..........  .  .......  |  miHimetre. 

i  kilogramme  per  square  centimetre  14.2232  pounds  per  square  inch. 

,0335  K^^^c^  J  14.7  pounds  per  square  inch. 

,  pound  per  square  inch  ...............  {  ^°^  **£S^-*^ 

i  pound  per  square  foot  ...............  {  ^83    kil°^rmmes    Per 


158  WEIGHTS   AND   MEASURES. 

Weight,  Pressure,  and  Measure  (continued}. 

i  kilogramme  per  square  metre .205  pounds  per  square  foot. 

i  centimetre  of  mercury .394  inch  of  mercury. 

i  inch  of  mercury 2.540  centimetres  of  mercury. 

i  centimetre  of  mercury 193  pound  per  square  inch. 

i  pound  per  square  inch 5.170  centimetres  of  mercury. 

i  gramme  per  litre 70. 1 16  grains  per  gallon. 

i  grain  per  gallon 0143  gramme  per  litre. 

i  kilogramme  per  cubic  metre 0624  pound  per  cubic  foot. 

i  pound  per  cubic  foot 16.020  kilogrammes  per  cubic  metre. 

. .  (      .084  ton  per  cubic  metre. 

.  tonne  per  cub.c  metre |       ^  ton  ^er  cubic  yard 

i  kilogramme  per  litre 10.016  pounds  per  gallon. 

i  pound  per  gallon 998  kilogramme  per  litre. 

i  ton  per  cubic  metre i.oi 6  tonnes  per  cubic  metre. 

i  ton  per  cubic  yard I>329  tonnes  per  cubic  metre. 

i  cubic  metre  per  kilogramme 16.019  cubic  feet  per  pound. 

i  cubic  foot  per  pound .0624  cubic  metre  per  kilogramme. 

j"    1.329  cubic  yards  per  ton. 
i  cubic  metre  per  tonne <     J  •  7  94  cubic  feet  per  cwt. 

(35.882  cubic  feet  per  ton. 

i  cubic  yard  per  ton 752  cubic  metre  per  tonne. 

i  cubic  foot  per  cwt 557  cubic  metre  per  tonne. 

i  cubic  foot  per  ton .0279  cubic  metre  per  tonne. 

Volume,  Area,  and  Length. 

i  cubic  metre  per  lineal  metre  1.196  cubic  yards  per  lineal  yard. 

i  cubic  yard  per  lineal  yard 836  cubic  metre  per  lineal  metre. 

i  cubic  metre  per  square  metre 3.281  cubic  feet  per  square  foot. 

i  cubic  foot  per  square  foot 3.048  cubic  metres  per  square  metre. 

i  litre  per  square  metre .0204  gallon  per  square  foot. 

i  gallon  per  square  foot 48.905  litres  per  square  metre. 

(      .405  cubic  metre  per  acre, 
i  cubic  metre  per  hectare \      .529  cubic  yard  per  acre. 

(89.065  gallons  per  acre. 

i  cubic  metre  per  acre 2.47 1  cubic  metres  per  hectare. 

i  cubic  yard  per  acre 1.902  cubic  metres  per  hectare. 

i ooo  gallons  per  acre 11.226  cubic  metres  per  hectare. 

Work. 

i  kilogrammetre  (k  x  m) 7. 233  foot-pounds. 

i  foot-pound 1382  kilogrammetre. 

i  cheval-vapeur  or  cheval  (75  k  x  m  )         g      horse-power, 
per  second) / 

horse-power 1.0139  chevaux. 

kilogramme  per  cheval 2.235  pounds  per  horse-power. 

pound  per  horse-power 447  kilogramme  per  cheval. 

square  metre  per  cheval 10.913  square  feet  per  horse-power. 

square  foot  per  horse-power .0916  square  metre  per  cheval. 

cubic  metre  per  cheval 3  5. 80 6  cubic  feet  per  horse-power. 

i  cubic  foot  per  horse-power 0279  cubic  metre  per  cheval. 


FRENCH   AND   ENGLISH  COMPOUND   UNITS. 


159 


Heat. 

i  calorie,  or  French  unit 3.968  English  heat-units. 

i  English  heat-unit .252  calorie. 

French  mechanical  equivalent  (425  )   3074    foot-pounds  =  774.70    foot- 

kilogrammetres) J  pounds  per  English  unit. 

English  mechanical  equivalent  (772  )  ,       ,  ., 

foot-pounds) ...  |       Ia6?    kilogrammetres. 

i  calorie  per  square  metre .369  heat-unit  per  square  foot. 

i  heat-unit  per  square  foot 2-  7  *  3  calories  per  square  metre. 

i  calorie  per  kilogramme i .  800  or  9/5  heat-units  per  pound. 

i  heat-unit  per  pound -5555   °r  5/9  calorie  per  kilo- 

gramme. 
Speed,  &c. 

(  3.281  feet  per  second, 

i  metre  per  second <  196.860  feet  per  minute. 

(  2.236  miles  per  hour, 

i  kilometre  per  hour .621  mile  per  hour. 

i  foot  per  second,  or  per  minute I  '3°5  metre.  Per  second.  OT  P« 

(  minute. 

i  mile  per  hour J  -447  metre  per  second. 

(          1.609  kilometres  per  hour. 

i  cubic  metre  per  second..  ..  ]        35-3i6  cubjc  feet  per  second. 

(    2119         cubic  feet  per  minute. 

i  cubic  foot  per  second,  or  per  minute  i  '0283  cubic  metre  Per  second> 

I  or  per  minute. 

i  cubic  metre  per  minute 1.308  cubic  yards  per  minute. 

i  cubic  yard  per  minute .765  cubic  metre  per  minute. 

Money. 

{4.320  pence  per  pound. 
.360  shilling  per  pound. 
40.320  shillings  per  cwt.,  or 
£40. 3  2  per  ton. 

i  penny  per  pound .23 1  franc  per  kilogramme. 

i  shilling  per  pound 2.772  franc  per  kilogramme. 

i  shilling  per  cwt,  or  £i  per  ton ...  I        24.802  francs  per  tonne 

|  2.48  francs  per  quintal, 

i  franc  per  quintal .403  shilling  per  cwt. 

i  franc  per  tonne (  «484  penny  per  cwt. 

(  .806  shilling  per  ton. 

i  franc  per  metre ...  . .  /  -726  shillinS  Per  ^d 

(          8.709  pence  per  yard. 

i  shilling  per  yard  1.378  francs  per  metre. 

i  franc  per  kilometre ...  . .  {  '°6386  £  Per  mil.^ 

(        15.326  pence  per  mile. 

^"i  per  mile 15.660  francs  per  kilometre. 

i  penny  per  mile .0652  franc  per  kilometre. 

i  franc  per  square  metre /         7'^3/?^  Per  *^K  ^  A 

\  .6636  shilling  per  square  yard. 


l6o  WEIGHTS  AND   MEASURES. 

i  shilling  per  square  yard  1.510  francs  per  square  metre. 

£\  per  square  yard 30. 1 94  francs  per  square  metre. 

{.270  penny  per  cubic  foot. 
7.281  pence  per  cubic  yard. 
.607  shilling  per  cub.c  yard. 
•°3°3  £  Per  cubic  yard. 

i  penny  per  cubic  foot 3. 708  francs  per  cubic  metre. 

i  penny  per  cubic  yard 137  franc  per  cubic  metre. 

i  shilling  per  cubic  yard i  .648  francs  per  cubic  metre. 

£i  per  cubic  yard 32.962  francs  per  cubic  metre. 

i  franc  ner  litre  I  43'27°  pence  per  gallon< 

"\    3.606  shillings  per  gallon. 

i  franc  per  hectolitre  I-%93  shillings  per  hogshead  (wine). 

i  shilling  per  hogshead 528  franc  per  hectolitre. 


GERMAN    EMPIRE.  —  WEIGHTS  AND    MEASURES.  —  Tables  No.  43. 

From  the  ist  January,  1872,  the  French  metric  system  of  weights  and 
measures  became  compulsory  throughout  the  German  Empire,  as  follows  :  — 

I.  GERMAN  MEASURES  OF  LENGTH. 

French  Measure. 

Strich  i  millimetre. 

i  o  Strichs  ................       New-Zoll  =            i  centimetre. 

i  oo  New-Zolls  ............  Stab  i  metre. 

10  Stabs  ..................       Kette  i  dekametre. 

i  oo  Kettes  ................  Kilometre  =  i  kilometre. 


7  Kilometres  ...........   ,  Mile 


II.  GERMAN  MEASURES  OF  SURFACE. 


mile, 


i  Quadrat-Stab  =  i  square  metre. 

i  oo  Quadrat-Stabs  ........   i  Ar  =        i  oo  square  metres. 

ioo  Ars  .....................   i  Hectar  =  |IO'°00  sc*uare  metres'  or 

(         2.47  acres. 

III.  GERMAN  MEASURES  OF  CAPACITY. 

i  Schoppen  J4  litre. 

(Beer  Measure.) 

2  Schoppens  .........   i  Kanne  =        i  litre. 

50  Kannes.  .  .  .   .  Scheffel  (bushel)  =  {  *  g^^tf  bushels. 

-,-.       /      ,  v  (    i  hectolitre,  or 

2  Scheffels  ...........   '  Fass  («*>  =  1  M.oi  gallons. 

The  kanne  is  further  divided  into  measures  of  %  kanne,  j^  kanne,  and 
/    kanne. 


GERMAN    EMPIRE.  —  WEIGHTS,   THE   FUSS.  l6l 

IV.  GERMAN  MEASURES  OF  WEIGHT. 

i  Milligramm      =          i  milligramme. 
10  Milligramms  ......   i  Centigramm    =          i  centigramme. 

10  Centigramms  .....   i  Dezigramm     =          i  decigramme. 

ioo  Dezigramms  .....  x  New-Loth        =  { 

j  500  grammes,  or 
50  New-Loths  ........   i  Pfund  j^  kilogramme,  or 

(      1.1023  pounds  avoirdupois. 

ioo  Pfunds  ............   i  Centner  L  {    5°  kilogrammes,  or 

( 


110.23  pounds  avoirdupois. 
=      I00°  kilogrammes,  or 
2000  Pfunds.          /  '  i  2204.6  pounds  avoirdupois. 


20  Centners,  or  )  Tonne  =  \  I00°  kilogrammes,  or 

i 


OLD   WEIGHTS   AND   MEASURES   OF  THE   GERMAN   STATES. 

These  vary  for  every  state.  The  chief  measures  of  length  are  the  Fuss, 
and  the  Elle,  of  which  the  second  is  in  general  twice  the  first.  The 
following  are  the  values  of  the  Fuss,  which  is  the  German  foot,  in  the 
principal  states. 

VALUES  OF  THE  GERMAN   Fuss  IN  THE  STATES  AND  FREE  TOWNS  OF 
THE  GERMAN  EMPIRE. — Table  No.  44. 

Prussia 12.356  inches. 

Bavaria 11.491  „ 

Wiirtemberg 11.279  » 

Saxony 11.149  » 

Baden 11.811  „ 

Mecklenburg-Schwerin 1 1.457  „ 

Hesse-Darmstadt 9.843  „ 

Hesse-Cassel < 11-328  „ 

Oldenburg 11.649  » 

Brunswick IJ-235  „ 

Hanover 11.500  „ 

Mecklenburg-Strelitz n-457  „ 

Anhalt 12.356  „ 

Saxe-Coburg-Gotha  11-324  „ 

Saxe-Altenburg 11.122  „ 

Waldeck 11.512  „ 

Lippe -. ii. 398  „ 

Schwarzburg-Rudolstadt 15.047  „ 

Schwarzburg-Sondershausen : — 

(1)  High  Sovereignty  and  Arnstadt ...  11.149  „ 

(2)  Low  Sovereignty  and  SonderShausen  1 1.33 1  „ 

Reuss 11.280  „ 

Schaumburg-Lippe 11.421  „ 

Hamburg  11.283  „ 

Liibeck 11.324  „ 

Bremen 11.392  „ 

11 


162 


WEIGHTS  AND   MEASURES. 


KINGDOM   OF   PRUSSIA.— OLD  WEIGHTS  AND  MEASURES.- 
Tables  No.  45. 

I.  PRUSSIAN  MEASURES  OF  LENGTH. 

English  Measure. 

i  Linie       =  .0858  inch. 

12  Linien i  Zoll  1.0297  inches. 

1.0297  feet. 
2  Fuss i  Elle        =  2.0596  feet. 

4.1192  yards. 

82, 

Used  by  Miners. 

i  Lachterlinie  =   .0927  inch. 

10  Lachterlinien i  Lachterzoll    =    .9268  inch. 

i o  Lachterzoll i  Achtel  =    .7723  foot. 

6  -p^g  > i  Lachter          =2.0596  yards. 

9  Fuss  i  Spanne  =6.1788  yards. 

Surveyors'  Measure. 

Scrupel  =    .0148  inch. 

10  Scrupel  Linie  =    .1483  inch. 

i  o  Linien Zoll  =  1.4828  inches. 

10  Zoll Land-Fuss  =  1.2356  feet. 

i  o  Land- Fuss Ruthe  =  4. 1 1 92  yards. 

2000  Ruthen Meile  =4.6809  miles. 

II.  PRUSSIAN  MEASURES  OF  SURFACE. 

i  Square  Linie  =     .00736  square  inch. 

144  Square  Linien i  Square  Zoll  =    1.0603  square  inches. 

144  Square  Zoll  i  Square  Fuss  =    1.0603  square  feet. 

144  Square  Fuss i  Square  Ruthe  =16.967  square  yards. 

180  Square  Ruthen...   i  Morgen  =      .63103  acre. 

30  Morgan i  Hufe  =  18.931  acres. 

III.  PRUSSIAN  MEASURES  OF  VOLUME. 

Cubic  Measure. 


1728  Cubic  Linien.... 
1728  Cubic  Zoll 
1728  Cubic  Fuss  ...... 


i  Cubic  Linie  =     .000632  cubic  inch. 

i  Cubic  Zoll  =    1.092  cubic  inches. 

i  Cubic  Fuss  =    1.092  cubic  feet. 

i  Cubic  Ruthe  =69.893  cubic  yards. 


For  measuring  stone  and  brickwork,  earth,  peat,  fascines,  and  firewood, 
the  following  are  used  :  — 


PRUSSIA.  —  CAPACITY,   WEIGHTS. 


163 


i  Cubic  Klafter,  or  )  ,  . 

108  Cubic  Fuss          }="7.93  cubic  feet 

4^  Klafters i  Haufe  =530.70        „ 

i  Schachruthe  (in  architecture)  144  Cubic  Fuss  =  157.25         „ 

IV.  PRUSSIAN  MEASURES  OF  CAPACITY. 
Dry  Measure. 

i  Maasche     =          .7560  quart. 

4  Masschen,  or  )  AT  , 

7  v*«  > i  Metze         =       3.0242  quarts. 

4  Metzen i  Viertel        =        3.0242  gallons. 

4  Viertel, or  )  Scheff  1        -  /    I-5121  DUSnelsJ  or 

48  Quarts       f  (1.941  cubic  feet. 

4  Scheffeln  i  Tonne        =        6. 0484  bushels. 

>  i  Maker  2.26815  quarters. 

i  Last  =      11.3407  quarters. 

The  Tonne  in  the  table  is  the  measure  for  salt,  lime,  and  charcoal. 
A  Tonne  of  flax-seed  is  2.354  Scheffeln. 

Liquid  Measure  (for  Wine  and  Spirits). 

32  Cubic  Zoll i  Ossel  =      1.0079  pints. 

2  Ossel i  Quart  =      1.0079  quarts. 

30  Quarts,  or     )  .    ,  ,, 

60  Ossel  }  r  Anker  7-559  gallons. 

2  Ankers i  Eimer  =    15.118       „ 

2  Eimers i  Ohm  =    30.237        „ 

-?  Eimers,  or     )  ~  ,    * 

i^Ohm          \  I0xhoft  '    45-355        „ 

4  Oxhoft,  or  )  „    , 

6  Ohm          I  lFuder  ='«'-4«       » 

V.  PRUSSIAN  MEASURES  OF  WEIGHT. 

Corn  4.115  grains. 

10  Corns Cent  .09406  dram. 

i  o  Cents Quentche  =  .9406  dram. 

i  o  Quentchen Loth  .  5  88  ounce. 

30  Loth Zollpfund  =  1.1023  pounds. 

100  Zollpfund Centner  =        1 10. 23  pounds. 

20  Zollpfund Stein  2  2.046  pounds. 

3  Centners i  Schiffspfund  =  < 

40  Centners i  Schiffslast    =  < 

The  Tonne  of  coals  is  2270  pounds  avoirdupois,  or  1.013  tons. 


1 64  WEIGHTS   AND    MEASURES. 

KINGDOM   OF   BAVARIA. — OLD  WEIGHTS  AND  MEASURES. — 
Tables  No.  46. 

I.  BAVARIAN  MEASURES  OF  LENGTH. 


12  Linien 

12  Zoll 

6  Fuss  

10  Fuss  .. 


Linie      =    .0798  inch. 
Zoll         =    .95756  inch. 
Fuss       =    .95756  foot. 
Klafter  =  5.74536  feet. 
Ruthe  =9.5756  feet. 


In  surveying,  the  Fuss  is  divided  into  10  Zoll,  and  i  Zoll  into  10  Linien. 
The  Elle  contains  2  Fuss  10%  Zoll,  =  2.733 


II.  BAVARIAN  MEASURES  OF  SURFACE. 

i  Square  Zoll  .91692  square  inch. 

144  Square  Zoll  ....     i  Square  Fuss  .91692  square  foot. 

100  Square  Fuss  ...     i  Square  Ruthe  10.  188  square  yards. 

400  Square  Ruthen  {  <  ^J&f^  }  T  {  ^.g  aST 


III.  BAVARIAN  MEASURES  OF  VOLUME. 

i  Cubic  Zoll  =  .878  cubic  inch. 

1728  Cubic  Zoll  .....................     i  Cubic  Fuss  -  .878  cubic  foot. 

126  Cubic  Fuss  (6x6x3^  Fuss)  i  Klafter        =  I  II0'628  cu^c  feet'  or 

(      4.097  cubic  yards. 

IV.  BAVARIAN  MEASURES  OF  CAPACITY. 

Dry  Measure. 

Dreisiger=    .12745  peck. 


4  Dreisigers 
4  Maassls . . . 
2  Viertel  ... 
6  Metzen... 
4  Schaffel . . 


Maassl  =    .12745  bushel. 

Viertel  =    .5098  bushel. 

Metze  =  1.0196  bushels. 

Schaffel  =  6. 1 1 76  bushels. 

Muth  =  3.0588  quarters. 


Liquid  Measure. 

i  Maaskanne  ~        .23529  gallon. 

64  Maaskannen i  Eimer  15.05856  gallons. 

25  Eimer i  Fass  =376.464  gallons. 

The  Schenk-Eimer,  ordinarily  used  in  the  Wine  trade,  contains  only 
60  Maaskannen,  equal  to  14.1174  imperial  gallons. 

V.  BAVARIAN  MEASURES  OF  WEIGHT. 

i  Quentchen^  •I5433  ounce. 

4  Quentchen i  Loth  .6173  ounce. 

32  Loth i  Pfund  1.23457  pounds. 

zoo  Pfund i  Centner      =  {  I23-«7  pounds  or 

(       1. 102  hundredweights. 


WURTEMBERG.— LENGTH,    SURFACE,    ETC.  165 

KINGDOM   OF  WURTEMBERG.— OLD  WEIGHTS  AND  MEASURES.— 

Tables  No.  47. 

I.  WURTEMBERG  MEASURES  OF  LENGTH. 

Punkte  .01128  inch. 

ip  Punkte Linie  .1128  inch. 

10  Linien Zoll  =  1.128  inches. 

10  Zoll Fuss  =  .93995  foot. 

10  Fuss Ruthe  9-3995  feet- 

2.144  Fuss EHe  2.015  feet. 

6  Fuss Klafter  =  5.6397  feet. 

26,000  Fuss i  Meile  (  8146.25  yards  or 

)        4.6285  miles. 

II.  WURTEMBERG  MEASURES  OF  SURFACE. 

i  Square  Zoll     =  1.272  square  inches. 

100  Square  Zoll i  Square  Fuss  -8835  square  foot. 

100  Square  Fuss i  Square  Ruthe  =          88.3506  square  feet. 

384  Square  Ruthen. . .  i  Morgen  =  {  3769-626  square  yards,  or 

(          .779  acre. 

..j  III.  WURTEMBERG  MEASURES  OF  VOLUME. 

i  Cubic  Linie      =       .001434  cubic  inch. 

i  ooo  Cubic  Linien i  Cubic  Zoll        =      1.434  cubic  inches. 

i  ooo  Cubic  Zoll i  Cubic  Fuss       =        .83045  cubic  foot. 

144  Cubic  Fuss i  Cubic  Klafter  =  119.583  cubic  feet. 

IV.  WURTEMBERG  MEASURES  OF  CAPACITY. 
Dry  Measure. 

i  Viertlein  =    .305  pint. 

4  Viertlein i  Ecklein  =  1.219  pints. 

8  Ecklein i  Vierling  =1.219  gallons. 

4  Vierling i  Simri  =  4.876  gallons. 

8  Simri i  Scheffel  =  4.876  bushels. 

Liquid  Measure. 

Quart  or  Schoppen  =        .4043  quart. 

4  Quarts Helleich  Maass       =      1.6173  quarts. 

10  Helleich  Maass Imi  =      4.0433  gallons. 

16  Imi Eimer  =    64.6928  gallons. 

6  Eimer Fuder  =  388. 1568  gallons. 

V.  WURTEMBERG  MEASURES  OF  WEIGHT. 

i  Quentchen  —         .1289  ounce. 

4  Quentchen iJLoth  •5I5^>  ounce. 

32  Loth i  Light  Pfund  =       1.03115  pounds. 

100  Heavy  Pfund,  or  )      ^  ,  , 

104  Light  Pfund......  }  '  Centner  '  I07-2396  P°™ds. 

ioo  Light  Pfund =  103.115  pounds. 


i66 


WEIGHTS  AND   MEASURES. 


KINGDOM   OF  SAXONY.— OLD  WEIGHTS  AND  MEASURES.— 
Tables  No.  48. 

I.  SAXON  MEASURES  OF  LENGTH. 

Linie  =  .07742  inch. 

2  Linien Zoll  =  .9291  inch. 

12  Zoll Fuss  =  .9291  foot. 

2  Fuss Elle  =  1.8582  feet. 

2  Ellen Stab  =  3.7165  feet. 

1 5  Fuss,  2  Zoll i  Ruthe  (Land  Measure)  =  4. 69 7  2  yards. 

16  Fuss i  Ruthe  (Road  Measure)  =  4.9553  yards. 

i  Lachter  (Mining)  =  2.1873  yards. 

1324.987  Ellen i  Meile  Post  =  4.6604  miles. 

II.  SAXON  MEASURES  OF  SURFACE. 

i  Square  Zoll  .8632  square  inch. 

144  Square  Zoll i  Square  Fuss    =     .863  2  square  foot. 

300  Square  Ruthen i  Acker  =  1.4865  acres. 

III.  SAXON  MEASURES  OF  VOLUME. 

i  Cubic  Zoll  =        .8021  cubic  inch. 

1728  Cubic  Zoll i  Cubic  Fuss  =        .8021  cubic  foot. 

108  Cubic  Fuss i  Klafter  =    86.624  cubic  feet. 

3  Klafter i  Schragen  =  259.873  cubic  feet. 

The  Klafter  is  6  Fuss  by  6  Fuss  by  3  Fuss.     The  Schragen  is  used  in 
the  measurement  of  firewood. 

IV.  SAXON  MEASURES  OF  CAPACITY. 
Dry  Measure. 


i  Maasche 

4  Maaschen i  Metze 

4  Metzen i  Viertel 

4  Viertel i  Scheffel 

12  Scheffel i  Malter 

2  Malter i  Wispel 

Liquid  Measure. 

i  Quartier 

4  Quartier i  Nossel 

2  Nossel  i  Kanne 

36  Kannen i  Anker 

2  Anker i  Eimer 

3  Eimer i  Oxhoft 

6  Eimer  . .  .   i  Fass  or  Barrel 


=  1.4463  quarts. 
=  1.4463  gallons. 
=  5-7852  gallons. 
=  2.8926  bushels. 
=  34.7124  bushels. 
=  69.4249  bushels. 


.2059  pint. 

.8237  pint. 

1.6474  pints. 

7.4237  gallons. 

14.8262  gallons. 

44.4687  gallons. 

88.9374  gallons. 


V.  SAXON  MEASURES  OF  WEIGHT. 
The  old  Saxon  measures  of  weight  are  the  same  as  those  of  Prussia. 


BADEN.— LENGTH,   SURFACE,   ETC. 


I67 


GRAND   DUCHY  OF  BADEN.— OLD  WEIGHTS  AND  MEASURES.— 

Tables  No.  49. 


I.  BADEN  MEASURES  OF  LENGTH. 


10  Punkte. 
10  Linien. 
loZoll  .... 
2  Fuss.... 
10  Fuss... 


i  Punkte 
Linie 
Zoll 
Fuss 
Elle 
Ruthe 


6  Fuss 

14814.815  Fuss 
2  Stunden.. 


.0118  inch. 

.118  inch. 
1.181  inches. 

.9842  foot. 
1.9685  feet. 
9.8427  feet. 


Klafter  =  5.9055  feet. 
Stunde  =4860.59  yards. 
Meile  =  5.5234  miles. 


II.  BADEN  MEASURES  OF  SURFACE. 

1.3951  square  inches. 
.9688  square  foot. 
10.7643  square  yards. 
=      1076.43  square  yards. 

I  4305.72  square  yards,  or 
"  (          .8896  acre. 


i  Square  Zoll 

100  Square  Zoll i  Square  Fuss 

100  Square  Fuss i  Square  Ruthe 

100  Square  Ruth  en...  i  Viertel 

4  Viertel i  Morgen 


III.  BADEN  MEASURES  OF  VOLUME. 

i  Cubic  Fuss   =         -95335  cubic  foot. 
144  Cubic  Fuss i  Klafter          =    137.28  cubic  feet. 

IV.  BADEN  MEASURES  OF  CAPACITY. 

Liquid  Measure. 

i  Glass  -  1.0563  gills. 

i  o  Glass i  Maass  =  1.3204  quarts. 

10  Maass i  Stutze  =  3.3014  gallons. 

10  Stutzen i  Ohm  =  33.014  gallons. 

10  Ohm i  Fuder  =  330.14  gallons. 

Dry  Measure. 

i  Becher  =  .2643  Pmt- 

10  Becher i  Maasslein  =  .1652  peck. 

10  Maasslein i  S ester  —  .4127  bushel. 

10  Sester  i  Malter  =  4.1268  bushels. 

10  Maker i  Zuber  =  41.2679  bushels. 

V.  BADEN  MEASURES  OF  WEIGHT. 

i  As  =        .7716  grain. 

10  As i  Pfennig  =      7.716  grains. 

10  Pfennig i  Centas  =        .1764  ounce. 

10  Centas i  Zehnling  =      1.7637  ounces. 

10  Zehnling i  Pfund  =      1.1023  pounds. 

100  Pfund i  Centner  =  110.230  pounds. 


i68 


WEIGHTS   AND   MEASURES. 


THE    HANSE    TOWNS. — OLD   WEIGHTS   AND   MEASURES. — 
Tables  No.  50. 

HAMBURG.— WEIGHTS  AND  MEASURES. 
I.  HAMBURG  MEASURES  OF  LENGTH. 


i  Achtel 

8  Achtel i  Zoll 

1 2  Zoll i  Fuss 

2  Fuss i  Elle 

6  Fuss i  Klafter,  or  Faden  = 

14  Fuss i  Marsch-Ruthe       = 

1 6  Fuss...  .  i  Geest-Ruthe 


inch. 

.9402  inch. 
.9402  foot. 
1.8804  feet. 
5.6413  feet. 
13.1629  feet. 
15.0434  feet. 

The  Hamburg  Elle  above  is  used  for  silk,  linen,  and  cotton  goods. 


The 


Brabant  Elle  is  equal  to  i  J/5  Hamburg  Elle;  and  4  of  them  are  reckoned 
equal  to  3  yards.  The  Prussian  Ruthe  is  also  used.  The  Prussian  Fuss  is 
used  in  surveying. 


II.  HAMBURG  MEASURES 


144  Square  Zoll... 
196  Square  Fuss.. 
256  Square  Fuss.. 
200  Square  Geest- 

Ruthen 

600  Sq.   Marsch- 
Ruthen . . 


i  Square  Zoll 

i  Square  Fuss 

i  Square  Marsch-Ruthe 

i  Square  Geest-Ruthe 

i  Scheffel  Geest-Land 


i  Morgen 


OF  SURFACE. 

.8840  square  inch. 
.8840  square  foot. 
173.26  square  feet. 
226.30  square  feet 
5028.98  square  yards,  or 

1.039  acres. 

11550.93  square  yards,  or 
2.386  acres. 


III.  HAMBURG  MEASURES  OF  VOLUME. 

i  Cubic  Zoll  =      .8311  cubic  inch. 

1728  Cubic  Zoll i  Cubic  Fuss          =     .8311  cubic  foot. 

88.9  Cubic  Fuss....  i  (Cubic)  Klafter  -  73.88  cubic  feet. 
120  Cubic  Fuss i  Tehr  =99-73  cubic  feet. 

IV.  HAMBURG  MEASURES  OF  CAPACITY. 

Liquid  Measure. 


2  Ossel 

Ossel 
Quartier 

= 

.09965  gallon. 
.  iQQT.  gallon. 

2  Quartier  
2  Kannen  
i  Stubchen  
4  Viertel  ...  . 

Kanne 
Stubchen 
Viertel 
Eimer 

— 

•3987  gallon. 
.7974  gallon. 
1.5947  gallons. 
6.3788  gallons. 

5  Viertel  
6  Eimer  
4  Anker  

Anker 
Tonne 
Ohm 

= 

7-9735  gallons. 
38.2728  gallons. 
3.1.804.0  gallons. 

6  Anker  
6  Ohm... 

Oxhoft 
Fuder.  or 

Tonneau  = 

^  A  .  *_*  **JL^\J      C^Cll.LVyi.lhJ. 

47.8410  gallons. 
TO  1.76/10  gallons. 

The  above  are  measures  for  Wines  and  Spirits.     For  Beer,  there  are 
three  sizes  of  Tonne,  containing  respectively  48,  40,  and  32  Stubchen. 


HAMBURG.— WEIGHTS.  169 


2  Small  Maass. 
4  Large  Maass 

4  Spint 

2  Himten 

2  Fass... 


Dry  Measure. 

Small  Maass  =  .0236  bushel. 

Large  Maass  =  .0473  bushel. 

Spint  =  .1890  bushel. 

Himten  =  .7560  bushel. 

Fass  =  1.5121  bushels. 

Scheffel  =  3.0242  bushels. 


10  Scheffeln i  Wispel  =30.2416  bushels. 

3  Wispel i  Last  =  90.7248  bushels. 

For  barley  and  oats,  the  Scheffel  contains  3  Fass. 

V.  HAMBURG  MEASURES  OF  WEIGHT. 

i  Half  Gramme  =              .0011  pound    =.5  gramme. 

10  Half  Grammen  i  Quint  .01102  pound  =  5  grammes. 

10  Quinten i  (New)  Unze  .11023  pound  =  50      » 

10  (New)  Unzen..  i  (New)  Pfund  1. 10232 pounds  =  500    „ 

100  (New)  Pfund  i  Centner  =        110.232  pounds     =      50  kilog. 

60  Centners. i  (Commercial)  Last  -  {  ^^o^tons8' }  =3oookil°g- 

This,  it  is  apparent,  is  a  metric  system  of  weights,  which  was  comparatively 
recently  introduced  and  adopted  at  Hamburg.  It  is  now,  of  course,  over- 
ruled by  the  French  metric  system  enforced  for  the  German  Empire. 


BREMEN. — OLD  WEIGHTS  AND  MEASURES. 

The  Fuss  is  equal  to  11.392  inches,  and  the  Klafter  is  equal  to  5.696 
feet.  The  Morgen  =  .6368  acre.  The  principal  measures  for  wines  and 
spirits  are  the  Viertel=  1.56  gallons;  the  Anker  =  5  Viertels  =  7.80  gallons; 
the  Oxhoft  =  46.8o  gallons.  The  Scheffel,  for  dry  goods  =  2.0388  bushels. 
The  old  weights  are  the  same  as  those  of  Hamburg. 


LUBEC. — OLD  WEIGHTS  AND  MEASURES. 

The  Fuss  is  equal  to  11.324  inches.  The  Viertel=i.6o  gallons;  the 
Anker  =  8  gallons;  the  Oxhoft  =  48.04  gallons.  The  Scheffel,  for  dry  goods, 
=  .9545  bushel.  The  old  Pfund  =1.0725  pounds,  and  the  Centner  = 
1.0725  cwts. 


GERMAN    CUSTOMS    UNION.— OLD  WEIGHTS  AND  MEASURES.— 

Table  No.  51. 

Centner 110.23  pounds  (50  kilogrammes). 

Ship-Last  of  timber about  80  cubic  feet. 

Scheffel 1.512  bushels. 

Klafter 6  feet. 

In  Oldenburg,  Hanover,  Brunswick,  Saxe-Altenbourg,  Birkenfeld,  Anhalt, 
Waldeck,  Reuss,  and  Schaumburg-Lippe,  the  old  system  of  weights  is  the 
same  as  that  of  Prussia. 


I/O  WEIGHTS   AND   MEASURES. 

AUSTRIAN    EMPIRE.  —  WEIGHTS   AND   MEASURES.  —  Tables  No.  52. 

I.  AUSTRIAN  MEASURES  OF  LENGTH. 

i  Punkte  .0072  inch. 

12  Punkte  ...............   i  Linie  .0864  inch. 

12  Linien  ...............   i  Zoll  1.0371  inches. 

12  Zoll  ..................   i  Fuss  1.0371  feet. 

2  Fuss  ..................   i  Elle  2.0742  feet. 

6  Fuss  ..................   i  Klafter  6.2226  feet 

'  4=o=  Klafter  ..............   .  Meile(post)  = 


II.  AUSTRIAN  MEASURES  OF  SURFACE. 

i  Square  Zoll  1.0756  square  inches. 

1  44  Square  Zoll  ...........   i  Square  Fuss       =  1.0756  square  feet. 

36  S^are  Fuss  ........    :  Square  Klafter    -  {      ^jS£g 

":  Sf-  or  }  '  Square  Ruthe      35-854  square  yards- 

64  Square  Ruthen  i  Metze  =      2294.7  square  yards. 


3  Metzen,  or  1  i  T     h  I  ^^4  scluare  yars,  or 

1600  Square  Klafter  ......  J  1        1.4223  acres. 

III.  AUSTRIAN  MEASURES  OF  VOLUME. 

Cubic  Measure. 

i  Cubic  Zoll       =         i*  i1  55  cubic  inches. 
1728  Cubic  Zoll  .....  i  Cubic  Fuss       =          1-1155  cubic  feet. 

.,6  Cubic  Fuss....   x  Cubic  Klafter  -  { 


IV.  AUSTRIAN  MEASURES  OF  CAPACITY. 
Dry  Measure. 

i  Probmetzen     =  \ 

8  Probmetzen i  Becher  .8460  pint. 

4  Becher  i  Futtermassel   =        1.6920  quarts. 

2  Futtermassel ..      .   i  Muhlmassel     -  (    3'fj°  qu,f ts'  or 

[      .8460  gallon. 

2  Muhlmassel i  Achtel  =        1.6920  gallons. 

IT-       i  (    3.3840  gallons,  or 

2  Achtel i  Viertel  =  <         2o  bushel 

4  Viertel i  Metze  =        1.6918  bushels. 

f  So.7S^6  bushels,  or 
3°  Metzen i  Muth  =  |  5^75^  ^^ 


AUSTRIAN   EMPIRE. — CAPACITY,  WEIGHTS. 


Liquid  Measure. 


i  Pfiff 


2  Pfiff '...    .   i  Seidel 


2  Seidel  

Kanne 

2  Kannen  

Mass 

10  Mass  

Viertel 

4  Viertel 

Eimer 

32  Eimer  

Fuder 

1.246  gills,  or 
10.781  cubic  inches. 
2.491  cubic  inches,  or 
.6229  pint. 
1.2457  pints. 
1.2457  quarts. 
3.1143  gallons. 
12.4572  gallons. 
398.6304  gallons. 


V.  AUSTRIAN  MEASURES  OF  WEIGHT. 


4  Pfenning 

4  Quentchen.. 

2  Loth 

4  Unzen i  Vierdinge 

2  Vierdinges . . .   i  Mark 


i  Pfenning 
i  Quentchen 
i  Loth 
i  Unze 


(  270.1  grains,  or 
(        .6173  dram. 


i 


2.4694  drams. 
9.8776  drams,  or 

.6173  ounce. 
1.2347  ounces. 
4.9388  ounces. 
9.8776  ounces,  or 

.6173  pound  avoirdupois. 

1.2347  pounds  avoirdupois. 

{123.47  pounds  avoirdupois,  or 
1.1024  hundredweights. 

In  1853,  a  pfund  of  500  grammes,  with  decimal  subdivisions,  was  adopted 
for  customs  and  fiscal  purposes. 


2  Marks,  or ) 
16  Unzen       j  " 


i  Pfund 


100  Pfund...        .   i  Centner 


RUSSIA. — WEIGHTS  AND  MEASURES. — Tables  No.  53. 
I.  RUSSIAN  MEASURES  OF  LENGTH. 

English  Equivalent. 

i  Vershok  =  1.75  inches. 

16  Vershpks i  Arschine  =          28  „ 

3  Arschines    i  Sajene      =  7  feet. 

C  3500  feet,  or 

500  Sajenes i  Verst       =  <  1166^3  yards,  or 

(        0.6629  mile. 

The  Fuss,  or  Russian  foot,  is  13.75  inches;  but,  since  1831,  the  English 
foot  of  1 2  inches  has  been  used  as  the  ordinary  standard  of  length,  each 
inch  being  divided  into  12  parts. 


i  Lithuanian  Meile 5-55 74  English  miles. 

i  Rhein  Fuss,  used  in  calculating  )  ^     r  ,  f 

export  duties  on  timber  /      *-°3  En&llsh  feet' 


172  WEIGHTS   AND   MEASURES. 

II.  RUSSIAN  MEASURES  OF  SURFACE. 

i  Square  Arschine  =  (       ^  stluare  inchf  '  or 

5.444  square  feet. 

9  Square  Arschines..   i  Square  Sajene      =  I         ^  sc*uare  feet'  or  ^ 

5.444  square  yards. 

2400  Square  Sajenes  .....   i  Desatine  I  (  :3'°66  *<&***  yards>  or 

[  2.70  acres. 

For  earthworks,  masonry,  &c.,  the  Sajene  is  divided  into  tenths  (dessiatka), 
hundredth*  (sotka),  and  thousandths  (tisiatchka),  which  are  used  as  a  basis 
for  lineal,  superficial,  and  cubic  measurements,  similarly  to  the  French 
metre  with  its  sub-multiples. 

III.  RUSSIAN  MEASURES  OF  CAPACITY. 

Liquid  Measure. 

iTscharkey  .8656  gill  or 

(         .2104  pint. 

10  Tscharkeys  ......    i  Kruschka  1.0820  quarts. 

100  Tscharkeys  ......   i  Vedro  2.7049  gallons. 

3  Vedros  ............   i  Anker  8.1147        „ 

^3  V6  Ankers  j  •'••   *  SarokowaJa  Boshka    =        108.196          „ 

Dry  Measure  (Grain). 

Garnietz  2.885  quarts. 

2  Garnietz  .........      Tschetwerka  =  1.4424  gallons. 

4  Tschetwerkas  .  .  .      Tschetwerik  =  -72I3  bushel. 

2  Tschetweriks.  .  .  .      Pajak  i  .4426  bushels. 

2  Pajaks  ............       Osmin  2.8852        „ 

2  Osmins  ..........      Tschetwert*   =  5.7704       „ 


16  Tschetwerts...  .   i  Last  =      II'^°.  quar*f,  or 

}    1.154  imperial  lasts. 

*  A  Tschetwert  is  usually  reckoned  as  5^  bushels,  and  100  Tschetwerts  as  72  quarters, 
though  they  are  more  exactly  72.1308  quarters. 
loo  quarters  are  equal  to  138.637  Tschetwerts. 

IV.   RUSSIAN  MEASURES  OF  WEIGHT. 
i  Dolis  -68576  grain. 

rZolotnick 


3  Zolotnicks...   i  Lotti  .4514      „ 

8  Zolotnicks...   i  Lana  1-2037  ounces. 

12  Lanas,  or    *\  f          .90285  pound  avoirdupois,  or 

32  Lottis,  or     >  i  Funt,  or  pound  =          14.446  ounces,  or 

96  Zolotnicks  )  (6320  grains. 

40  pounds i  Pood  36.114  pounds  avoirdupois. 

T,    T  f    ^61.14  pounds  avoirdupois,  or 

IoPoods- iBerkovitz  3  3.224  hundredweights. 

3  Berkovitz i  Packen  9.672  hundredweights. 


HOLLAND,  BELGIUM,  NORWAY,  ETC.  1/3 

62.0257  Poods i  English  ton. 

2481.0268  Russian  pounds i          „ 

The  Pood  is  commonly  estimated  at  36  pounds  avoirdupois. 
The  Nuremberg  pound,  used  for  apothecaries'  weight,  weighs  5527  grains, 
or  about  .96  pound  troy. 

The  Ship-Last  is  equal  to  2  tons  nearly. 

The  Carat,  for  weighing  pearls  and  precious  stones,  is  about  3  x/e  grains. 


HOLLAND. 

The  metric  system  was  adopted  in  Holland  in  1819;  the  denominations 
corresponding  to  the  French  are  as  follows : — 

Length. — Millimetre,  Streep;  centimetre,  Duim;  decimetre,  Palm;  metre, 
El;  decametre,  Roede;  kilometre,  Mijle. 

Surface. — Square  millimetre,  Vierkante  Streep;  square  centimetre,  Vier- 
kante  Duim;  and  so  on.  Hectare,  Vierkante  Bunder. 

Cubic  Measure. — Millistere,  Kubicke  Streep,  and  so  on. 

Capacity. — Centilitre,  Vingerhoed;  decilitre,  Maatje;  liquid  litre,  Kan; 
dry  litre,  Kop;  decalitre,  Schepel;  liquid  hectolitre,  Vat  or  Ton;  dry 
hectolitre,  Mud  or  Zak;  30  hectolitres  =  i  Last=  10.323  quarters. 

Weight. — Decigramme,  Korrel;  gramme,  Wigteje;  decagramme,  Lood; 
hectogramme,  Onze;  kilogramme,  Pond. 


BELGIUM. 


The  French  metric  system  is  used   in    Belgium.      The  name  Livre  is 
substituted  for  kilogramme,  Litron  for  litre,  and  Aune  for  metre. 


DENMARK. 

•:     WEIGHTS  AND  MEASURES.  —  Tables  No.  54. 

I.  DANISH  MEASURES  OF  LENGTH. 

i  Linie  .0858  inch. 

12  Linier  ........   i  Tomme  1.0297  inches. 

12  Tommer  .....   i  Fod  1.0297  feet. 

2  Fod  ...........   i  Alen  2.0594    „ 


or   '  ^ 

2,000  Roder,  or  )     x  Mn  f  8237.77  yards,  or 

24,000  Fod  J  (        4.68055  miles. 

23,642  Fod  ...........   i  nautical  mile=  4.61072  English  miles. 

II.  DANISH  MEASURES  OF  SURFACE. 

144  Square  Linie  ......   i  Square  Tomme  =    1.0603  square  inches. 

144  Square  Tomme...   i  Square  Fod        =    1.0603  square  feet. 
144  Square  Fod  ........   i  Square  Rode     =  16.966  square  yards. 


WEIGHTS  AND   MEASURES. 


III.   DANISH  MEASURES  OF  VOLUME. 

1728  Cubic  Linier i  Cubic  Tomme  =     1.0918  cubic  inches. 

1728  Cubic  Tomme....   i  Cubic  Fod         =    1.0918  cubic  feet. 
The  Favn  of  firewood  measures  6x6x2  Fod  =  72  cubic  Fod  =  78.60 
cubic  feet.     In  forest  measure  it  is  6^x6^x2  Fod  =  84^  cubic  Fod  = 
92.26  cubic  feet. 

IV    DANISH  MEASURES  OF  CAPACITY. 
Liquid  Measure. 

Paegle  =  .4248  pint. 

4  Paegle Pot  =  1.6991  pints. 

2  Potter Kande  3-39^3     » 

38  Potter Anker  =  8.0709  gallons. 

136  Potter Tonde  =  28.885         » 

6  Ankerne Oxehoved  ==  48.4256       „ 

4  Oxehoveder Fad  =193.7027       „ 

Dry  Measure. 

Pot  =    1.6991  pints. 

18  Potter Skeppe  =    3.8232  gallons. 

2  Skepper Fjerdingkar  .9558  bushel. 

4  Fj  erdingkar Tonde  =    3.8231  bushels. 

12  Tender Laest  =45.8769       „ 

V.   DANISH  MEASURES  OF  WEIGHT. 

i  Ort  7-7J63  grains. 

10  Ort Kvint  77-l63        „ 

100  Kvinten Pund  1.1023  pounds. 

100  Pund Centner  =110.23  „ 

40  Centner Last  1.9684  tons. 

52  Centner Skip-Last  2-559°    •>•> 

1 6  Pund i  Lispund  17.637  pounds. 

320  Pund i  Skippund      =      3.149  cwts. 


SWEDEN.  —  WEIGHTS  AND  MEASURES.  —  Tables  No.  55. 
I.   SWEDISH  MEASURES  OF  LENGTH. 


i  o  Linier 
10  Turner 
10  Fot 
10  Stanger 


2  Fot 
6  Fot 


i  Linie       = 
i  Turn 
i  Fot 
i  Stang 
i  Ref 

i  Men. 


i  Aln 

i  Faden    = 


.1169  inch. 

1.1689  inches. 
11.6892     „ 

9.7411  feet. 
32.4703  yards. 


1.942  feet. 
5-845    » 


SWEDEN,   SWITZERLAND.  175 

II.  SWEDISH  MEASURES  OF  SURFACE. 

ioo  Square  Linier...   i  Square  Turn     =  1.3666  square  inches, 

ioo  Square  Turner.,   i  Square  Fot        =  .9489  square  foot, 

ioo  Square  Fot i  Square  Stang    =  3.5146  square  yards. 

ioo  Square  Stanger     i  Square  Ref       = 


4  Square  Fot i  Square  Aln       =  3.7956  square  feet. 

,6  Square  Ref....   ,  Tunn,and         =  {  *fmy+S*>  w 

III.  SWEDISH  MEASURES  OF  VOLUME. 

Cubic  Measure. 

i  Cubic  Turn   =1.5972  cubic  inches. 

1000  Cubic  Turner i  Cubic  Fot     =    .9263  cubic  foot. 

8  Cubic  Fot i  Cubic  Aln     =  7.4104  cubic  feet. 

Liquid  and  Dry  Measure. 

1000  Cubic  Linier i  Cubic  Turn  =      .  1 843  gill. 

ioo  Cubic  Turner i  Kanna  =    2.3096  quarts. 

10  Kanna i  Cubic  Fot  =    5.774  gallons. 

8  Cubic  Fot i  Cubic  Aln  =46.192       „ 

IV.  SWEDISH  MEASURES  OF  WEIGHT. 

i  Korn  .6564  grain. 

ioo  Korn i  Ort  =        2.4005  drams. 

ioo  Ort i  Skalpund    =          .9377  pound. 

ioo  Skalpund :  Centner      .  {  93;77»9  P-nds,  or 

ioo  Centner i  Ny-Last      =        4.1892  tons. 

The  metric  system  will  become  obligatory  in  1889. 


NORWAY. 

The  French  metric  system  is  in  force  in  Norway. 


SWITZERLAND. — WEIGHTS  AND  MEASURES. — Tables  No.  56. 
I.  Swiss  MEASURES  OF  LENGTH. 

i  Striche .01181  inch. 

i  o  Striche i  Linie =  .11811     „ 

10  Linien i  Zoll =  1.18112  inches. 

10  Zoll i  Fuss  (3  decimetres)....  —  11.81124      „ 

2  Fuss i  Elle =  1.9685  feet. 

6  Fuss i  Klafter =  5-9°56     ,, 

i  o  Fuss i  Ruthe =  9.8427     ,, 

1600  Ruth  en  ....   i  Schweizer-stunde,  or  Lien=  < 


176  WEIGHTS   AND   MEASURES. 


II.  Swiss  MEASURES  OF  SURFACE. 

i  Square  Zoll  I-3947  square  inches. 

100  Square  Zoll i  Square  Fuss       =          .9688  square  foot. 

36  Square  Fuss i  Square  Klafter  =       34.8768  square  feet. 

100  Square  Fuss i  Square  Ruthe    =      10.7643  square  yards 

400  Square  Ruthen..   i  Juchart  .8694  acre. 

6400  Jucharten i  Square  Stunde   =  5693.52  acres. 

350  Square  Ruthen i  Juchart,  of  meadow  land. 

450  Square  Ruthen i  Juchart,  of  woodland. 

III.  Swiss  MEASURES  OF  VOLUME. 

i  Cubic  Zoll        :    1.6476  cubic  inches. 

1000  Cubic  Zoll i  Cubic  Fuss      =     .9535  cubic  foot. 

216  Cubic  Fuss i  Cubic  Klafter  =    7.6172  cubic  yards. 

1000  Cubic  Fuss i  Cubic  Ruthe  =35.3166         „ 

IV.  Swiss  MEASURES  OF  CAPACITY. 

Dry  Measure. 

i  Imi  =    1.3206  quarts. 

10  Imi i  Maass  =     .4127  bushel. 

10  Maass i  Malter  =    4. 1268  bushels. 

Liquid  Measure. 

2  Halbschoppen i  Schoppen  =    2.6412  gills. 

2Schoppen..  i  Halbmaass  =    1.3206  pints. 

2  Halbmaass i  Maass  =    2.6412     „ 

i oo  Maass i  Saum  =33.015  gallons. 

V.  Swiss  MEASURES  OF  WEIGHT. 


4  Quntli 

2  Loth 

1 6  Unzen 

100  Pfund... 


Quntli  =          2.2048  drams. 

Loth  .55 ii  ounce. 

Unze  1.1023  ounces. 

Pfund  1.1023  pounds. 

Centner  =      110.233  pounds,  or  .9842  cwt. 


The  Pfund  is  divided  into  halves,  quarters,  and  eighths.  It  is  also 
divided  into  500  Grammes,  and  decimally  into  Decigrammes,  Centi- 
grammes, and  Milligrammes. 


SPAIN. — WEIGHTS  AND  MEASURES. — Tables  No.  57. 

The  French  metric  system  was  established  in  Spain  in  1859.  The  metre 
is  named  the  Metro;  the  litre,  Litro;  the  gramme,  Grammo;  the  are,  Area; 
the  tonne,  Tonelada.  The  metric  system  is  established  likewise  in  the 
Spanish  colonies.  The  old  weights  and  measures  are  still  largely  used. 


SPAIN — LENGTH,  SURFACE,  ETC. 


177 


I.  OLD  SPAIN 

12  Puntos  
12  Lineas 

FISH  MEASURES  OF 

Punto 
Linea 
Pulgada 
Sesma 
Pies  de  Burgos 
Vara 
Estado 
Estadal 
Legua  (Castilian) 
Leeua  (Spanish) 

LENGTH. 

=      .00644  inch. 
=      -07725  inch. 
—      -927  inch 

6  Pulgadas  
2  Sesmas 

=    5.564  inches. 
-      .9273  foot. 
=    2.782  feet. 
=    5.564  feet. 
-11.128  feet. 
=    2.6345  miles. 
=    4.2ic;i  miles. 

3  Pies  de  Burgos 
2  Varas  

4  Varas  

5000  Varas 

8000  Varas...            .  i 

II.  OLD  SPANISH  MEASURES  OF  SURFACE. 


i  Square  Pies 

9  Square  Pies i  Square  Vara 

1 6  Square  Varas i  Square  Estadal 

50  Square  Varas  . . . .  i  Estajo 
576  Square  Estadals.  i  Fanegada 
50  Fanegadas i  Yugada 


=      .860  square  foot. 
=     .860  square  yard. 
=  13.759  square  yards. 
=  42.997  square  yards. 
=    1.6374  acres. 
=  81.870  acres. 


III.  OLD  SPANISH  MEASURES  OF  CAPACITY. 
Liquid  Measure. 

i  Capo  .888  gill. 

4  Capos i  Cuartillo  .  1 1 1  gallon. 

4  Cuartillos i  Azumbre  .444  gallon. 

2  Azumbres i  Cuartilla  .888  gallon. 

^      ,.,,          C   i  Arroba  Mayor,  or  Cantara  ) 

4Cuartillas...|                (for  We)                }=  3-55'gaH. 

16  Cantaras i  Mayo  =56.832  gallons. 

The  old  measure  for  oil  is  the  Arroba  Menor  =  2.7652  gallons. 

Dry  Measure. 


Ions. 


Ochavillo 

=        -00785  peck. 

4  Ochavillos. 

Racion 

=         O3IA  Deck 

4  Raciones  

Quartillo 

•  W^J-if.     J^/V^\_xJV. 

=       .03  1  4  bushel. 

2  Quartillos  

Medio 

=        .0628  bushel. 

2  Medios  

Almude 

=        .1256  bushel. 

12  Amuerzas  , 

Fanega 

1.5077  bushels. 

12  Fanegas  

Cahiz 

=    18.0920  bushels. 

IV.  OLD  SPANISH  WEIGHTS. 

Grano 

.771  grain. 

12  Granos  

Tomin 

=        9.247  grains. 

3  Tomines  .... 

Adarme 

=      2  7.  7  4  grains. 

2  Adarmes  .... 

Ochavo,  or  Drachma 

.1268  ounce. 

8  Ochavos  

Onza 

=        1.0144  ounces. 

8  Onzas  

Marco 

=        8.1154  ounces. 

2  Marcos  

Libra  (Castilia  a) 

=        1.0144  pounds. 

roo  Libras  

Quintal 

=    101.442  pounds. 

10  Quintals  

Tonelada 

=  1014.42  pounds. 

12 

178  WEIGHTS  AND   MEASURES. 

PORTUGAL. 

The  French  metric  system  of  weights  and  measures  was  adopted  in  its 
entirety  during  the  years  1860-63,  and  was  made  compulsory  from  the  ist 
October,  1868.  The  chief  old  measures  still  in  use  are,  the  Libra  =  1.012 
pounds;  Almude,  of  Lisbon  =  3.7  gallons;  Almude,  of  Oporto  -  5.6  gallons; 
Alquiere  =  3.6  bushels;  Moio  =  2.78  quarters. 


ITALY. 


The  French  metric  system  is  used  in  Italy.  The  metre  is  named  the 
Metra;  the  are,  Ara;  the  stere,  Stero;  the  litre,  Litro;  the  gramme,  Gramma; 
the  tonneau  metrique,  Tonnelata  de  Mare.  The  various  old  weights  and 
measures  of  the  different  Italian  States  are  still  occasionally  used. 


TURKEY. 

Length. — i  Pike  or  Dra=27  inches,  divided  into  24  Kerats;  i  Forsang 

—  3.116  miles,  divided   into  3  Bern;   the  Surveyor's  Pik,  or  the  Halebi 

—  27.9  inches;  and  5*^  Halebis  =  i  reed. 

Surface. — The  squares  of  the  Kerat,  the  Pike,  and  the  Reed.  The 
Feddan  is  an  area  equal  to  as  much  as  a  yoke  of  oxen  can  plough  in  a 
day. 

Capacity,  Dry. — The  Rottol  =  1.411  quarts,  contains  900  Dirhems; 
22  Rottols  =  i  Killow  =  7.762  gallons,  or  .97  bushel,  the  chief  measure  for 
grain. 

Liquid. — i  Oka  =  1.152  pints;  8  Oke  =  i  Almud  =1.152  gallons;  i  Rottol 
-2.5134  pints;  100  Rottols  =  i  Cantar  =  31.417  gallons. 

Weights. — The  Oke  =  2.8342  pounds,  divided  into  4  Okiejehs,  or  400 
Dirhems  of  1.81  drams;  i  Rottolo  =  1.247  pounds;  100  Rottolos=  i  Cantar 
=  124.704  pounds. 


GREECE   AND    IONIAN    ISLANDS. 

The  French  metric  system  is  employed  in  Greece.  The  metre  is  named 
the  Pecheus;  kilometre,  Stadion;  are,  Stremma;  litre,  Litra;  gramme, 
Drachme.  i^  kilogrammes  =  i  Mna;  i^  Quintals  =  i  Tolanton 
i  y?,  Tonneaux  =  i  Tonos  =  29.526  cwts. 

In  the  Ionian  Islands,  whilst  they  were  under  the  protection  of  Great 
Britain  (1830  to  1864),  the  British  weights  and  measures  were  those  in  use, 
with  Italian  names.  The  foot  was  named  the  Piede;  the  yard,  the  Jarda; 
the  pole,  the  Carnaco;  the  furlong,  the  Stadio;  the  mile,  the  Miglio.  The 
gallon  was  the  Gallone;  the  bushel,  the  Chilo;  the  pint,  the  Dicotile;  the 
pound  avoirdupois,  the  Libra  Grossa;  the  pound  troy,  the  Libra  Sottile. 
The  Talanto  consisted  of  100  pounds,  and  the  Miglio  of  1000  pounds. 


MALTA. 


In  round  numbers,  3^  Palmi  =  i  yard;  i  Canna  =  2  2/7  yards. 
The   Salma  =  4.964    acres.     Approximately,    543    Square   Palmi  =  400 
square  feet;  16  Salmi  =  71  acres. 


EGYPT. — LENGTH,  SURFACE,  ETC.  179 

i  Cubic  Tratto  =  8  cubic  feet;  144  Cubic  Palmi  =  96  cubic  feet;  i  Cubic 
Canna  =  343  cubic  feet. 

Approximate  weights: — 15  Oncie=i4  ounces;  i  Rotolo=i^  pounds; 
4  Rotoli  =  7  pounds;  64  Rotoli  =  i  cwt. ;  i  Cantaro  =  175  pounds;  i  Quintal 
=  199  pounds;  64  Cantari  =  5  tons. 


EGYPT.  —  WEIGHTS  AND  MEASURES.  —  Tables  No.  58. 
I.  EGYPTIAN  MEASURES  OF  LENGTH. 

Pik,  or  cubit  of  the  Nilometre  ...............   20.  65  inches. 

Pik,  indigenous  .................................  22.37      „ 

Pik,  of  merchandise  ...........................   25.51      „ 

Pik,  of  construction  ............................  29.53      „ 

6  Palms  ...........  .  .........   i  Pik. 

24  Kirats  ....................   i  Pik  or  Draa. 

4.73  Piks  of  construction...   i  Kassaba  in  surveying,    =11.65 

II.  EGYPTIAN  MEASURES  OF  SURFACE. 

i  Square  Pik          -    6.055  square  feet. 
22.41  Square  Piks  ......   i  Square  Kassaba  =  15.07  square  yards 

333.33  Square  Kassaba,    i  Feddan  -9342  acre. 

III.  EGYPTIAN  MEASURES  OF  CAPACITY. 

i  Kadah  1.684  pints. 

2  Kadahs  ........................    i  Milwah  6.735      » 

2  Milwahs  ........................   i  Roobah  =  1.684  gallons. 

2  Roobahs  ........................   i  Kelah      =  3.367       „ 

2  Kelehs  .........................   i  Webek    =  6.734       „ 

6  Webeks  ........  .   i  Ardeb     =  /  4°'4°4  S&™.™ 

\    6.48  cubic  feet. 

The  Guirbah  of  water  (a  government  measure)  is  I/I5  cubic  metre  =  66  2/z 
litres,  or  11.772  cubic  feet. 

IV.  EGYPTIAN  MEASURES  OF  WEIGHT. 

i  Kamhah    =       .746  grain. 

4  Kamhahs  .............    i  Kerat. 

1  6  Kerats  .................    i  Dirhem     =    1.792  drachms. 

24  Kerats  ................   i  Mitkal. 

8  Mitkals  ...............   i  Okieh. 


ioo  Rottols  ...............   i  Kantar     =  98.207  pounds. 


400  Dirhems  ...............    i  Oke  =    2.728       „ 

36  Okes  ..................   i  Kantar      =98.207       „ 


180  WEIGHTS  AND   MEASURES. 

MOROCCO. 

Length. — The  Tomin  =  2.81025  inches;  the  Dra'a-8  Tomins  =  22.482 
inches. 

Capacity. — The  Muhd  =  3.08135  gallons;  the  Saa  =  4  Muhds— 12.3254 
gallons. 

Weights. — The  Uckia  =  392  grains;  the  Rotal  or  Artal  =  20  Uckieh  = 
1. 1 2  pounds;  the  Kintar=  100  Rotales  =  112  pounds. 

Oil  is  sold  by  the  Kula  =  3.3356  gallons.  Other  liquids  are  sold  by 
weight. 


TUNIS. 

Length. — The  Dhraa,  or  Pike,  is  the  unit  of  length.  The  Arabian  Dhraa, 
for  cotton  goods  =19.224  inches;  the  Turkish  Dhraa,  for  lace  =  25. 0776 
inches;  the  Dhraa  Endaseh,  for  woollen  goods  =  26.4888  inches. 

The  Mil  Sah'ari  =  .9i49  mile. 

Capacity. — For  dry  goods  the  Sa£=  1.2743  pint;  12  Saa-i  Hueba^ 
6.8228  gallons. 

For  liquids,  the  Pichoune  =  .4654  pint;  4  Pichounes=i  Pot  =i. 8616 
pints;  15  Pots  =  i  Escandeau,  and  4  Escandeaux=i  Millerole=  13.9623 
gallons. 


ARABIA. 

The  weights  and  measures  of  Egypt  are  used  in  Arabia. 


CAPE    OF    GOOD    HOPE. 


The  standard  weights  and  measures  are  British,  with  the  exception  of  the 
land  measure.  To  some  extent,  the  old  British  and  the  Dutch  measures 
are  in  use.  The  general  measure  of  surface  is  the  old  Amsterdam  Morgen, 
reckoned  equal  to  2  acres;  though  the  exact  value  is  equal  to  2.11654 
acres.  1000  Cape  feet  are  equal  to  1033  British  feet. 


INDIAN    EMPIRE. — WEIGHTS   AND   MEASURES. 

An  Act  "  to  provide  for  the  ultimate  adoption  of  an  uniform  system  of 
weights  and  measures  of  capacity  throughout  British  India  "  was  passed  in 
October,  1871.  The  ser  is  adopted  under  the  Act  as  the  primary  standard 
or  unit  of  weight,  and  is  a  weight  of  metal  in  the  possession  of  the  Govern- 
ment, equal,  when  weighed  in  a  vacuum,  to  one  kilogramme.  The  unit  of 
capacity  is  the  volume  of  one  ser  of  water  at  its  maximum  density,  equiva- 
lent to  the  litre.  Other  weights  and  measures  are  to  be  multiples  or  sub- 
multiples  of  the  ser,  and  of  the  volume  of  one  ser  of  water. 

The  following  are  the  weights  and  measures  in  common  use  in  India: — 


BENGAL— LENGTH,   SURFACE,   ETC.  l8l 

BENGAL. — WEIGHTS  AND  MEASURES. — Tables  No.  59. 

I.  BENGAL  MEASURES  OF  LENGTH. 

T  Jow,  or  Jaub =          ^  inch. 

3  Jow lUngulee =          %     „ 

4  Ungulees i  Moot =  3  inches. 

3  Moots i  Big'hath,  or  Span       =  9      „ 

2  Big'haths i  Hat'h,  or  Cubit. . .      =          1 8      „ 

2  Hat'h i  Guz i  yard. 

2  Guz i  Danda,  or  Fathom      =  2  yards. 

T-.      ,  f  2000  yards,  or 

roooDandas i  Coss =|        J^'^ 

4  Coss i  Yojan =  4.5454  miles. 

II.  BENGAL  MEASURES  OF  SURFACE. 

i  Square  Hat'h  =  2.25  square  feet. 

4  Square  Hat'hs i  Cowrie  i  square  yard. 

4  Cowries ...   i  Gunda  4  square  yards. 

20  Gundas i  Cottah  80  „ 

20  Cottahs i  Beegah  =  {  l6o°  ^ e  ^ards'  or 

|          .3306  acre. 

For  land  measure,  the  following  table  is  used  for  Government  surveys: — 
i  Guz  33  lineal  inches. 

3  Guz i  Baus,  or  Rod  =  8^  lineal  feet. 

9  Square  Guz i  Square  Rod  =          68  I/IQ  square  feet. 

400  Square  Rods r  Beegah          =  I  3o»S  square  yards,  or 

^  .025  acre. 

III.  BENGAL  MEASURES  OF  CAPACITY. 

The  Seer  is  a  measure  common  to  liquids  and  dry  goods.  It  is  taken 
at  68  cubic  inches,  or  1.962  pints,  in  volume.  But  it  varies  in  different 
localities.  5  Seer=  i  Palli,  and  8  Palli  =  i  Maund,  or  9.81  gallons.  The 
Sooli  =  3.065  bushels,  and  16  Soolis  =  i  Khahoon,  or  49.05  bushels. 

IV.  BENGAL  MEASURES  OF  WEIGHT. 

The  Tolah,  or  weight  of  a  Rupee,  180  grains,  is  the  unit  of  weight. 

i  Tolah         =  1 80  grains. 

5  Tolahs i  Chittak       =  900      „ 

16  Chittaks i  Seer  =      2.057  pounds. 

5  Seers i  Passeeree  =    10.286       „ 

8  Passeerees...  .   i  Maund       =    82.286 


MADRAS. — WEIGHTS  AND  MEASURES. — Tables  No.  60. 
I.  MADRAS  MEASURES  OF  LENGTH. 

The  English  foot  and  yargl  are  used.  The  Guz  is  33  inches.  The  Baum 
or  fathom  is  about  6%  feet.  A  Nalli-Valli  is  a  little  under  i^  miles. 
7  Nalli-Valli  =  i  Kadam,  or  about  10  miles.  The  following  are  native 
measures : — 


1 82  WEIGHTS   AND   MEASURES. 

8  Torah  i  Vurruh  .4166  inch. 

24  Vurruh i  Mulakoli  =  10  inches. 

4  Mulakoli  i  Dumna     =  40      „ 

II.  MADRAS  MEASURES  OF  SURFACE. 

The  English  acre  is  generally  known.  The  native  measures  are  uncer- 
tain. In  Madras  and  some  other  districts,  the  following  native  measures 
are  used : — 

i  Coolie    :  64  square  yards. 

4  z/e  Coolies  i  Ground  =        266^/3  square  yards. 

24  Grounds,  or  )  r  /  6400  square  yards,  or 

100  Coolies         ) =\        1.3223  acres. 

1 6  Annas  (each  400  yards),  i  Cawnie. 

III.  MADRAS  MEASURES  OF  CAPACITY. 

i  Olluck  =      .361  pint. 

8  Ollucks i  Puddee  ==    1.442  quarts. 

8  Puddees  i  Mercal  ==    2.885  gallons. 

5  Mercals   i  Parah  =14.426        „ 

80  Parahs i  Garce  =  18.033  quarters. 

This,  though  the  legal  system,  is  not  used.  The  "customary"  Puddee  is 
still  in  general  use;  it  has,  when  slightly  heaped,  a  capacity  of  1.504  quarts. 
The  Mercal  has  a  capacity  of  3.0006  gallons;  but,  when  heaped,  it  is  equal 
to  8  heaped  Puddees.  The  Seer-measure  is  the  most  common;  its  cubic 
contents  are  from  66^4  to  67  cubic  inches. 

IV.  MADRAS  MEASURES  OF  WEIGHT. 

i  Tola  1 80  grains. 

3  Tolas i  Pollum  =  1.234  ounces. 

8  Pollums  i  Seer       =  9.874       „ 

5  Seers i  Viss  3.086  pounds. 

8  Viss i  Maund  =  24.686        „ 

-Maunds ,  Candy    =    {« 

In  commerce,  the  Viss  is  reckoned  as  3^  pounds;  the  Maund,  25 
pounds;  and  the  Candy,  500  pounds. 


BOMBAY. — WEIGHTS  AND  MEASURES. — Tables  No.  61. 

I.  BOMBAY  MEASURES  OF  LENGTH. 

i  Ungulee  =   9/l6  inch. 

2  Ungulee i  Tussoo    =    i^i  inches. 

8  Tussoos i  Vent'h     =9         „ 

16  Tussoos i  Hat'h      =  18 

24  Tussoos i  Guz         =27         „ 

The  Builder's  Tussoo  =  2.3625  inches  in  Bombay;  and  i  inch  in  Surat. 


BOMBAY,  CEYLON,  BURMAH.  183 

II.  BOMBAY  MEASURES  OF  SURFACE. 

34  z/6  Square  Hat'h...   i  Kutty    =  9-8175  square  yards. 

20  Kutties  ..............   i  Fund     =        196.35  „ 

,        [  3927  square  yards,  or 
2oPund  ................   iBeegah=<*     r 


120  Beegah  ..............   i  Chahur=  97.368  acres. 

In  the  Revenue  Field  Survey,  the  English  acre  is  used. 

III.  BOMBAY  MEASURES  OF  CAPACITY. 

i  Tippree=  .2800  pint. 

2  Tipprees  ...............   i  Seer       =  .5600     „ 

4  Seers  ..................   i  Pylee      =  2.2401  pints. 

1  6  Pylees  ..................   i  Parah     =  4.4802  gallons. 

8  Parahs  ..................   i  Candy   =  35.8415        „ 

i  Mooda  = 


Another  liquid  measure  is  the  Seer  of  60  Tolas  =  1.234  pints. 
In  timber  measurement  in  the  Bombay  dockyards,  a  Covit  or  Candi 
12.704  cubic  feet. 


CEYLON. 

The  British  weights  and  measures  are  used. 


BURMAH. 


The  English  yard,  foot,  and  inch  are  being  adopted;  also  the  English 
Measures  of  Capacity.  Weights. — The  Piakthah  or  Viss  is  3.652  pounds, 
and  contains  100  Kyats  of  252  grains  each. 


CHINA. — WEIGHTS  AND  MEASURES. — Tables  No.  62. 
I.  CHINESE  MEASURES  OF  LENGTH. 

i  Fen  (line)  .141  inch. 

10  Fen i  Ts'un  (punto  or  inch)  =          1.41  inches. 

10  Ts'un i  Ch'ih  (covid  or  foot)  =        14.1         „ 

i°  Ch'ih i  Chang  (rod)  =  { '^.75  feet'.     " 

10  Chang i  Yin  =        39.17  yards. 

The  Ch'ih  of  14.1  inches  is  the  legal  measure  at  all  the  ports  of  trade. 

At  Canton,  the  values  of  the  Ch'ih  are  as  follows: — 

Tailor's  Ch'ih 14.685  inches. 

Mercer's  Ch'ih  (wholesale) 14.66  to  14.724  inches. 

Mercer's  Ch'ih  (retail) 14.37  to  14.56        „ 

Architect's  Ch'ih 12.7  inches. 

At  Pekin  there  are  thirteen  different  Ch'ihs. 


1 84  WEIGHTS   AND   MEASURES. 

ITINERARY  MEASURE. 

5  Ch'ih  (covids) i  Pii  (pace). 

360  Pii i  Li  -  about  y$  mile. 

250  Li  (geographical) i  Tii  (degree)  =       „      83  miles. 

II.  CHINESE  LAND  MEASURE. 

25  Ch'ih  (covids) i  Kung  (bow)  =        30^  square  feet. 

24o  Kung. . .  .   ,  Mou  (rood)  =  {  |^  squar>> 

100  Mou c i  King  =        1 6  3/3  acres. 

The  principal  land  measure  is  the  Mou. 

III.  CHINESE  CUBIC  MEASURE,  AND  MEASURES  OF  CAPACITY. 

100  Cubic  Ch'ih  (covids) i  Fang  or  Ma. 

i  o  Ho  (gills) i  Sheng  (pint)  =  about  2  pints. 

10  Sheng i  Tou  (peck)  =      „      2^  gallons. 

5  Tou iHu(bushel)-      „    12^ 

Liquids  are  measured  by  vessels  containing  definite  weights,  as  i,  2,  4, 
and  8  Taels;  also  large  earthen  vessels  holding  15,  30,  and  60  Catties. 

IV.  CHINESE  MEASURES  OF  WEIGHT. 

i  Liang  or  Tael  =      ix/3  ounces. 

16  Liang i  Chin  or  Catty  =      ix/3   pounds. 

100  Chin i  Tan  or  Picul   =  i 


COCHIN-CHINA. 

Length. — The  Thuoc,  or  cubit,  19.2  inches,  is  the  chief  unit  of  measure 
of  length.  It  varies  considerably  for  different  places.  The  Li  or  mile  is 
486  yards;  2  Li  make  i  Dam;  and  5  Dam  make  i  League  =  2. 761  miles. 

Surface. — 9  Square  Ngu  make  i  Square  Sao  =  64  square  yards.  100 
Square  Sao  make  i  Square  Mao  =  6400  square  yards,  or  1.32  acres. 

Weights. — The  smallest  weight  is  the  Ai  =  . 0000006  grain.  The  weights 
ascend  by  a  decimal  scale,  until  10,000,000,000  Ai  are  accumulated  = 
i  Nen  =  .8594  pound.  The  greatest  weight  is  the  Quan  =  68y^  pounds. 

Capacity  for  Grain. — i  Hao  =  62/9  gallons.  2  Hao  =  i  Shita=i24/9 
gallons. 


PERSIA. 

Length. — The  Gereh  -  2^  inches;  16  Gerehs-  i  Zer='38  inches.  The 
Kadam  or  Step  =  about  2  feet;  12,000  Kadam  =  i  Fersakh  =  about  4^  miles. 

Surface  and  Cubic  Measures. — These  are  the  squares  and  cubes  of  the 
lengths. 

Capacity  (Dry  Goods}. — The  Sextario  =  .07236  gallon.  4  Sextarios  = 
i  Chenica;  2  Chenicas  =  i  Capicha;  3^  Capichas=  i  Collothun;  8  Collo- 
thun  =  i  Artata  -  1.809  bushels. 


PERSIA,   JAPAN. 


185 


Liquids  are  sold  by  weight. 

Weights. — The  Miscal^yi  grains;  16  Miscals=i  Sihr;  100  Miscals  = 
i  Ratel=  1.014  pounds;  40  Sihrs=  i  Batman  (Maund)  =  6.49  pounds;  100 
Batman  (of  Tabreez)  =  i  Karvvar  =  649.i42  pounds. 


JAPAN. — WEIGHTS  AND  MEASURES. — Tables  No.  63. 
I.  JAPANESE  MEASURES  OF  LENGTH. 

Rin  .012  inch. 

10  Rin Boo  .120  inch. 

10  Boo Sun  1.20  inches. 

10  Sun Shaku  =          23  J5/l6  inches. 

10  Shaku Jo  9  feet  1 1  s/l6  inches. 

6  Shaku Ken  5  feet  ix^i  inches. 

60  Ken i  Cho  =        119.4  yards. 

3^hu lRi    ii^asatr  • 

Rough  timber  is  sold  by  the  Yama-Ken-Zaii  =  63  Sun.    Cloth  is  measured 
by  the  Shaku  of  15  inches,  with  decimal  sub-multiples. 

II.  JAPANESE  MEASURES  OF  LAND. 

i  Shaku  =  .9885  square  foot. 

36  Square  Shaku i  Tsubo  =  3-954  square  yards. 

30  Tsubo i  Se         =  118.615  square  yards. 

10  Se i  Tan       =  39.212  square  poles. 

10  Tan i  Cho      =  2.451  acres. 

III.  JAPANESE  MEASURES  OF  CAPACITY. 

i  Kei      =       .0000318  pint. 

10  Kei Sat       =       .000318  pint. 

10  Sats Sai       =  .00318  pint. 

10  Sai Shaku  =       .0318  pint. 

10  Shaku Go       =       .3178  pint. 

10  Go Sho      =       .3973  gallon. 

10  Sho To       =  3.970  gallons. 

10  To Koku  =  39.703  gallons. 

IV.  JAPANESE  MEASURES  OF  WEIGHT. 

Shi  .0058  grain. 

10  Shi Mo  .058       „ 

10  Mo Rin  .5797     „ 

10  Rin Fun  =  5.7972  grains. 

10  Fun Momme  —  57.972        ,, 

100  Momme Hiyaku-me  =  .8282  pound. 

1000  Momme i  Kwam-me  =  8.2817  pounds. 

1 60  Momme i  Kin  =  i^i            „ 

100  Kin i  Hiyak-kin  —  132^            „ 


1 86  WEIGHTS   AND    MEASURES. 

STRAITS    SETTLEMENTS. 

The  unit  measure  of  length  is  the  yard;  land  is  measured  by  the  acre. 
The  Chupack  or  quart  of  4  Paus  =  8  imperial  gills ;  4  quarts  =  i  Gantang 
or  gallon  =  32  gills.  The  Kati=iI/3  pounds;  100  Kati  =  i  Picul  = 
J-SS'/s  pounds;  40  Picul=  i  Koyan  =  5333 V3  pounds. 


JAVA. 

Length. — The    Duim=i.3    inches.     12    Duims=i    foot.      The    Ell  = 
27.08  inches. 

Surface. — The  Djong  of  4  Bahu  -  7.015  acres. 

Capacity,  for  rice  and  grain. — The  measures  are  in  fact  measures  of 
definite  weights,  i  sack  =  61.034  pounds;  2  sacks  =  i  Pecul;  5  Peculs 
=  i  Timbang  =  5.45  cwts.;  6  Timbang  =  i  Coyau  =  32.7  cwts.  For  liquids : 
The  Kan  =  .328  gallon;  388  Kans=  i  Leager  =  127.34  gallons. 

Weights. — The  Tael=  593.6  grains;  i6Taels=i  Catty  =1.356  pounds: 
TOO  Catties  =  i  Pecul  =  135.63  pounds. 


UNITED    STATES    OF    AMERICA. 

Length. — The  measures  are  the  same  as  those  of  Great  Britain. 

In  Land  Surveying,  the  unit  of  measurement  is  the  chain,  and  it  is  deci- 
mally subdivided. 

In  City  Measurements,  the  unit  is  the  foot,  and  it  is  decimally  subdivided. 

In  Mechanical  Measurements,  the  unit  is  the  inch,  and  it  is  divided  into 
a  hundred  parts. 

Surface. — The  measures  are  the  same  as  those  of  Great  Britain. 

Capacity. — The  measures  of  capacity  for  dry  goods  and  for  liquids  are  the 
same  as  the  old  English  measures.  The  standard  U.  S.  gallon  is  equal  to 
the  old  English  wine  gallon,  or  231  cubic  inches;  it  contains  8^3  pounds 
of  pure  water  at  62°  F. 

Dry  Measure. — Table  No.  64. 

i  gill.  =     .96945  imperial  gill. 

4  gills i  pint  =     .96945  imperial  pint. 

2  pints  i  quart  —  1.9388  ,       pints. 

4  quarts i  gallon  =     .96945          ,       gallon. 

2  gallons i  peck  =  1.9388  ,       gallons. 

4  pecks i  bushel  -96945          ,       bushel. 

4  bushels i  coomb  =  3.8777  ,       bushels. 

2  coombs i  quarter  =     .96945          ,       quarter. 

5  quarters i  wey  or  load   =  4.8472  „        quarters. 

2  weys i  last  =  9.6945  ,,       quarters. 

For  the  Wine  and  Spirit  Measures,  and  the  Ale  and  Beer  Measures,  see 
the  Old  Measures  of  Great  Britain,  page  139. 

i  cord  of  wood  =128  cubic  feet  =  (4  feet  x  4  feet  x  8  feet). 

Weights. — The  Weights  are  the  same  as  those  of  Great  Britain.  (See 
page  140.) 


BRITISH   NORTH  AMERICA,   ETC.  l8/ 

There  are,  in  addition,  the  Quintal  or  Centner  of  100  pounds;  and  the 
New  York  ton  of  2000  pounds,  which  is  also  used,  for  retail  purposes  espe- 
cially, in  most  of  the  States.  The  old  hundredweight  and  old  ton  are,  for 
the  most  part,  superseded  by  the  quintal  and  the  New  York  ton.  The 
wholesale  coal  and  iron  ton  is  2240  pounds.  The  French  metric  system  of 
weights  and  measures  was  legalized  in  1866  concurrently  with  the  old  system. 


BRITISH    NORTH    AMERICA.— WEIGHTS   AND   MEASURES. 

Until  the  23d  May,  1873,  the  standard  measures  of  length  and  surface, 
and  the  weights,  were  the  same  as  those  of  Great  Britain;  whilst  the 
measures  of  capacity  were  the  old  British  measures  for  dry  goods,  for  wine, 
and  for  ale  and  beer.  At  the  above-named  date  a  new  and  uniform  system 
of  weights  and  measures  came  into  force,  in  which  the  imperial  yard,  pound 
avoirdupois,  gallon,  and  bushel,  became  the  standard  units,  and  the 
imperial  system  was  adopted  in  its  integrity,  with  two  important  exceptions : 
that  the  hundredweight  of  112  pounds,  and  the  ton  of  2240  pounds  were 
abolished;  and  the  hundredweight  was  declared  to  be  100  pounds,  and  the 
ton  2000  pounds  avoirdupois, — thus  assimilating  the  weights  of  Canada  to 
those  of  the  United  States. 

The  French  metric  system  of  weights  and  measures  has  been  made 
permissive  concurrently  with  the  standard  weights  and  measures. 


MEXICO. 

The  weights  and  measures  are  the  old  weights  and  measures  of  Spain. 


CENTRAL   AMERICA   AND   WEST    INDIES. 

WEST   INDIES   (British). 
The  weights  and  measures  are  the  same  as  those  of  Great  Britain. 

CUBA. 

The  old  weights  and  measures  of  Spain  are  in  general  use.  For  engineer- 
ing and  carpentry  work  the  Spanish,  English,  and  French  measures  are  in 
use.  The  French  metric  system  of  weights  and  measures  is  legalized,  and 
is  used  in  the  customs  departments. 

GUATEMALA   AND    HONDURAS. 

The  weights  and  measures  are  the  old  weights  and  measures  of  Spain. 

BRITISH   HONDURAS. 
In  British  Honduras,  the  British  weights  and  measures  are  in  use. 

COSTA  RICA. 

The  old  weights  and  measures  of  Spain  are  in  general  use.  But  the 
introduction  of  the  French  metric  system  is  contemplated. 


1 88  WEIGHTS  AND   MEASURES. 

ST.    DOMINGO. 

The  old  Spanish  weights  and  measures  are  in  general  use.  The  French 
metric  system  is  coming  into  use. 

SOUTH   AMERICA. 
COLOMBIA. 

The  French  metric  system  was  introduced  into  the  Republic  in  1857, 
and  is  the  only  system  of  weights  and  measures  recognized  by  the  govern- 
ment. In  ordinary  commerce,  the  Oncha,  of  25  Ibs.,  the  Quintal,  of 
100  Ibs.,  and  the  Carga,  of  250  Ibs.,  are  generally  used.  The  Libra  is 
1. 1 02  pounds.  The  yard  is  the  usual  measure  of  length. 

VENEZUELA. 

The  system  and  practice  are  the  same  as  those  of  Colombia. 

ECUADOR. 

The  French  metric  system  became  the  legal  standard  of  weights  and 
measures  on  the  ist  January,  1858. 

GUIANA. 

In  British  Guiana,  the  weights  and  measures  are  those  of  Great  Britain. 
In  French  Guiana  or  Cayenne,  the  ancient  French  system  is  practised.  In 
Dutch  Guiana,  the  weights  and  measures  of  Holland  are  employed. 

BRAZIL. 

The  French  metric  system,  which  became  compulsory  in  1872,  was 
adopted  in  1862,  and  has  since  been  used  in  all  official  departments.  But 
the  ancient  weights  and  measures  are  still  partly  employed.  They  are,  with 
some  variations,  those  of  the  old  system  of  Portugal. 

Length. — The  Line  =  .0911  inch,  and  is  divided  into  tenths.  The  Polle- 
gada  =1.0936  inches.  The  Pe  =  13.1236  inches,  or  I/3  metre.  The  Vara  = 
1.215  yards;  and  iy£  Varas  =  the  geometrical  pace  =1.82 2 7  yards.  The 
Milha=  1.2965  miles;  and  3  Milhas=  i  Legoa  =  3.8896  miles. 

6  yards  are  reckoned  equal  to  5  Varas. 

Surface. 

64  Square  Pollegadas...  i  Square  Palmo  ==    .5315  square  foot. 

25  Square  Palmos  i  Square  Vara  =  1.4766  square  yards. 

4  Square  Varas i  Square  Bra^a  =5.9063  ,, 

4840  Square  Varas i  Geira  =  1.4766  acres. 

Capacity  (Dry  Goods}. — The  Salamine  =  .3808  gallon;  2  Salamines  = 
i  Oitavo;  2  Oitavo  =  i  Quarto;  4  Quartas=i  Alqueiro  =  .38o8  bushel; 
4  Alqueiras  =  i  Fangas;  15  Fangas  =  i  Moio  =  2.8560  quarters. 

Liquids. — The  Quartilho  =  .6i4i  pint;  4  Quartilhos=  i  Canada;  6  Cana- 
das  =  i  Pota  or  Cantaro;  2  Potas=  i  Almuda-  3.6846  gallons. 


PERU,   CHILI,   BOLIVIA,   ETC.  189 

Weights.  —  The  Arratel=  1.0119  pounds,  is  divided  into  16  Onc.as,  and 
then  into  8  Oitavos.  32  Arratels=i  Arroba;  4  Arrobas  =  i  Quintal  = 
129.5181  pounds;  and  13^  Quintals  =  i  Tonelada=  15.6116  cwts. 

There  is  also  the  Quintal  of  100  Arratels.  Ships'  freight  is  reckoned  by 
the  English  ton  =  70  Arrobas. 

PERU. 

The  French  metric  system  was  established  in  1860,  but  is  not  yet  gener- 
ally used.  The  weights  and  measures  in  common  use  are  :  —  The  ounce  = 
1.014  ounce;  the  Libra=  1.014  pound;  the  Quintal  =  101.44  pounds;  the 
Arroba  =  25.  36  pounds,  or  6.70  gallons;  the  gallon  =  .74  imperial  gallon; 
the  Vara-,927  yard;  the  square  Vara  =  .859  square  yard. 


CHILI. 

The  French  metric  system  has  been  legally  established;  but  the  old 
weights  and  measures  are  still  in  general  use.  These  are  the  same  as  those 
of  Peru. 

BOLIVIA. 

The  weights  and  measures  are  the  same  as  the  old  weights  and  measures 
of  Peru  and  Chili. 

ARGENTINE   CONFEDERATION. 

The  French  metric  system  has  recently  been  established.  The  old 
weights  and  measures  are  commonly  used  :  —  the  Castilian  standards  of  the 
old  Spanish  system.  The  Quintal  =  101.4  pounds;  the  Arroba  =  25.35 
pounds;  the  Fanega  =  1.5  bushels. 

URUGUAY. 

The  French  metric  system  was  established  in  1864.  The  old  weights 
and  measures  are  the  same  as  those  of  the  Argentine  Confederation.  The 
weights  and  measures  of  Brazil  are  in  general  use. 

PARAGUAY. 

The  weights  and  measures  are  the  same  as  the  old  ones  of  the  Argentine 
Confederation. 

AUSTRALASIA. 

In  New  South  Wales,  Queensland,  Victoria,  South  Australia,  West 
Australia,  Tasmania,  and  New  Zealand,  the  legal  weights  and  measures  are 
the  same  as  those  of  Great  Britain.  But  the  old  British  measures  of 
capacity  are  also  much  used. 

In  land  measurement,  a  "section"  is  an  area  equal  to  80  acres. 


MONEY. 


GREAT    BRITAIN    AND    IRELAND. 

COINS.  MATERIAL.         WEIGHT. 

Grains. 

farthing bronze 43-750 

y2d. halfpenny do.        ...   87.500 

4  farthings i  penny do.        ...145.833 

$d.  threepenny  piece silver 21.818 

4^. groat,  or  fourpenny  piece do.         ...   29.091 

6d.  sixpence do.         ...  43.636 

12  pence i  shilling do.         ...   87.273 

2  shillings i  florin do.         . . . 1 74.545 

2  y2s i  half-crown do.          ...  2 1 8. 1 82 

IQS i  half-sovereign gold 61.6372 

2os i  sovereign,  or  pound  sterling  do.  . . .  1 23. 2  745 

The  bronze  coins  are  made  of  an  alloy  of  copper,  tin,  and  zinc;  the 
silver  coins  contain  92^  per  cent,  of  fine  silver,  and  7^  per  cent,  of  alloy; 
the  gold  coins,  91^/3  per  cent,  of  fine  gold,  and  8y$  per  cent,  of  alloy. 

The  Mint  price  of  standard  gold  is  ^£3,  i*js.  io^d.  per  ounce. 

One  pound  weight  of  silver  is  coined  into  66  shillings.  The  intrinsic 
value  of  22  shillings  is  equal  to  £i  sterling. 

The  intrinsic  value  of  480  pence  is  equal  to  £i  sterling. 

FRANCE. — MONEY. 

Copper. 
COINS.  WEIGHT.  VALUE  IN  ENGLISH  MONEY. 


Grammes* 

£ 

s.       d. 

i/ 

/  100 

franc  

i 

centime  .. 

.    i  

0 

o        'A- 

'/SO 

franc  

2 

centimes 

.     2  

o 

o        z/5 

'/«, 

franc  

5 

centimes 

(sou)  

•  5  

0 

o        YZ 

•Ac 

franc  

10 

centimes 

(gros  sou)  .  . 

,.10  

o 

0        I 

Silver. 

V, 

franc  

20 

centimes  . 

..    I  

0 

0       2 

franc  

5o 

centimes. 



.-  2.5  

0 

o     4^4 

I 

franc  

IOO 

centimes. 

..  5    .... 

o 

o     9^ 

more 

exactly 

9.524^. 

2 

francs  

..10  

o 

1      7 

5 

francs  

••25  

o 

3   ii  f6 

GERMANY,   HANSE   TOWNS.  IQI 

Gold. 


Grammes. 


"" 


5  francs  ..................   1-61290  .....................  o  3  n^j 

10  francs  ..................   3-22580  .....................  o  7  n^ 

20  francs  (Napoleon)...  6-45161  (99-56  grains).  ..o  15  10^ 

50  francs  .................  16*12902  .....................  i  19  8  1/5 

100  francs  ..................  32*25805  .....................  3  19  4  4/IO 

The  English  value  is  calculated  at  the  rate  of  25  francs  20  centimes  to 
j£i.  The  bronze  coins  consist  of  an  alloy  of  95  parts  of  copper,  4  of  tin, 
i  of  zinc.  The  standard  fineness  of  the  gold  pieces,  and  of  the  silver 
5-franc  pieces  is  90  per  cent.,  with  10  per  cent,  of  copper;  of  the  other 
silver  coins,  83.5  per  cent.;  and  of  the  bronze  coins,  95  per  cent. 

GERMANY.—  MONEY. 

The  following  system  of  currency  was  established  throughout  the  Ge 
Empire  in  1872:  — 

ENGLISH  VALUE. 

s.       d. 
i  Pfennig  ..............    =      o       .1175 

10  Pfennig  ...............  i  Groschen  ............    —      o     1.175 

i  o  Groschen  ............  i  Mark  .................    =      o  1  1  ^ 

i  o  Marks  (gold)  ..................................    =      9     9^ 

20  Marks  (gold)  ...................................    =    19     7 

The  2o-mark  gold  piece  weighs  122.92  grains,  and  the  standard  fineness 
of  the  gold  pieces  is  90  per  cent,  of  gold. 

Before  1872,  accounts  were  reckoned  in  the  following  currency  in  North 
Germany  :  — 

s.       d. 
12  Pfennig  ...............  i  Silbergroschen  ......    =      i     i  T/5 

30  Silbergroschen  ......  i  Thaler  ................    =      3     o 

In  South  Germany:  — 

4  Pfennig  .................  i  Kreutzer  .............    =      o       y$ 

60  Kreutzers  .......  .......  i  Florin  .................    =      i     8 

HANSE    TOWNS.  —  MONEY. 

The  monetary  system  is  that  of  the  German  Empire. 
Hamburg.  —  According  to  the  old  monetary  system,  in  which  silver  was 
the  standard,  12  Pfennig  =i  Schilling  =  s/6  d.;  and  16  Schillings  =i  Mark 


Bremen.  —  Old  system:  —  5  Schmaren=  i  Groot  =  II/20^.;  and  71  Groots  = 
i    Rix-dollar  =  3  j.  3  3/s  d.     The   Rix-dollar,   or  Thaler,  was  a  money  of 
account. 

Lubec.  —  The  old  system  was  the  same  as  that  of  Hamburg,  and,  in 
addition,  3  Marks  =  i  Thaler  =  35-.  ^d. 


1 92  MONEY. 

AUSTRIA.— MONEY. 


s.     d. 


i  Kreutzer  (copper) o       I/5 

4  Kreutzers    (do.) o       4/s 

i  o  Kreutzers  (silver) o     2^6 

20  Kreutzers    (do.) o     4^ 

#  Florin          (do.) o     5^ 

1  Florin          (do.) i  n% 

2  Florins         (do.) 3  11^2 

4  Florin  piece  (gold) 7  n 

8  Florin  piece   (do.) 15  10 

100  Kreutzers  make  i  Florin. 

The  4-florin  gold  piece  weighs  49.92  grains,  and  the  standard  of  fineness 
is  90  per  cent,  of  gold. 

RUSSIA. — MONEY. 

s.      d. 

i  Copeck =    o       .38 

100  Copecks i  Silver  Rouble =32 

The  copper  coins  are  pieces  of  ^,  ^,  i,  2,  3,  5  Copecks.  The  silver 
coins  are  pieces  of  5,  10,  15,  20,  25  Copecks,  the  Half  Rouble,  and  the 
Rouble;  the  gold  coins  are  the  Three-rouble  piece,  the  Half  Imperial  of 
five  Roubles,  and  the  Imperial  of  10  Roubles.  The  5-rouble  gold  piece 
weighs  10 1  grains,  and  the  standard  of  fineness  is  91^3  per  cent,  of  gold. 
Paper  currency: — i,  3,  5,  10,  25,  50,  TOO  Roubles. 

HOLLAND.— MONEY. 

s.     d. 

i  Cent =    o       Vs 

100  Cents i  Guilder  or  Florin =    i     8 

BELGIUM. — MONEY. 
The  monetary  system  is  exactly  the  same  as  that  of  France. 

DENMARK. — MONEY. 

S.        d, 

i  Skilling =  o       .2745 

16  Skillings i  Mark =  o     4.392 

96  Skillings,  or  6  Marks i  Rigsdaler,  or  Daler =  22  7/20 

SWEDEN. — MONEY. 

s.       d. 

i  Ore =   o       .133 

100  Ore i  Riksdaler =    i  ij^ 

NORWAY.— MONEY. 

s.  d. 

i  Skilling =?    o  .444 

24  Skillingen i  Ort  or  Mark ••--   o  iojg 

5  Ort i  Species-Daler =4  5^$ 


SWITZERLAND,   SPAIN,   ETC.  193 

SWITZERLAND. — MONEY. 

The  monetary  system  of  Switzerland  is  the  same  as  that  of  France.  The 
Centime  is  called  a  Rappe. 

SPAIN. — MONEY. 

d. 

i  Centimo =    95 

100  Centimes i  Peseta =    i  franc,  or     9^ 

The  bronze  coins  are  pieces  of  i,  2,  5,  and  10  centimes.  The  silver 
coins  are  pieces  of  20  and  25  centimos,  and  i,  2,  and  5  pesetas.  The  gold 
coins  are  pieces  of  5,  10,  20,  25,  50,  and  100  pesetas.  The  piece  of 
5  pesetas  is  35-.  n^/.,  English  value.  The  25  peseta  piece  is  19^.  9^^., 
English  value. 

The  old  monetary  system  was  based  on  the  Real-Vellon,  2~L/>d.  English 
value;  it  was  the  2oth  part  of  the  Silver  Hard  Dollar,  ^s.  zd.  English  value, 
and  of  the  Gold  Dollar  or  Coronilla.  The  Duro  was  identical  with  the 
American  Dollar. 

PORTUGAL. — MONEY. 

The  unit  of  account  is  the  Rei,  of  which  18^  Reis  make  i  penny;  and 
4500  Reis  make  i  sovereign.  The  Milreis  is  1000  Reis,  43.  $Vzd.  English 
value.  The  Corda  is  the  heaviest  gold  coin,  of  10,000  Reis,  £2,  ^s.  5  *^. 
English  value,  and  weighs  17.735  grammes. 

ITALY. — MONEY. 

d. 

i  Centime =  .95 

100  Centimes i  Lira =    i  franc,  or  9^ 

Copper  coins  are  pieces  of  i,  3,  and  5  Centimes;  silver  coins,  20  and 
50  Centimes,  and  i,  2,  and  5  Lire;  gold  coins,  5,  10,  20,  50,  and  100  Lire. 
These  coins  are  the  same  in  weight  and  fineness  as  the  coins  of  France. 

TURKEY. — MONEY. 

s.      d. 

i  Para =     o        '/is.s 

40  Paras i  Piastre.... =     o     2.16 

100  Piastres i  Medjidie,  or  Lira  Turca   =    1 8     o 

The  Piastre  is  roughly  taken  equal  to  2d.  sterling. 

GREECE   AND    IONIAN    ISLANDS.—  MONEY. 

100  Lepta i  Drachma =    i  franc,  or  ^yzd. 

The  currency  of  Greece  is  the  same  as  that  of  France. 

In  the  Ionian  Islands,  whilst  they  were  under  British  protection  (1830- 
1864),  accounts  were  kept  by  some  persons  in  Dollars,  of  100  Oboli  =  4^.  2d.; 
by  others  in  Pounds,  of  20  shillings,  of  12  pence,  Ionian  currency;  the 
Ionian  Pound  being  equal  to  205-.  q.6d.  sterling.  By  other  persons  accounts 
were  kept  in  Piastres  of  40  Paras  =  2  */45d. 

13 


IQ4  MONEY. 

MALTA. — MONEY. 


20  Grani  

Grano 

s.        d. 

—    o         */ 

the 

Taro... 

~     O       1  3^ 

12  Tari  

Scudo  

-    i     8 

60  Piccioli 

Or, 
Carlino  . 

•  •-    o        185 

9  Carlini  . 

Taro... 

—    O      I  'Z/s 

1  2  Tari  .  . 

SrnHo  . 

-    i      8 

>h  money  is  in  general  circulation. 

The  Sovereign  =12  Scudi  ; 

40  Paras 

100  Piastres 

5  Egyptian  Guineas., 

1000  Purses 

97.22  Piastres 


Shilling  =  7  Tari  4  Grani. 

EGYPT. — MONEY. 

£    s.    d. 

Para t          o     o       .0615 

Piastre  (Tariff ) -         o     o     2.461 

Egyptian  Guinea =          i     o     6.84 

Kees,  or  Purse =          52   10.2 

Khuzneh,  or  Treasury  =  5142   10     o 
English  Sovereign. 

The  Egyptian  guinea  weighs  132  grains,  and  the  standard  of  fineness  is 

per  cent,  of  gold. 
Two  piastres  (current)  are  equal  to  one  piastre  (tariff). 

MOROCCO. — MONEY. 

s.      d. 

I    Flue  :      O  37/g6o 

24  Flues i  Blankeel   =  •    o       37/40 

4  Blankeels i  Ounce       -    o     3.7 

10  Ounces i  Mitkul        =    3     i 

TUNIS. — MONEY. 

s.  d. 

i  Fel  -    o  35/288 

3  Fels i  Karub        =    o  35/96 

1 6  Karubs i  Piastre        :    o  55/5 

ARABIA. — MONEY. 

s.     d. 
80  Caveers i  Piastre  or  Mocha  Dollar ==    3     5 

CAPE    OF    GOOD    HOPE. — MONEY. 

Public  accounts  are  kept  in  English  money;  but  private  accounts  are 
often  kept  in  the  old  denominations,  as  follows : — 

s.  d. 
i  Stiver            =    o      3/8 

6  Stivers i  Schilling     =    o     2^ 

8  Schilling i  Rix-dollar   =    i     6 

The  Guilder  is  equal  to  6d. 


INDIAN   EMPIRE,    CHINA,    ETC.  195 

INDIAN    EMPIRE. — MONEY. 

Throughout  India,  accounts  are  kept  in  the  following  moneys: — 

s.      d. 

i  Pie =F    o     o>y%     nominal  value. 

12  Pies i  Anna :=    o     i^  do. 

1 6  Annas i  Rupee 20  do. 

The  intrinsic  value  of  the  Rupee  is  is.  lo^d.;  it  weighs  180  grains. 
The  English  Sovereign  is  equal  to  10  Rupees  4  Annas. 

i  Lac  of  Rupees     =  100,000  rupees  =  ,£10,000. 
i  Crore  of  Rupees  =  100  lacs  =  ,£1,000,000. 

In  Ceylon,  the  Rupee  is  divided  into  100  Cents. 

The  gold  coin,  Mohur,  is  equal  to  15  rupees;  it  weighs  180  grains,  and 
the  standard  fineness  is  91.65  per  cent,  of  gold. 

CHINA. — MONEY. 

s.      d. 

i  Cash  (Le) =  o  7/IOO 

10  Cash i  Candareen  (Fun) =  o  7/IO 

10  Candareens i  Mace  (Tsien) =  o  7 

10  Mace i  Tael  (Le'ang) =  5  10 

COCHIN-CHINA. — MONEY. 

s.     d. 

i  Sapek,  or  Dong,  or  Cash =    o       */l8 

60  Sapeks i  Mas,  or  Mottien si    o     3^ 

10  Mas i  Quan,  or  String =    2     9^ 

PERSIA. — MONEY. 

s.      d. 

i  Dinar =    o  J/8o 

50  Dinars i  Shahi =    o  ^ 

20  Shahis i  Keran =   o  n}s£ 

10  Kerans i  Toman =    9  3^ 

JAPAN. — MONEY. 

s.     a. 

10  Rin i  Sen =  ^ 

100  Sen i  Yen =    42 

There  are  gold  coins  of  the  value  of  i,  2  and  5  yen,  with  a  standard 
fineness  of  90  per  cent.  The  5~yen  piece  weighs  128.6  grains.  The  silver 
yen  weighs  416  grains,  with  the  same  standard  of  fineness. 

JAVA. — MONEY. 
The  money  account  of  Java  is  the  same  as  that  of  Holland. 

UNITED    STATES    OF    AMERICA. — MONEY. 

s.     d. 

i  Cent =o       y2 

10  Cents i  Dime =05 

100  Cents i  Dollar =   4     2 


196  MONEY. 

CANADA.—  BRITISH    NORTH   AMERICA.  —  MONEY. 

s.     d. 
i  Mil  ..................    =     o       Yao   sterling. 

10  Mils  ..................  i  Cent  ................    =     o       yz         do. 

i  oo  Cents  ................  i  Dollar  .......  ,  .......    =     4     i%         do. 

4  Dollars  .......................................    =    20     o     currency. 

Or, 

i  Penny  currency       —      o       ^  sterling. 
12  Pence  .................  i  Shilling     do  .....    =     o     9  4/s      do. 

20  Shillings  ...............  i  Pound       do  .....    =    16     5^       do. 

The  Dollar  of  Nova  Scotia,  New  Brunswick,  and  Newfoundland,  is  equal 
to  4s.  2d.  sterling.  In  the  Bermudas,  accounts  are  kept  in  sterling  money. 

MEXICO.  —  MONEY. 

Accounts  are  kept  in  dollars  of  100  cents.  The  dollar  is  equal  to  ^s.  2d. 
sterling. 

CENTRAL   AMERICA   AND   WEST    INDIES.—  MONEY. 

WEST   INDIES   (British). 

Accounts  are  kept  in  English  money;  and  sometimes  in  dollars  and 
cents,  i  dollar  =  4$.  2d. 

CUBA.  —  MONEY. 

The  moneys  of  various  nations  were  in  circulation  before  the  current 
war  (1875).  But  the  principal  silver  currency  was  the  10  cent  and  5  cent 
pieces  of  the  United  States.  The  gold  currency  consists  of  the  Ounce,  of 
the  value  of  1  6  dollars,  yz  ounce,  y^  ounce,  Y%  ounce. 

GUATEMALA,   HONDURAS,   COSTA   RICA. 

The  moneys  of  account  are  the  same  as  those  of  Mexico. 

ST.    DOMINGO. 

Accounts  are  kept  in  current  dollars  (called  Gourdes]  and  cents.  The 
cent=  Va^'j  and  ioo  cents  =  i  dollar  = 


SOUTH    AMERICA.  —  MONEY. 

COLOMBIA,  VENEZUELA,  ECUADOR. 


The  moneys  of  account  are,  the  Centavo  =  %</.;  and  ioo  Centavos  = 
i  Peso  =  4-f.  2d. 

GUIANA. 

In  British  Guiana  the  dollar  of  4^.  zd.  is  used,  divided  into  ioo  cents. 
In  French  Guiana,  French  money  is  used.  In  Dutch  Guiana,  the  money  of 
Holland  is  used. 


BRAZIL,   PERU,   ETC.  197 

BRAZIL.—  MONEY. 


i  Rei  .......................    =   o      2-7/ioo 

1000  Reis  ..................  i  Milreis  ..................    ^23 

PERU.  —  MONEY. 

s.     d. 

i  Centesimo  =    o       .37 

100  Centesimos  ........   i  Dollar,  or  Peso         =31 

CHILI.  —  MONEY. 

s.     d. 

i  Centavo  —    o       .45 

100  Centavos  ...........   i  Dollar,  or  Peso         =39 

BOLIVIA. 

i  Centena  =   o      .37 

100  Centenas  ..........   i  Dollar  =31 

ARGENTINE   CONFEDERATION. 

i  Centesimo  =   o       .25 

100  Centesimos  ........  i  Dollar,  or  Patercon  =    2     i 

URUGUAY. 

i  Centime 
100  Centimes  ...........   i  Dollar 

PARAGUAY. 

i  Centena  =   o       .37 

100  Centenas  ...........   i  Dollar  =31 

AUSTRALASIA. 
Accounts  are  kept  in  pounds,  shillings,  and  pence  sterling. 


WEIGHT    AND    SPECIFIC    GRAVITY. 


The  specific  gravity,  or  specific  weight  of  a  body,  is  the  ratio  which  the 
weight  of  the  body  bears  to  the  weight  of  another  body  of  equal  volume 
adopted  as  a  standard  for  comparison  of  the  weights  of  bodies.  For  solids 
and  liquids,  pure  water  at  the  mean  temperature  62°  R,  is  adopted  as  the 
standard  body  for  comparative  weight.  For  gases,  dry  air  at  32°  F.,  and 
under  one  atmosphere  of  pressure,  or  14.7  Ibs.  per  square  inch,  is  the  body 
with  which  they  are  compared. 

The  specific  gravity  of  bodies  is  found  by  weighing  them  in  and  out  of 
water,  according  to  the  following  rules. 

RULE  i. — To  find  the  specific  gravity  of  a  solid  body  heavier  than  water. 
Weigh  it  in  pure  water  at  62°  F.,  and  divide  its  weight  out  of  water  by  the 
loss  of  weight  in  the  water.  The  quotient  is  the  specific  gravity. 

Note. — The  loss  of  weight  in  water  is  the  difference  of  the  weight  in  air 
and  the  weight  in  water,  and  it  is  equal  to  the  weight  of  the  quantity  of 
water  displaced,  which  is  equal  in  volume  to  the  body. 

RULE  2. — To  find  the  specific  gravity  of  a  solid  body  lighter  than  water. 
Load  it  so  as  to  sink  it  in  pure  water  at  62°  F.,  and  weigh  it  and  the  load 
together,  out  of  water,  and  in  water;  weigh  the  load  separately  in  and  out 
of  water;  deduct  the  loss  of  weight  of  the  load  singly  from  that  of  the 
combined  body  and  load ;  the  remainder  is  the  loss  of  weight  of  the  body 
singly,  by  which  its  weight  out  of  water  is  to  be  divided.  The  quotient  is 
the  specific  gravity. 

RULE  3. — To  find  the  specific  gravity  of  a  solid  body  which  is  soluble  in 
water.  Weigh  it  in  a  liquid  in  which  it  is  not  soluble;  divide  the  weight 
out  of  the  liquid  by  the  loss  of  weight  in  the  liquid,  and  multiply  by  the 
specific  gravity  of  the  liquid.  The  product  is  the  specific  gravity  of  the 
body. 

RULE  4. — To  find  the  specific  gravity  of  a  liquid.  Weigh  a  solid  body  in 
the  liquid  and  in  water,  as  well  as  in  the  air,  and  divide  the  loss  of  weight 
in  the  liquid  by  the  loss  of  weight  in  water.  The  quotient  is  the  specific 
gravity. 

RULE  5. — To  find  the  weight  of  a  body  when  the  specific  gravity  is  given. 
Multiply  the  specific  gravity  by 

MULTIPLIER.  WEIGHT    OF 

62-355  (tne  weight  in  pounds  of  a  cubic  foot  of 

pure  water  at  62°  F.) =  i  cubic  foot,  in  Ibs. 

1683.60 =i  cubic  yard,  in  Ibs. 

15.0      =i  „  incwts. 

•  75    =i          »          in  tons. 


WEIGHT   AND   SPECIFIC  GRAVITY.  199 

Note. — As  one  cubic  foot  of  water  at  62°  F.  weighs  about  1000  ounces 
(exactly  997.68  ounces),  the  weight  in  ounces  of  a  cubic  foot  of  any  other 
substance  will  represent,  approximately,  its  specific  gravity,  supposing 
water  =  1000. 

If  the  last  three  places  of  figures  be  pointed  off  as  decimals,  the  result 
will  be  the  specific  gravity  approximately,. water  being  =  i. 

In  France,  the  standard  temperature  for  comparison  of  the  density  of 
bodies,  and  the  determination  of  their  specific  gravities,  is  that  of  the 
maximum  density  of  water, — about  4°  C.,  or  39°.  i  F.,  for  solid  bodies;  and 
32°  F.,  or  o°  C.,  for  gases  and  vapours,  under  one  atmosphere  or  .76  centi- 
metres of  mercury.  In  practice,  it  is  usual  to  adopt  the  cubic  decimetre  or 
litre  as  the  unit  of  volume,  since  the  cubic  decimetre  of  distilled  water,  at 
4°  C.  weighs,  by  the  definition,  i  kilogramme.  Consequently  the  specific 
gravity  of  a  body  is  expressed  by  the  weight  in  kilogrammes  of  a  cubic 
decimetre  of  that  body. 

The  densities  of  the  metals  vary  greatly.  Potassium  and  one  or  two 
others  are  lighter  than  water.  Platinum  is  more  than  twenty  times  as 
heavy.  Lead  is  over  eleven  times  as  heavy;  and  the  majority  of  the  useful 
metals  are  from  seven  to  eight  times  as  heavy  as  water. 

Stones  for  building  or  other  purposes  vary  in  weight  within  much 
narrower  limits  than  metals.  With  one  exception,  they  vary  from  basalt  and 
granite,  which  are  three  times  the  weight  of  water,  to  volcanic  scoriae  which 
are  lighter  than  water.  The  exception  referred  to  is  barytes,  which  is  con- 
spicuously the  heaviest  stone,  being  4^  times  as  heavy  as  water.  The 
sulphate  of  baryta  is  known  as  heavy  spar. 

Amongst  other  solids,  flint-glass  has  three  times  the  weight  of  water;  clay 
and  sand,  twice  as  much;  coal  averages  one  and  a  half  times  the  weight  of 
water;  and  coke  from  one  to  one  and  a  half  times.  Camphor  has  about 
the  same  weight  as  water. 

Of  the  precious  stones,  zircon  is  the  heaviest,  having  four  and  a  half 
times  the  weight  of  water;  garnet  is  four  times  as  heavy,  diamond  three 
and  a  half  times  as  heavy,  and  opal,  the  lightest  of  all,  has  just  twice  the 
weight  of  water. 

Peat  varies  in  weight  from  one-fifth  to  a  little  more  than  the  weight  of 
water. 

The  heaviest  wood  is  that  of  the  pomegranate,  which  has  one  and  a  third 
times  the  weight  of  water.  English  oak  is  nearly  as  heavy  as  water,  and 
heart  of  oak  is  heavier;  the  densest  teak  has  about  the  same  weight  as 
water;  mahogany  averages  about  three-fourths,  elm  over  a  half,  pine  from 
a  half  to  three-fourths,  and  cork  one-fourth  of  the  weight  of  water.  Of  the 
colonial  woods,  the  average  of  22  woods  of  British  Guiana  weighs  74  per 
cent,  of  the  weight  of  water;  of  36  woods  of  Jamaica,  83  per  cent.;  and  of 
1 8  woods  of  New  South  Wales,  96  per  cent. 

Wood-charcoal  in  powder  averages  one  and  a  half  times  the  weight  of 
water;  in  pieces  heaped,  it  averages  only  two-fifths.  Gunpowder  has  about 
twice  the  weight  of  water. 

Of  animal  substances,  pearls  weigh  heaviest,  two  and  three-quarter  times 
the  weight  of  water;  ivory  and  bone  twice,  and  fat  over  nine-tenths  the 
weight  of  water. 

Of  vegetable  substances,  cotton  weighs  about  twice  as  much  as  water; 
gutta-percha  and  caoutchouc  nearly  the  same  weight  as  water. 


200  WEIGHT  AND   SPECIFIC   GRAVITY 

Mercury,  the  heaviest  liquid  at  ordinary  temperatures,  has  over  thirteen 
and  a  half  times  the  weight  of  water;  and  bromine  nearly  three  times  the 
weight.  The  water  of  the  Dead  Sea  is  a  fourth  heavier,  and  ordinary  sea- 
water  two  and  a  half  per  cent,  heavier  than  water;  whilst  olive-oil  is  about 
one-tenth  lighter,  and  pure  alcohol  and  wood-spirit  a  fifth  lighter  than 
water. 

Turning  to  gaseous  bodies,  water  at  62°  F.  has  772.4  times  the  weight  of 
air  at  32°  F.,  under  a  pressure  of  one  atmosphere;  and  the  specific  gravity 
of  air  at  32°  F.  is  .001293,  that  of  water  at  62°  F.  being  =  i.  Oxygen  gas 
weighs  a  tenth  more  than  air,  gaseous  steam  weighs  only  five-eighths  of  air, 
and  hydrogen,  the  most  perfect  type  of  gaseity,  has  only  seven  per  cent,  of 
the  weight  of  air.  Water  has  upwards  of  11,000  times  the  weight  of 
hydrogen. 

One  pound  of  air  at  62°  F.  has  the  same  volume  as  a  ton  of  quartz. 

The  following  Tables,  Nos.  65  to  69,  contain  the  weights  and  specific 
gravities  of  solids,  liquids,  and  gases  and  vapours.  The  specific  gravities 
have  been  derived  from  the  works  of  Rankine,  Ure,  Wilson,  Claudel,  and 
Peclet,  Delabeche  and  Playfair,  Fowke,  and  others  whose  names  are  men- 
tioned in  the  body  of  the  tables.  Columns  containing  the  bulks  of 
bodies  have  been  added  to  the  tables. 

The  specific  gravity  of  alloys  does  not  usually  follow  the  ratios  of  those 
of  their  constituents ;  it  is  sometimes  greater  and  sometimes  less  than  the 
mean  of  these.  Ure  gives  the  specific  gravities  of  some  alloys  of  copper,  tin, 
zinc,  and  lead,  examined  by  Crookewitt.  The  following  are  the  specific 
gravities  of  the  alloys,  as  ascertained  by  Crookewitt;  and,  for  the  purpose 
of  comparison,  they  are  preceded  by  the  specific  gravities  of  the  particular 
samples  of  the  elementary  metals  employed. 

SPECIFIC  GRAVITY. 

Copper 8.794 

Tin 7.305 

Zinc 6.860 

Lead TI-354 

Alloys: — Copper  2,  tin     5 7-652 

Copper  i,  tin    i 8.072 

Copper  2,  tin    i 8.512 

Copper  3,  zinc  5 7.939 

Copper  3,  zinc  2 8.224 

Copper  2,  zinc  i 8.392 

Copper  2,  lead  3 10.753 

Copper  i,  lead  i IO-375 

Tin         i,  zinc  2 7.096 

Tin        i,  zinc  i 7-IJ5 

Tin        3,  zinc  i 7.235 

Tin        i,  lead  2 9-965 

Tin         i,  lead  i 9-394 

Tin        2,  lead  i 9.025 

The  following  binary  alloys  have,  on  the  one  side,  a  density  greater  than 
the  mean  density  of  their  constituents;  and,  on  the  other  side,  a  density 
less  than  the  mean  density  of  the  constituents. 


OF   METALS  AND  ALLOYS. 


2O I 


Alloys  having  a  density  greater  than  the  mean. 

Gold  and  zinc. 
Gold  and  tin. 
Gold  and  bismuth. 
Gold  and  antimony. 
Gold  and  cobalt. 
Silver  and  zinc. 
Silver  and  lead. 
Silver  and  tin. 
Silver  and  bismuth. 
Silver  and  antimony. 
Copper  and  zinc. 
Copper  and  tin. 
Copper  and  palladium. 
Copper  and  bismuth. 
Lead  and  antimony. 
Platinum  and  molybdenum. 
Palladium  and  bismuth. 


Alloys  having  a  density  less  than  the  mean. 

Gold  and  silver. 
Gold  and  iron. 
Gold  and  lead. 
Gold  and  copper. 
Gold  and  iridium. 
Gold  and  nickel. 
Silver  and  copper. 
Iron  and  bismuth. 
Iron  and  antimony. 
Iron  and  lead. 
Tin  and  lead. 
Tin  and  palladium. 
Tin  and  antimony. 
Nickel  and  arsenic. 
Zinc  and  antimony. 


202 


VOLUME,   WEIGHT,  AND   SPECIFIC   GRAVITY 


TABLE   No.  65.— VOLUME,   WEIGHT,   AND    SPECIFIC    GRAVITY 
OF   SOLID    BODIES. 


T7ATV/TTT  TAT?      A/rP"PAT  Q 

Weight  of  one 
cubic  foot. 

Specific  Gravity. 

Platinum  

pounds. 
I  ^4.2 

Waters  i. 
21  522 

Gold                                         

I2OO 

TQ  2AC 

Mercury,  fluid  ...             

84.Q 

ly.z,^.^ 
I  3  ^0,6 

Lead  milled  sheet 

712 

I  I  4l8 

Do    wire                               

7O/L 

II  282 

Silver  

6cc 

IO  ^CX 

Bismuth  

/•" 

617 

Q  QO 

Copper  sheet                                    

C  AQ 

8  805 

Do      hammered  

i)4V 
cc6 

8  QI7 

Do.     wire  

CCA 

8880 

Bronze  :  —  84  copper,  16  tin,  gun  metal  

CT.A. 

8.1?6 

8^                   17    „ 

528 

846 

528 

846 

7Q               2  1         mill-bearinsrs  . 

CAA 

871 

-2C                6s    ,      small  bells  . 

;>44 
co? 

806 

21                     7Q    ,. 

4.61 

7  ^JQ 

15       „       85    „    speculum  metal... 

4.6  c 

7.4.C 

Nickel  hammered                                     

CA  T 

867 

Do     cast     

S4* 

c  16 

828 

Brass  :  —  cast  

CQC 

8.10 

75  copper,  25  zinc,  sheet  

C27 

8.4.5 

66                ^4.           vellow  .  .  . 

ci8 

8  ^o 

60        ,,       40     „      Muntz's  metal,  .  .  . 

«JH_- 

c  1  1 

8.20 

Brass  wire 

r  -30 

8  CA8 

Manganese                                         

JJJ 

4.QQ 

800 

Steel  •  —  Least  and  greatest  density  

4.-5C  to  4.0^ 

7  72Q  to  7  QO4. 

Homogeneous  metal  

7.QO4. 

Blistered  steel                                  

xRR 

7  82^ 

Crucible  steel    

488  to  490 

7.821?  to  7  8^0 

Do.        average  

4.80 

7.84.2 

Cast  steel                                            .  . 

4.8Q  tO  4.8Q  CJ 

7  8A4.  tO  7  8^1 

Do       average              ..        

480  ^ 

7  84.8 

Bessemer  steel  

.     4.8Q  tO  4.QO. 

7.8A4.  tO  7.8^7 

Do                average 

A  go  6 

7  8?2 

Mean  for  ordinary  calculations.  .     .  . 

4896 

7  852 

Iron,  wrought  :  —  Least  and  greatest  density... 
Common  bar  

466  to  487 
4.71 

7.47  to  7.808 

7.cc 

Puddled  slab    .     . 

460  Z  tO  A7d 

yy    ^ 

7  ?^  to  7  DO 

Various  —  Irons  tested  by  Mr.  Kirkaldy 
Do.                  average  

...468  to  486... 
4.77 

...7.5*07.8 

7.6i; 

Common  rails 

466  to  476 

7,4.7  to  7  64. 

Do.             average     .      ... 

4.70 

7.C4. 

Yorkshire  iron  bar  

4.84. 

7.758 

Lowmoor  plates,  i)4  to  3  ins.  thick  
Beale's  rolled  iron 

487 

4.76 

7.808 
7.6^2 

Pure  iron   (exceptional),  by  electro-  1 
deposit  (Dr.  Percy)  J 

508 

8.140 

Mean  for  ordinary  calculations 

4.80 

7.698 

OF   SOLID   BODIES. 


203 


FAMILIAR  META 

LS  (continued 
•reatest  densit 

)• 

Weight  of  one 
cubic  foot. 

Specific  Gravity. 

pounds. 
378.25^467.66 
468 

Water  =  i. 

6.900  to  7.500 

7.50 
7  2O 

White 



GrlTcLV 

Eglinton  hot-blast,  ist  melting... 
2d       do.    ... 
I4th    do.     ... 

*¥+  y 
435 

4.7C 

6.969 
6  Q7O 

470 

7-530 
6.977t07.II3 
7.094 
7  217 

Mallett 

442 

ACO 

Mean,  for  ordinary  calculations.. 
Tin                                      

462 

7.409 
7  2O 

Zinc,  sheet     

Do   cast 

428 

Al8 

.Z«J 

6.86 
671 

Antimony  

Aluminium  wrought 

167 

160 

2.67 
2  c6 

Do.         cast  

Magnesium 

108.5 
1  165  o 

1.74 
...    18.68 

18.40 

17  60 

OTHER    METALS. 

Indium 

Uranium  .            

1147.0 

IOQ7  O     . 

Tungsten  

Thallium 

742.6 
7^  8 

11.91 
II  80 

Palladium       .  . 

Rhodium  

660.9 
6236 

1  0.60 

JO  OO 

Osmium 

Cadmium     ...    . 

542.5 

c-27  c 

8.70 
862 

Molybdenum  

Ruthenium  

->J/O      ••• 

536.2 

C3O  O 

8.60 

8.50 
6.1  1 
.  .      6.00 

Cobalt 

Tellurium  

381.0 
774..  I        , 

Chromium  

Arsenic 

J'/T'1 
361.5 
•3-3O  ? 

5.80 

C    -2Q 

Titanium  

Strontium  

158.4 

131  o 

2.54 

2  IO 

Glucinum        

Calcium  

98.5 

Q4..8 

I.58 

I.C2 

Rubidium  

Sodium.        

^ 
60.5 

C-2  6 

0.97 
086 

Potassium  

Lithium  

37-0 
ES. 

id  Pure 

°-59 

Specific  Gravity. 

-3    C2 

Zircon 

PRECIOUS 

Specific  Gravity. 

A     CO 

>    STON 

Diamoi 
Boart 

Garnet  

3.60  to  4.20 

A  OI 

•     »•«       OO^ 

3-50 
1  C.O 

Malachite.  

Topaz 

Sapphire 

3.98 
•3  or 

Tournic 
Lapis  l'< 
Turquo 
Jasper, 
Beryl 

iline 

3.07 
206 

Emerald          

izuli 

Do.      Aqua  marine.. 
Amethyst 

j-yj 

2.73 

-J  Q2 

ise                

^.•yw 

2.84 

2.6  to  2.7 
2.68 

2.OQ 

Onyx,  Agate... 

Ruby.        .               .... 

yy* 

3-95 
3-50  to  3.53 

Diamond  

Opal... 

204 


VOLUME,   WEIGHT,   AND   SPECIFIC  GRAVITY 


Q'PO'M'R*^ 

Cubic  feet  to  one 
ton,  solid. 

Weight  of  one  cubic! 
foot,  solid. 

Specific  Gravity. 

Specular,  or  red  iron  ore  
Magnetic  iron  ore 

cubic  feet. 
...         6.84     ... 
7  CK 

pounds. 
...        327.4        ... 

•317  5 

Water  =  i. 
...       5.251 

r  OQA 

Brown  iron  ore  .           

/.u;> 

Q  ID 

2/Ld  6 

3«wyi' 

•3   Q22 

Spathic  iron  ore 

Q  38 

238  8 

0  820 

Clydesdale  iron  ores  

V-  J° 
11.76     . 

IQO.I; 

j.o^y 
•2  O^  tO  1.^80 

Barytes                                  

807 

277  £ 

A   AC 

Basalt     ..             

14  7  to  12.0 

152  8  to  187  i 

*y  A:  to  3.OO 

Mica  

14.0  to  12.3 

160.3  t°  182.7 

°.?7  to  2.Q3 

Limestone,  Magnesian  

12.6 

178.3 

...     2.86 

Do         Carboniferous 

T  -J     -J 

1  68  o 

2  60 

Marble  :  —  Paros  

1  >  J 
12  7 

177  I 

^.uy 
2  84. 

African          

12  8 

I7A  6 

2  8O 

Siberian  

13.2 

170.2 

2.73 

Pyrenean 

T  -3    2 

I7O  2 

2  73 

Carrara  

1  >•£ 

13.2 

.   .        169.6 

•*•/  J 

2.72 

Esrptian,  green... 

I  3   C 

1665 

267 

French  

136 

I6?  2 

26; 

Florentine,  Sienna  
Trap,  touchstone.  

14-3 

13.2 

I57.I 
I6Q.6       ... 

2.52 

2.72 

Granite   Sienite,  gneiss  

15  2  tO  12  I 

147  i  to  184  6 

2  36  tO  2.96 

Do       Gray 

12  8  tO  II  8 

1746  to  190  8 

2  80  tO  3  06 

Porphyry 

T  -2    C    tO    131 

1  66  5  to  171  5 

"*  67  to  2  75 

Alabaster,  Calcareous  

1JO   tw   1J'L 
I3.O 

172.1 

2.76 

Do.       Gypseous  

3*2 

i  c  6 

IAA  O 

2  31 

Chalk  Air-dried 

14  Q  tO  14  I 

i  ^o  to  1  1>9 

*»J* 

•7  46  to  2  55 

Slate  '                                

13  8  to  126 

162  i  to  177  7 

2  60  tO  2  85 

Serpentine  

.   .       12.8        ... 

I7C.2 

...      2.8l 

Potter's  Stone 

12  8 

I7A  6 

2  80 

Schist,  Slate  

128     ... 

1  7/1.6 

...       2.80 

Do.     Rough          

IQ  Q  tO  12  Q 

1  12  8  to  173  3 

i  81  to  2  78 

Lava,  Vesuvian  

->i  o  tO  128 

i  06  6  to  175  2 

171  to  2  8  1 

Talc  Steatite 

T  "2    -2 

1  68  4 

^  7O 

Rock  Crystal 

1  J'J 

136 

165  2 

26; 

Ouartz 

1  J.U 

1  3  8  to  133 

162  8  to  169  o 

^s.u^ 

2  6  1  to  271 

Do.    Crystalline  

136 

i6c  2 

2.6? 

Do.   for  paving  
Do    porous  for  millstones 

...     14.4     ... 
28  ; 

...      155-9     .- 

78  6 

...  2.50 

I  26 

Do.  flaky,  for           do  
Flint  

*"O 

...    14.1    ... 

137 

...      159.0     ... 
164.  o 

...  2.55 

2.63 

Felspar  

1  J./ 

i3.8 

162.1 

2.60 

Gypsum                         

*>" 

T  c  6 

143  A. 

2.30 

Lias         .  .         

l^.U 

ID  O  tO  147 

**Kr*r 

140  3  to  152  8 

2.21J  to  2.4.; 

Graphite  

16  * 

I  37.  2 

2.  2O 

Sandstone                         

17  3  tO  I  A.  1 

I2Q  7  to  I  ^7  I 

2  08  tO  2  52 

Tufa  volcanic             

29  7  to  26  i 

75.4  to  86  o 

1.  21  tO  1.38 

Scoria,     do      

4.-?  •? 

ci.7 

.83 

OF   SOLID   BODIES. 


205 


SUNDRY   MINERAL 

QTT'RQT'ATSJP'R'Q 

Cubic  feet  to  one 
ton,  solid. 

Weight  of  one  cubic 
foot,  solid. 

Specific  Gravity. 

Glass  :  —  Flint  ?  

cubic  feet. 

pounds. 
l87.0 

Water  =  i. 
1  OO 

Green  .                               

1684. 

27O 

Plate  

1684 

2  7O 

Crown                                    .... 

TCC  Q 

2  CO 

St   Gobain                      

^jj-y 

I  cc  •? 

•*•>*' 

2  AQ 

Common,  with  base  of  potash 
Fine,                do.            do. 
Common,  with  base  of  soda... 
Fine,                do.          do. 
Soluble                           

153-4 
...       152.8       ... 
152.8 
...       I52.I       ... 

77  Q 

2.46 

...     2.45 

2.45 
...    2.44 

Porcelain  •  —  China  

/  /'V 
14.8  A. 

2  -^8 

Sevres                   .... 

I  ^Q  7 

2  2A 

Portland  Cement          

28  7  to  23  8 

i^y-/ 

7o  tO  QA 

^..44 

i  25  to  i  51 

Concrete:  — 
P.  cement  i,  and  shingle  10 

...     16.1      ... 

1  39 

2.23 

P.  cement,  rubble,  and  sand 
P  cement  i,  and  sand  2  

1  6.6  to  1  6.0 
176 

135  to  140 

127 

2.17  to  2.25 

2  OA. 

Roman  cement  i,and  sand  2 
M  ortar  

18.7 
206 

120 

IOQ 

1.92 
I  71; 

Brick  

18  i  to  16  o 

12  A.  7  tO  I  ^C  3 

2  OO  to  217 

Brickwork  

20  4  to  195 

1  10  to  115 

i  76  to  i  84 

Masonry,  Rubble  

IQ.A   tO   I  C.6 

1  1  C.3  to  14.^  4. 

I  85  tO  2  30 

Marl 

22  4  tO  1  8  Q 

99  8  to  1185 

I  60  tO  I  QO 

Do.  very  tough                 

T  C   1 

14.6 

2  "34. 

Potash  

*j«O 

17  I 

1^1 

2  IO 

Sulphur 

A/.i 

18  o 

T2/1  7 

2  OO 

Tiles  

...     18.0      .  . 

I2A.7 

2.OO 

Rock  Salt  

171  tO  ICQ 

131  to  1  40  7 

2  IOOtO2  257 

Common  Salt,  as  a  solid  

l87 

I  IQ  7 

Q2 

Clay.... 

l87 

I  IQ  7 

Q2 

Sand,  pure  

18  Q 

118  c 

•:?•*• 
QO 

earthy 

±«j.y 

21  I 

106  o 

7O 

Earth  :—  Potter's  
Argillaceous  

...    18.9    ... 

22  A. 

...      118.5      ... 

QQ  8 

...         .90 
60 

Light  vegetable  
Mud  

...      25.7        ... 
22 

...     87.3    ... 

101  6 

...       1  .40 
I  6^ 

Materials  inthebed  of  the  Clyde  :—- 
Fine  sand  and  a  few  pebbles,  } 
laid    in  a    box,  loose,  not  > 

26           ^ 

87 

I.3Q 

pressed,  nearly  dry  ) 

Pressed 

2A 

Q2 

I  4.8 

Mud  at  Whiteinch,  dry,  and  ) 
firmly    packed,    containing  > 

2* 

y-4 
Q7 

1.56 

very  fine  sand  and  mica  j 
Wet  mud,  rather  compact  and 

IQ 

III 

I.QC 

firm,  well  pressed  into  the  box 
Wet,  fine,  sharp   gravel,  well 

18 

12A. 

I.QQ 

pressed  

Wet,  running  mud  

18  i 

\22l4 

I.Q7 

Sharp    dry   sand    deposit,   in 
harbour     

...    24.3     ... 

...          92 

...      1.48 

Port-Glasgow  bank  (sand),  wet, 

186 

1  2O  C 

I  Q3 

pressed  into  a  box  .  . 

206 


VOLUME,   WEIGHT,   AND   SPECIFIC  GRAVITY 


MINERAL  SUBSTANCES  (continued}. 

Materials  in  the  bed  of  the  Clyde: — 
Sand  opposite  Erskine  House,  ) 

wet,  pressed ) 

Alluvial  earth,  pressed 

Do.        do.    loose 

Plaster: — 24  hours  after  using 

2  months  after  using  . . . 
Coal,  Anthracite  (see  Sect.  COAL) 
Bituminous      do.          do. 
Boghead  (cannel)  do.     do. 

Coke 

Phosphorus 

Alum 

Camphor 

Melting  Ice 


Cubic  feet  to  one 
ton,  solid. 


cubic  feet. 


.-       19-3        ... 
24 

-     33 

22.6 

..       25.7        .. 
26.2  tO  22.6 

30  to  28.1 

30 

39  to  21.6 

20.3 
..     20.9 


36.3 
39 


(Veight  of  one  cubic 
foot,  solid. 


pounds. 


.      116 

93 
..       67 

99.2 

-       87.3     .. 
85.410  99.1 
74.8  to  81.7 

74-8 

57.4  to  103.5 

110.4 

107.2 

61.7 

57.5      .. 


Specific  Gravity. 


Water  =  i. 


..       1.86 

1.49 
..  1 .08 

1.59 
..  1.40 

1.37  to  1.59 

1.20  tO  1.31 
1.20 

.92  to 

1.77 
..  1.72 

•99 
..       .922 


.66 


COALS. 

(Delabeche  and  Playfair.} 

Welsh :— Anthracite 

Porth  Mawr  (highest) 

Llynvi  (one  of  the  lowest) 

Average  of  37  samples 

Newcastle:— Hedley's  Hartley  (highest)  ... 
Original  Hartley  (one  of  the  lowest) 

Average  of  18  samples 

Derbyshire  and  Yorkshire: — Elsecar 

Butterley 

Stavely 

Loscoe,  soft 

Average  of  7  samples 
Lancashire: — Laffack  Bushy  Park  (highest) 

Cannel,  Wigan  (lowest) 

Average  of  28  samples 

Scotch : — Grangemouth  (highest) 

Wallsend  Elgin 

Average  of  8  samples 

Irish : — Slievardagh  Anthracite 

Warlich's  artificial  fuel 

(Nicoll  and  Lynn.} 

South    Lancashire    and    Cheshire    Coals, 
average  of  14  samples 


Cubic  feet 
in  a  ton. 


Heaped. 


cubic  feet. 

38.4 
42.0 
42.0 
42.7 

45.6 

45-3 

474 
47-3 
44.9 
48.8 

47-4 
42.6 
46.4 
45.2 
40.1 
41.0 
42.0 
35-7 
324 


42 


Weight  of  one 
cubic  foot. 


Solid.      Heaped 


pounds. 
85.4 
86.7 
80.3 
82.3 
81.8 
78.0 

78.3 
80.8 
79-8 
79-8 
79.6 
79.6 
84.1 
76.8 

79-4 
80.5 
74-8 
78.6 
99.6 
72.2 


pounds, 
58.3 

53-3 
53-3 
53-i 
52.0 
49.1 
49-8 
47-2 
474 
49-9 
45-9 
45-9 
52.6 

48.3 
49-7 
54-3 
54.6 
50.0 
62.8 
69.6 


Specific 
Gravity. 


Water  =  i 
1-37 


.28 

•315 

•31 


.28 

.27 

.285 

.292 

•35 

•23 

•273 

.29 

.20 
.259 

•59 


OF   SOLID   BODIES. 


207 


PEAT. 

(Dr.  Sullivan?) 
Irish   peat    (comprising    an 
average  amount  of  water 
'from  20  to  25  per  cent)  :  — 
Lightest  upper  moss  peat  .  .  . 
Average  light  moss  peat  
Average  brown  peat  
Compact  black  peat 

Cubic  feet  per 
ton,  stalked. 

Weight  of  one 
cubic  foot, 
stalked. 

Weight  of  one 
cubic  foot, 
solid. 

Specific 
Gravity. 

cubic  feet. 
.  360.60..  . 

pounds. 
...      6.06 

pounds. 

62.5to8l.I 

I3.7t02I.O 

20.91025.3 
29.71041.7 
40.51044.5 

45.lto6l.3 
53.2to6l.8 

...    66.0    ... 

6.9  to  16.2 
15.01041.8 
25.61056.1 
38.7  to  64.2 

Water  =  i. 

i.o  to  1.3 

.2I9t0.337 

.335  to  .405 
.476  to  .669 

.650t0.7I3 
.72410.983 

.72510.991 
...    1.058 

.11  to    .26 
.24  to    .67 
.41  to    .90 
.62  to  1.03 

254.20 
...147.00... 
131.28 
...   99.36... 
2OO.29 

..  .188.0   ... 

155.5 

...141.75... 
5l.2t040.O 

8.81 

...  15-13 
17.06 
...  22.54 
ll.iS 

...   11.92 
14.40 

...  15.80 
43.75  to  56.8 

Densest  peat  

Mean  of  five  samples  

(Another  observation?) 
Average  upper  brown  peat  .. 
Moderately  compact  lower  ) 
brown  turf.             .    .    .     I 

Mean  of  two  classes  
Condensed  peat  

(Kane  and  Sullivan?) 
Excessively    light,    spongy  ) 
surface  peat  } 

Light  surface  peat 

Rather  dense  peat.. 

Very  dense  dark  brown  peat 
Very  dense  blackish  brown  ) 
compact  peat  ...              \ 

Exceedingly  dense  jet  black  > 
neat                                        ( 

Exceedingly    dense,    dark,  ) 
blackish  brown  peat...    .  ( 

(Karmarsch?) 
Turfy  peat,  Hanover  

Fibrous  peat,     do. 

Earthy  peat,      do. 

Pitchy  peat,       do  

FUEL    IN    FRANCE. 

(Claudel.) 
Pure  Graphite  

onSkfoot      Specific  Gravity. 

pounds.                 Water  =  i. 

T/l  C    "2                               -7    2-2 

Anthracite  8' 

—r_rj                        —  «jj 

,.5  to  91.0     1.34  to  1.46 
>.8  to  84.8      1.28  to  1.36 

84.8             1.36 

82  3     ....      1.32 

Rich  coal,  with  a  long  flame  7c 

Dry  coal,  with  a  long  flame...    . 

Rich  and  hard  coal  

Smithy  coal      ....                                                 7C 

).8  to  81.1      1.28  to  1.30 
.9  to  84.2      1.25  to  1.35 
,.3  to  74.8      1.16  to  1.20 
1.6  to  74.2      1.  10  to  1.19 
81.7                 1.31 

72  1                      I  l6 

Lignite  .  .        75 

Do     bituminous                                                         72 

Do      imperfect  6£ 

"Tayet"... 

Bitumen,  red           

Do.      black  

66.7                       1.07 
cj  7                      o  8^ 

Do      brown 

Asphalte  .    . 

66.1                i.  06 

208 


VOLUME,   WEIGHT,   AND   SPECIFIC   GRAVITY 


\A7f^OriQ 

Weight  of  one 
cubic  foot. 

Specific  Gravity. 

pounds. 
842 

Boxwood  

648 

I  O4 

Do        ofHolland              

82  1 

I   32 

C.67 

O  QI 

Lignum  vitcE                             .... 

4O  C  to  82  Q 

***y* 

6l  tO   I   33 

Ebony                        

7O  C 

>V-O  lw  *'JJ 
113 

/'-'O 
7;  c 

Do.      Black  

74.2 

I.IQ 

Oak   Heart  of                   

7-3  o 

I  17 

Do.   English    

c8o 

O  Q3 

Do    European 

A?  o  to  61  7 

^'VJ 

OQ  tO       QQ 

Do    American   Red       

CA  2 

.uy  LU      .yy 
87 

41  8  to  63  o 

67  to  i  01 

Rosewood 

64  2 

I  O3 

Satin-wood  

CQ.Q 

l.Uj 

O.q6 

Walnut,  Green  

C7  A 

O  Q2 

Do.      Brown  

42  4 

.  .      0.68 

Laburnum 

C7  A 

O  Q2 

Hawthorn  .                                    .        

V-4 
C.6  7 

O  QI 

Mulberry  

cc  c 

w.yi 

0.80 

Plum-tree 

CA    2 

*?y 

O  87 

Teak,  African                                                   

08 

Mahogany,  Spanish  

C3  O 

0.8; 

Do          St   Domingo 

46  8 

O  7C 

Do          Cuba                       

34  Q 

"•/i 

o  56 

Do          Honduras     

OH-V 

34  Q 

o.  c6 

Beech 

46  8  to  ^  30 

v.3w 
075  to  o  85 

Do      with  20  per  cent  moisture                   .    ... 

CT    I 

o  82 

Do      cut  one  year                           

31.1 

41  2 

066 

Ash 

C2  A 

o  84 

Do.  with  20  per  cent,  moisture  

$4.q. 

43.7 

0.70 

Acacia               .                           

CT    I 

082 

Do.    with  20  per  cent,  moisture  

J1'* 

44  Q 

0.72 

Holly                                                               

47  C 

o  76 

Hornbeam         .  .     .    .                       

^/O 

47  C. 

o  76 

Yew  

...        4/.} 
4O  I  tO  ?O  fj 

0.74  to  0.81 

Birch 

44  Q  to  46  I 

o  72  to  o  74 

Elm      ..  .                                              

34  3 

o  cc 

Do.  Green  

O^*"J 

47  ^ 

0.76 

Do  with  ''o  per  cent  moisture                        . 

yl/1    Q 

O  72 

Yoke-Elm      do.                do  

47.  c 

O.76 

Rock-Elm  

CQ  O 

0.80 

O.74 

Do  Red  pine                                              .... 

2Q  Q  to  43  7 

o  48  to  o  70 

Do.  Spruce  

2Q  Q  tO  43  7 

o  48  to  0.70 

31  1  8  tO  3Q.Q 

0.50  to  0.64 

Do  W^hite  pine  English 

34  3 

o  cc 

Do.          do           Scotch  

J^fO 
34  3 

O.C3 

Do.          do.             do.     20  per  cent,  moisture  

30  6 

O.4Q 

Do  Yellow  pine                              .... 

41  2 

066 

Do.          do         American  

287 

O.46 

American  Pine-wood,  in  cord  (heaped)  

21 

O.34 

Apple-tree 

AC   C 

O.73 

OF   SOLID   BODIES. 


209 


Weight  of  one 
cubic  foot. 

Specific  Gravity. 

Pear-tree  

pounds. 

AC   C 

O  7  "? 

Orange-tree  

TJO 

AA    0 

^•/  j 
o  71 

Olive-tree  

A2  A. 

068 

Maple  

o  6? 

Do.     20  per  cent  moisture      .    . 

AT  8 

o  67 

Service-tree  

4.1  8 

o  67 

Cypress,  cut  one  year  

A\  2 

066 

Plane-tree  

Ao  c 

o  6? 

Vine-tree  

07  A 

o  60 

Aspen-tree  

J/'*t 
•3,7  A 

o  60 

Alder-tree.     .        .              ... 

•3  A.  Q 

o  i>6 

Do.         20  per  cent,  moisture  

J4-V 
•37  A 

o  60 

Sycamore  

l68 

o  so 

Cedar  of  Lebanon    .       .    . 

"3.O  6  tO    "3.C    C 

^/.jy 

o  4.0  to  o  ^7 

Bamboo  

JU.U    IU    j}.} 

TQ  C  tO  2  A  Q 

o  31  to  o  4.0 

Poplar  

2A  -2 

O  3Q 

Do.     White  

20  o  to  "?i  8 

o  32  to  o  ^  i 

Do.     20  per  cent,  moisture  

2Q  O 

O.A8 

Willow  

"•y     •• 

•so  6 

O  AQ 

Cork  

ICQ 

O  2A 

Elder  pith  

A  7/1 

o  076 

INDIAN   WOODS. 

(Berkley.} 
Northern  Teak  

^tf/.t 

cc 

0882 

Southern  Teak 

A8 

O  77O 

Jungle  Teak  

AI 

o  658 

Blackwood  

c6 

0898 

Khair  

7-5 

I  171 

Erroul  

63 

I  014 

Red  Eyne  .... 

68 

I  OQI 

Bibla  

c6 

0.8Q8 

Poon..                   .  .. 

•3Q 

"•7 

o  625 

Kullum  .    .  . 

jy 
AI 

o6;8 

Hedoo  

•2Q 

o  62^ 

COLONIAL   WOODS. 

(Fowke.} 
JAMAICA:— 
Black  heart  ebony  

74-2 

I  IQ 

Lignum  vitas  

4.0.  c;  to  7  ^  o 

ny 

o  65  to  i  17 

Small  leaf.  

73.O 

I  17 

Neesberry  bullet-tree  .. 

/J.w 

6c  c 

I  CK 

Red  bully-tree  

62.36 

*-«**3 

I  OO 

Iron  wood.. 

;;    J     ' 

6l  7 

O  QQ 

Sweet  wood  

60  c 

^•vy 

O  Q7 

Fustic  

60  c 

O  Q7 

Satin  candlewood  

CQ  Q 

"•W 

o  96 

Bastard  cabbage  bark  

586 

O  QA 

White  dogwood  

?8.6 

O.QA 

Black        do  

j">vy 
c8  o 

O  Q3 

Gynip....         

«;8o 

O  Q3 

14 

210 


VOLUME,  WEIGHT,   AND   SPECIFIC   GRAVITY 


Weight  of  one 
cubic  foot. 

Specific  Gravity. 

COLONIAL  WOODS  (continued}. 

JAMAICA  (continued'];  — 
Wild  mahogany 

pounds. 
cy.4 

O.Q2 

Cashaw.... 

C7.A 

O.Q2 

W^ld  orange  

53.0  to  56.7 

0.85  to  0.91 

Sweet    do  

49-3 

0.70 

Bullet-tree  (bastard) 

...    56.1 

O.QO 

Tamarind  

54.2 

0.87 

Do       wild 

46.8 

O.7; 

Prune. 

53.6 

0.86 

Yellow  Sanders  

"...     53.6    ... 

...     0.86 

Beech  

52.4 

0.84 

French  Oak 

...     48.0    ... 

0.77 

Broad  Leaf  .        .    . 

48.0 

0.77 

Fiddle  Wood 

44.  7 

O.7I 

Prickle  Yellow 

4.7.O 

O.6o 
w.w 

Boxwood 

AT.  O 

O  OQ 

Locust-tree  

42.4 

0.68 

Lancewood 

4.2.4. 

.  .     0.68 

Green  Mahogany 

4.1.2 

0.66 

Yacca 

•3Q   ? 

o  63. 

Cedar  

36.2 

0.58 

Calabash 

•34.  q 

^ 
0.^6 

Bitter  Wood   ...                 .           .... 

•24.? 

O.CC 

Blue  Mahoe 

•3-1  7 

O  ?4. 

Average  of  36  woods  of  Jamaica  

C2.I 

0.831; 

NEW  SOUTH  WALES:— 
Box  of  Ilwarra  

7-2.0 

.17 

Do.  Bastard  

6Q.8 

1  2 

Do.  True  of  Camden 

^ 

60  5 

O.Q7 

Mountain  Ash 

.11 

Kakaralli  

68  6 

.IO 

Iron  Bark  

64.2 

.03 

Do.       broad-leaved 

63.6 

.   .          .02 

Woolly  Butt  

^j       ... 
63.0 

.OI 

Black      Do  

cc.c 

0.8o 

Water  Gum. 

636 

I.OO 

Blue       Do  

CJ2.4. 

0.84 

Cog  Wood  

CQ.Q 

0.0,6 

Mahogany.. 

CQ  2 

O  Q$ 

Do.         swamp  

536 

j& 

0.86 

Grav  Gum  

;8.o 

O.Q1 

Stringy  Bark  

536 

0.86 

Hickory  

468 

o  71; 

Forest  Swamp  Oak  

4.1  2 

0.66 

Mean  of  18  woods  of  New  South  Wales.. 
BRITISH  GUIANA:  — 
Sipiri,  or  Greenheart  

...        59.9      ... 

65  5  to  68  o 

0.96 
1.05  to  1.09 

Wallaba  

64.8 

1.04. 

Brown  Ebony  

64.2 

1.03 

Letter  Wood  

62  36 

I.OO 

Cuamara  or  Tonka  

;:  'Ju 
6i.7 

O.QQ 

Monkey  Pot  

58.6 

0.04 

Mora  

C7  4 

O.Q2 

OF   SOLID   BODIES. 


211 


COLONIAL  WOODS  (continued}. 
BRITISH  GUIANA  (continued)  :— 

Weight  of  one 
cubic  foot. 

Specific  Gravity. 

pounds. 
56.7 

Cabacalli  

55-5 

54.2 

0.89 
0.87 

Kaieeri-balli  

Sirabuliballi                            .  .. 

52.4 
5O.  C. 

0.84 

0.81 

Buhuradda  

Buckati 

50.5 

c.o.  i; 

0.8  1 
o  81 

Houbaballi  . 

Baracara  

50.5 
480 

0.8  1 

O  77 

White  Cedar 

Locust-tree  . 

44-3 
47.7 

0.71 
O.7O 

Cartan  

Purple  Heart  

42.4 
3Q.Q 

0.68 
o  64 

Bartaballi 

Crabwood  

374 

34.3 

•;:4 
0.60 

O  C£ 

Silverballi 

Mean  of  22  woods  of  British  Guiana. 

JT-J    '•' 

46.1 

96.7 

0.74 
I.  re 

WOOD-CHARCOAL  (as  powder). 

(Claudel.) 
Willow  

Oak  

954 

Q2.Q 

1-53 

I  4Q 

Alder  

Lime-tree  

91.0 

QO  4 

.^y 
1.46 

I  AC, 

Poplar  

Average  of  5  charcoals   

93-5 
IQ.'Z 

±.^.5 
1.50 

O.63 

WOOD-CHARCOAL  (in  small  pieces,  heaped). 

(Claudel.) 
Walnut  

Ash  

34-3 
32.t; 

0.55 
O  C2 

Beech  

Yoke-elm  

O-4-O 

28.7 

287 

**y> 

0.46 

o  46 

Apple-tree  

White  Oak  .... 

26.2 
215.6 

0.42 

O.4I 

Cherry  -tree  

Birch  

22.5 

.  .       22.5 

0.36 

o  36 

Elm  

Yellow  Pine  

20.6 

17  S 

o-33 
o  28 

Chestnut-tree  

Poplar  

I5.6 
IC..O      , 

0.25 

O.24 

Cedar  

Average  of  1  3  charcoals  

25.3 

109.110114.7 

15  to  15.6 
13.7  to  14.3 
12.5  to  13.1 

14. 

0.405 

1.75  to  1.84 
0.24  to  0.25 

O.22  tO  0.23 
0.20  tO  0.21 
O.225 

Gunpowder  .    . 

WOOD-CHARCOAL  (as  made,  heaped). 
Oak  and  Beech  

Birch  

Pine  

.  L 

_  

212 


WEIGHT  AND  VOLUME  OF 


A  WTTVT  A  T        QlfRCiT  A  IMPI?  Q 

Weight  of  one 
cubic  foot. 

Specific  Gravity. 

(Claudel.) 
Pearls  

pounds. 
...     169.6 

2  72 

Coral                                     

l67  7 

2  60 

Ivory  .  ..        

IIQ  7 

**fy 

I  Q2 

Bone                                         .               .  .             .  . 

1  1  2  2  to  124  7 

I  80  tO  2  OO 

Wool                                             

IOO  A. 

61 

Tendon  .                           

698 

12 

Cartilage  

..       680      ... 

OQ 

Crystalline  humour 

67  3 

08 

Human  body                          .     .        

667 

O7 

Nerve  

6d  Q 

OA 

Wax  

CQ.Q 

O  Q6 

White  of  whalebone                           

S87 

O  QA 

Butter  

587 

O  QA 

Pork  fat 

C8  7 

O  QA 

Mutton  fat  

5°'/ 

c.7.4. 

O.Q2 

Animal  charcoal  in  heaps 

CO  to  C2 

o  80  to  081 

VEGETABLE    SUBSTANCES. 
Cotton  

D^  lu  D^ 
.   .      1  2  1.  6 

QC 

Flax 

1  1  1  6 

7Q 

Starch 

•/V 

C-3 

Fecula  .        

Q-J    C 

Gum  —  Myrrh  

848 

36 

Do      Dragon 

82    1 

»2 

Do      Dragon's  blood                           

74.8 

2O 

Do.     Sandarac                    

680 

OQ 

Do.     Mastic  

66.7 

.07 

Resin  —  Jalap                                           .    .        ... 

76  i 

22 

Do.      Guayacum                 ..        .        

748 

2O 

Do.      Benzoin  

68.0 

.OQ 

Do       Colophany                                               .    .  . 

667 

O7 

Amber,  Opaque                       .               

680 

OQ 

Do.       Transparent  .    .    .        

67  l 

...        I  08 

Gutta-percha 

60  c 

O  Q7 

Caoutchouc                                                

v^.-j 
58  o 

^.y/ 
O  Q3 

Grain  WTieat,  heaped        

4.6  7 

O  7C, 

Do      Barley,       do  

36.6 

o.co 

Do      Oats           do 

31  2 

O  C.O 

j1-^ 

VARIOUS   SUBSTANCES. 


213 


TABLE   No.  66.— WEIGHT  AND   VOLUME   OF   VARIOUS 
SUBSTANCES.     (Tredgold.} 


SUBSTANCE. 

Cubic  feet  per 
ton,  in  bulk. 

Weight  of  one 
cubic  foot,  in  bulk. 

Lead  (cast  in  pigs)              

cubic  feet. 

4 

Ibs. 
^7 

Iron  (cast  in  pigs) 

62? 

3W 

360 

Limestone  or  marble  (in  blocks) 

13 

172 

Granite  (Aberdeen,  in  blocks)  

I7.C 

1  66 

Granite  (Cornish  in  blocks) 

164. 

Sandstone  (in  blocks)                 

16 

I4.I 

Portland  stone  (in  blocks)  

17 

132 

Potter's  clay 

17 

I  3O 

Loam  or  strong  soil 

18 

126 

Bath  stone  (in  blocks)  

18 

I23.tr 

Gravel  

21  V;» 

IOQ 

Sand  . 

23  5 

QC 

Bricks  (common  stocks  dry) 

24. 

Q3 

Culm  

36 

63 

\Vater  (river) 

^6 

62  c 

Splint  coal 

3Q.C 

****J 
57 

Oak  (seasoned)  

4? 

C2 

Coal  (Newcastle  caking)  :.. 

AC 

CQ 

Wheat  

4.7 

48 

Barley  ... 

en 

38 

Red  fir  

cq 

...  ;  38 

Hay  (compact,  old) 

280 

8 

TABLE   No.    67.— WEIGHT    AND   VOLUME    OF   GOODS    CARRIED    ON 
THE  BOMBAY,    BARODA,    AND   CENTRAL  INDIA   RAILWAY. 

By  Colonel  J.  P.  KENNEDY,  Consulting  Engineer  of  the  Railway. 


No.  of 
kind. 

CLASSIFICATION  OF  GOODS  CONVEYED. 

Cubic  feet 
per  ton. 

Weight  per 
cubic  foot. 

Cubic  feet 
per  ton, 
in  bulk 
(estimated). 

Unpressed  cotton 

cubic  feet. 
224. 

Ibs. 
..      IO    ... 

cubic  feet. 
...280 

2 

Furniture  

2OO 

II 

2?O 

3    ••• 

A 

Half-pressed  cotton  
Cotton  seeds  

...186... 
1  86 

...    12    ... 
12 

•••233 
233 

Wool  

I4.O  ... 

...  16  ... 

...  17; 

6 

Fruit  and  vegetables  

IOO 

22 

12$ 

7     . 

EP-O-S  ... 

QO  ... 

.    2C    . 

..113 

Class  i. 

Averages  

..  174.  .. 

.    13    .. 

...  2iy 

214  WEIGHT  AND  VOLUME  OF   GOODS. 

GOODS    CONVEYED    OVER   THE    INDIAN    RAILWAY    (continued}. 


No.  of 
kind. 

CLASSIFICATION  OF  GOODS  CONVEYED. 

Cubic  feet 
per  ton. 

Weight  per 
cubic  foot. 

Cubic  feet 
per  ton, 
in  bulk 
estimated). 

8    ... 

Grass  

cubic  feet. 
80   ... 

Ibs. 
...    28 

cubic  feet. 
IOO 

Q 

80 

28 

IOO 

IO    ... 

Bagging  .. 

7O 

12 

87 

I  I 

Commissariat  stores  

7O 

87 

12    ... 

Full-pressed  cotton  

..    7O    .. 

,.    12    .. 

°/ 

87 

Flax  and  hemp  . 

•32 

14   . 

Groceries  

60 

J^ 

.    17 

7C 

I  ^ 

Grains  and  seed  

60 

...    /  5 

ll... 

Twist  

60 

17 

17 

Sugar 

...    /  ^ 

18  ... 

Soap  

..  c6 

,.   AO   . 

7O 

Firewood 

50 

7O 

20  ... 

Salt  

5° 

AA   . 

/u 
64. 

21 

Lime    

* 

AA 

6A 

22    ... 

Dry  Fruits  

CO   . 

AC 

U4 

^ 

Class  2. 

Averages  

60 

17 

'  •* 

23  ... 

2A 

Jagree  (Molasses)  
Kupas  (Seed  cotton)        * 

...  45  ». 

AC 

...     50    ... 

...    56 

Mowra  (flowers  which  produce  spirit) 
Timber  

•  45 
...  45  ... 

...     50    ... 

...     56 

c6 

27    . 

Ghee  (clarified  butter)  

AO 

Lj 

>v 

CO 

28 

Oil 

c6 

...    5^ 

2Q    .. 

Piece  goods  

.   AO     . 

Lj 

CO 

Rape  

AO 

c6 

11 

Beer  and  Spirits  

36 

62 

AC 

12 

Coal 

28 

80 

11    . 

Paper  

...    28    .   . 

...  80  ... 

T.C 

Tobacco  

28 

80 

•3C 

•3C      . 

Opium  

26 

86 

36 

Machinery 

° 

•3  T 

JW 

.   •> 

JL 

Class  3 

Averages  

AI 

CA 

C  i 

37   .. 

Cutlery  

...    2O    ... 

...  112  ... 

.    2C 

Potash  

2O 

1  12 

2C 

•3.Q 

Sand  

2O 

.  112  ... 

2O 

AO 

Colour       ... 

18 

I2A 

22 

AI 

Bricks  

17 

1  12 

21 

A2 

Stone  

I  c 

IA8 

IQ 

Metal  

c    . 

..  AA.1  .  . 

6* 

Class  4 

Averages  

1  1 

2O1  . 

14 

Averages  of  all  classes  

...64.4... 

...35.4..- 

...3o 

Note. — The  last  column  has  been  added  by  the  author;  the  quantities  are  calculated  by 
adding  one-fourth  to  the  quantities  in  the  third  column,  to  give  approximate  estimate  of 
the  volume  occupied  in  waggons  by  the  goods,  or  the  space  required  to  load  a  ton  of  each 
kind.  Sand,  No.  39,  lies  solid  in  any  situation. 


WEIGHT  AND   SPECIFIC   GRAVITY  OF   LIQUIDS. 


215 


TABLE    No.   68.— WEIGHT  AND   SPECIFIC   GRAVITY   OF 

LIQUIDS. 


LIQUIDS  AT  32°  F. 

Weight  of  one 
cubic  foot. 

Weight  of 
one  gallon. 

Specific 
Gravity. 

Mercury  
Bromine  

pounds. 
..848.7    ... 

i8;.i 

pounds. 
..  136.0   ... 
2Q.7 

Water  =  i. 

..13-596 
2.966 

Sulphuric  acid,  maximum  concentration- 
Nitrous  acid                          

..  114.9  ••• 
06.8 

..     18.4   ... 
ICC 

..      .84 
.cc 

Chloroform  

.    QS.S    . 

,      1C.  7     ., 

,C7 

Water  of  the  Dead  Sea  

77.4 

12.4 

.24 

Nitric  acid,  of  commerce  
Acetic  acid,  maximum  concentration  

/  /  -t 
..    76.2   ... 
67.4. 

..     12.2    ... 

10.8 

..      .22 
1.  08 

Milk  

.    64.^  .. 

,    10.3  . 

.    I.O^ 

Sea  water  ordinary          ...    ...        

"•H3 

04.  o? 

IO.  ^ 

J  - 
I.O26 

Pure  water  (distilled)  at  39°.  i  F  

;rr    -> 
.    62.4.25 

..      IO.O    ... 

..    I.OOO 

VV^ine  of  Bordeaux  

62.1 

Q.Q 

O.QQ4 

Do       Burgundy               ..          

6l  Q 

Q  Q 

..    O.QQI 

Oil,  lintseed  

..          Vi.V/       ... 

eg  7 

Q.4. 

O.Q4 

Do    DODDV 

3^./ 
s8  i 

Q  ^ 

O  Q3 

-L-'U.    P^JJJJ^    

Do  rape-seed                              . 

..        JU.l       ... 

57  4. 

y-j   ••• 

Q  2 

O.Q2 

Do  whale    

C7  4. 

Q.2    ., 

..    O.Q2 

Do.  olive  

">7.I 

Q.IC 

O.QII; 

Do  turpentine 

CA  7 

87 

.    0.87 

Do  potato   

C  I  2 

8.2 

0.82 

Petroleum  

CA.Q  . 

..     8.8  ... 

..  0.88 

Naphtha       ... 

c  ?  i 

8  ; 

0.8; 

Ether  nitric  

6q  T- 

i  i.i  ... 

..  i.  ii 

Do      sulphurous  

•*      ~7'J    '•' 

67.4. 

10.8 

i.  08 

Do.     nitrous  
Do      acetic   

...    55.6  ... 
a  6 

...     8.9  ... 

8.Q 

...  0.89 
0.89 

Do     hydrochloric  

C4..-2    . 

87  ., 

...  0.87 

Do      sulphuric     . 

AA  Q 

7  2 

O.72 

Alcohol  proof  spirit  

*Ft-y 
1:7  A 

Q.2 

..    O.Q2 

Do      pure  

4.Q.-2 

7.Q 

0.70 

.       13.I      . 

8.5    ... 

...    0.85 

^^ood  spirit     

4.Q  Q 

80 

0.80 

H-y-y 

216 


WEIGHT,    ETC,    OF   GASES   AND   VAPOURS. 


TABLE    No.  69.— WEIGHT  AND   SPECIFIC   GRAVITY   OF 
GASES   AND   VAPOURS. 


GASES    AT   32°    F.    AND   UNDER   ONE 
ATMOSPHERE   OF    PRESSURE. 

Volume  of  one 
pound  weight. 

Weight  of  one  cubic  foot. 

Specific 
Gravity. 

Vapour  of  mercury  (ideal)  
Vapour  of  bromine 

cubic  feet. 
...    1.776... 
2.236 
.    2.3T7 

in  pounds. 
...0.563     ... 
0.447 
...0.428     ... 
0.378 
...0.245     ••• 
0.217 
...0.209     ... 
0.206 
...0.197     ... 
0.1814 
...0.1302  ... 
0.12344 
...0.12259 
0.089253 
...0.080728 
0.078596 
...0.0781  ... 
0.0795 

o  05022 

in  ounces. 
...9.008     .. 
7.156 
...6.846     .. 
6.042 
...3.927     .. 
3480 
..  1  34.O 

Air  =  i. 
...6.9740 
5.5400 
...  5.3000 
4-6978 
...  3.0400 
2.6943 
2  5860 

Chloroform  

Vapour  of  turpentine 

2.637 

A  O7  C 

Acetic  ether 

Vapour  of  benzine  

4.598 
...    4790... 

4-777 
...    5.077  ... 

5-5I3 
...    7.679  ••• 

8.101 
...  8.157  ... 
11.205 
...12.387... 

12.723 
...12.804... 
12.580 

...IQ  QI3 

Vapour  of  sulphuric  ether  
Vapour  of  ether  (?) 

3-302 
...3.IS2     . 

2.5563 

..     2.44.OO 

Chlorine  

Sulphurous  acid  

2.902 
...2.083     ••• 

i-975 
...1.961    ... 
1.428 
...1.29165 
1.258 
...1.250    ... 
1.272 
...0.8035  ••• 
0.7613 
...0.7139... 
0.5658 
...0.0895 

2.2470 
...  1.6130 
1.5290 
...  1.5186 
I.I056 
...  1.0000 

0.9736 
...0.9674 
0.9847 

...  0.6220 

0.5894 
...0.5527 
0.4381 

.  .  .  0.0692 

Alcohol  
Carbonic  acid  (actual).      ..  . 

Do.           (ideal)  

Oxygen 

Air  
Nitrogen  

Carbonic  oxide  
Olefiant  gas  

Gaseous  steam  

Ammoniacal  gas  

21.017 
...22.412  ... 

0.04758 
,..0.04462 

Light  carburetted  hydrogen  .. 

Coal-eras  (page  4?  8) 

28.279 
178.83    ... 

0.03536 
...0.005592 

Hvdrogen  

TABLES   OF   THE   WEIGHT    OF    IRON    AND 
OTHER    METALS. 


Wrought  Iron.  —  According  to  Table  No.  65  of  the  Weight  and  Specific 
Gravity  of  Solids,  the  weight  of  a  cubic  foot  of  wrought  iron  varies,  for 
various  qualities,  from  466  pounds  to  487  pounds  per  cubic  foot,  and  the 
average  weight,  taken  for  purposes  of  general  calculation,  is  480  pounds  per 
cubic  foot.  This  average  weight  is  equivalent  to  a  weight  of  40  pounds  per 
square  foot,  i  inch  in  thickness  —  a  convenient  unit,  which  is  usually 
employed  in  the  development  of  tables  of  weights  of  iron  for  engineering 
and  manufacturing  purposes.  The  extremes  of  variation  from  this  medium 
unit,  extend  from  7/%  pound  less,  to  about  $/$  pound  more  than  40  pounds 
per  square  foot,  or  from  2.2  to  1,5  per  cent,  either  way  —  a  deviation,  the 
extent  of  which  is  of  little  or  no  practical  consequence,  and  which,  at  all 
events,  is  comprehended  in  the  percentages  allowed  in  the  framing  of 
estimates. 

The  average  weight  of  a  cubic  inch  of  wrought  iron  is 

.*77  pound,     ••;-: 


or  one-tenth  more  than  a  quarter  of  a  pound.  For  a  round  number,  when 
cubic  inches  are  dealt  with,  it  may  be,  and  is  usually,  taken  as  .28'  pound, 
which  is  only  four-fifths  of  i  per  cent,  more  than  the  medium  weight,  and 
corresponds  to  a  weight  of  483.84  pounds  per  cubic  foot,  or  to  40.32 
pounds  per  square  foot,  i  inch  thick,  or  to  10  pounds  per  lineal  yard, 
i  inch  square. 

The  volume  of  i  pound  of  wrought  iron  is  3.6  cubic  inches. 

Steel.  —  The  weight  of  a  cubic  foot  of  steel  varies  from  435  pounds  to 
493  pounds  per  cubic  foot,  and  the  average  weight  is  about  490  pounds 
per  cubic  foot.  For  convenience  of  calculation,  the  average  weight  is  taken 
in  the  following  tables,  as  489.6  pounds  per  cubic  foot,  for  which  the 
specific  weight  is  1.02,  when  that  of  wrought  iron--  i.oo.  The  weight  of  a 
square  foot,  i  inch  thick,  is  40.8  pounds;  of  a  lineal  yard,  i  inch  square, 
10.2  pounds;  and  of  a  cubic  inch,  .283  pound. 

The  volume  of  i  pound  of  steel  is  3.53  cubic  inches. 

Cast  Iron.  —  The  weight  of  a  cubic  foot  of  cast  iron  varies  from  378^ 
pounds  to  467^5  pounds  per  cubic  foot,  and  the  average  weight  is  taken  as 
450  pounds.  The  weight  of  a  square  foot,  i  inch  thick  is,  therefore,  37.5 
pounds;  of  a  lineal  yard,  i  inch  square,  9.375  pounds;  and  a  cubic  inch, 
.26  pound.  The  specific  weight  is  .9375. 

The  volume  of  i  pound  of  cast  iron  is  3.84  cubic  inches. 

The  following  data,  for  the  weight  of  iron,  are  abstracted  for  readiness 
of  reference  :  — 


218 


WEIGHT   OF   METALS. 


WROUGHT  IRON,  ROLLED. 

cubic  foot, 480  pounds,  or  4.29  cvvts. 

square  foot,  i  inch  thick, 40  pounds. 

square  foot,  3  inches  thick, 120  pounds,  or  1.07  cwts. 

3  square  feet,  i  inch  thick, 120  pounds,  or  1.07  cwts. 

lineal  foot,  i  inch  square, 3^  pounds,  or  .03  cwt. 

cubic  inch,  say 0.28  pound. 

3.6  cubic  inch, i  pound. 

lineal  yard,  i  inch  square, 10  pounds. 

lineal  foot,  3  inches  square, 30  pounds. 

lineal  foot,  6  inches  square, 120  pounds,  or  1.07  cwts. 

lineal  foot,  3  inches  by  i  inch  thick,  10  pounds. 

lineal  foot,  ^  inch  in  diameter,. ...  2  pounds. 

lineal  foot,  2  inches  in  diameter,...  10.5  pounds. 

lineal  foot,  6^  in.  in  diameter,  about  i  cwt. 

CAST  IRON. 

i  cubic  foot, 450  pounds,  or  4  cwts. 

5  cubic  feet, i  ton. 

i  square  foot,  i  inch  thick, 37.5  pounds. 

i  square  foot,  3  inches  thick  (^  cub.  ft.),  112.5  pounds,  or  i  cwt. 

3  square  feet,  i  inch  thick, 112.5  pounds,  or  i  cwt. 

i  cubic  inch, 0.26  pound. 

3.84  cubic  inches,  i  pound. 

The  Table  No.  70  contains  the  weight  of  iron  and  other  metals  for  the 
following  volumes : — 

i  cubic  foot. 

i  square  foot,  i  inch  thick,  or  T/I2th  of  a  cubic  foot. 

i  lineal  foot,  i  inch  square,  or  T/12th  of  a  square  foot. 

i  cubic  inch,  or  x/I2th  of  a  lineal  foot. 

A  sphere,  i  foot  in  diameter. 

The  specific  gravity  due  to  the  respective  weights  per  cubic  foot  is  also 
given,  and  likewise  the  specific  weight  or  heaviness,  taking  the  weight  of 
wrought  iron  as  i,  or  unity. 

The  next  Table,  No.  71,  contains  the  volumes  of  iron  and  other  metals 
for  the  following  weights : — 

ton,  in  cubic  feet. 

cwt.,  in  square  feet,  i  inch  thick. 

cwt.,  in  lineal  feet,  i  inch  square. 

pound,  in  cubic  inches. 

ton,  as  a  sphere,  in  feet  of  diameter. 

ton,  as  a  cube,  in  feet  of  lineal  dimension. 

The  next  Table,  No.  72,  contains  the  weight  of  i  square  foot  of  metals  of 
various  thickness,  advancing  by  sixteenths  and  by  twentieths  of  an  inch,  up 
to  i  inch  in  thickness. 

The  fourth  Table,  No.  73,  contains  the  weight  of  prisms  or  bars  of  iron, 
and  other  metals,  or  metals  of  any  other  uniform  section,  for  given  sectional 
areas,  varying  from  .1  square  inch  to  10  square  inches  of  section,  advancing 
by  one-tenth  of  an  inch,  for  i  foot  and  i  yard  in  length. 


TABLES   OF   WEIGHT   AND   VOLUME   OF   METALS. 


219 


This  table  is  useful  in  calculations  of  the  weights  of  bars  of  every  form, 
rails,  joists,  beams,  girders,  tubes,  or  pipes,  &c.,  when  the  sectional  area 
is  given. 

The,  table  is  available  for  finding  the  weight  of  a  metal  for  any  sectional 
area  up  to  100  square  inches,  by  simply  advancing  the  decimal  points  one 
place  to  the  right;  or,  in  round  numbers,  up  to  1000  square  inches,  by 
advancing  the  decimal  points  two  places.  For  example,  to  find  the  weight 
of  wrought  iron  having  a  sectional  area  of  17  square  inches: — 

For  1.7  square  inches,  the  weight  per  foot  is  5.67  pounds. 
For  17  square  inches,  the  weight  per  foot  is  56.7  pounds. 
For  170  square  inches,  the  weight  per  foot  is  567  pounds. 

Table  No.  70. — WEIGHT  OF  METALS. 


METAL. 

Cubic  Foot. 

Square  Foot, 
i  Inch  Thick. 

Lineal 
Foot, 
i  Inch 
Square. 

Cubic 
Inch. 

Sphere, 
i  Foot 
Dia- 
meter. 

Specific 
Gravity. 

Specific 
Weight. 

Wrought  Iron  
Cast  Iron  

Ibs.    or    cwts. 

480  or    4.29 
450  or   4.02 

Ibs.     or     cwts. 
40     or    .357 

37.  5  or  .77C, 

Ibs. 

3-333 
3.121; 

Ib. 

.278 
.260 

Ibs. 

25I 
236 

Water 

7^698 
7.217 

Wro'ght 
Iron=i. 
1.  000 
.9771: 

Steel  

489.  6  or   4.37 

40.  8  or    .364 

3.400 

.283 

257 

7.852 

.O2O 

Copper,  Sheet  
Copper,  Hammered 
Tin 

549  or   4.90 
556  or   4.96 
462  or   4.13 

45.  8  or   .409 
46.3  or    .413 
38.  c,  or    .  344 

3-813 
3.861 
3.208 

.318 
.322 
.268 

287 
291 
242 

8.805 
8.917 
7.4O9 

.144 
.158 
.962 

Zinc  

437  or    3.90 

36.4  or    .325 

3.035 

.253 

229 

7.008 

.910 

Lead               .       .... 

712  or   6.36 

59.  3  or    .  53O 

4.Q44 

.412 

777 

II.4I8 

.48? 

Brass,  Cast  
Brass    \Vire. 

505  or   4.51 
Wl  or   4.76 

42.  i  or    .375 
44.  4  or   .396 

3-507 
^.701 

.292 
.308 

31  J 
264 
279 

8.099 
8.348 

.052 

.  no 

Gun  Metal  ... 
Silver  
Gold  

5  24  or   4.68 

655or    5-85 
1  200  or  10.72 

43.  7  or   .390 
54.  6  or   .488 
loo.oor   .893 

3.639 

4-549 
8.333 

•304 

•379 
.604 

274 

343 
628 

8.404 
10.505 
19.24^ 

.092 
.365 

2.500 

Platinum  

1342  or  12.  oo 

in.  Son.  ocx} 

9.320 

•777 

703 

21.522 

••J*~ 

2.796 

Table  No.  71. — VOLUME  OF  METALS  FOR  GIVEN  WEIGHTS. 


METAL. 

Cubic  Feet 
to  a  Ton. 

Square  Feet, 
i  Inch  Thick, 
,to  a  cwt. 

Lineal  Feet, 
i  In.  Square, 
to  a  cwt. 

Cubic  Inches 
to  a  Ib. 

Diameter 
of  a  Sphere 
of  i  Ton. 

Side  of  a 
Cube  of 
i  Ton. 

Wrought  Iron  
Cast  Iron  
Steel  
Copper,  Sheet  
Copper,  Hammered 
Tin 

cubic  feet. 
4.67 
4.98 
4.58 
4.08 
4-03 

4  86 

square  feet. 
2.80 
2.99 
2-75 

2-44 
2.42 

2  QI 

feet. 

33-6 
35-8 
32.9 
29.4 
29.0 

7.4  Q 

cubic  inches. 
3-60 
3-84 

3-53 
3-15 
3-H 

1  74 

feet. 
2.07 
2.12 
2.26 
I.98 
I.98 
2.  IO 

feet. 
-67 
•71 

.66 
.60 

•59 
.69 

Zinc  
Lead 

5-13 

•3    JC 

^..yi 

3.08 

i  So 

36.8 

22  7 

3-95 

2  47 

2.14 

1.81 

•73 

•47 

Brass  Cast  

4..4J 

2  67 

V.Q 

7.42 

2.04 

.64 

Brass  ^Vire 

4  2O 

2  7.O 

TO  7 

^.24 

2.OO 

.61 

Gun  Metal  
Silver  

4.28 
3.42 

*-J" 

2.56 
2.O=; 

30.8 
24.6 

3-30 
2.64 

2.02 
1.87 

.62 
•51 

Gold  
Platinum 

1.87 
1.67 

1.  12 
I  OO 

13-4 
12.  0 

1.44 
1.29 

r-59 

1.47 

.28 
.19 

22O 


WEIGHT   OF   METALS. 


Table  No.  72. — WEIGHT  OF  i  SQUARE  FOOT  OF  METALS. 

Thickness  advancing  by  Sixteenths  of  an  Inch. 


THICK- 
NESS. 

WRO'T 
IRON. 

Specific 

Wt.=I. 

CAST 
IRON. 

Specific 

STEEL. 
Specific 

Wt.=  1.02. 

COPPER. 

Specific 
wt.=i.i6. 

TIN. 

Specific 
wt.=-962. 

ZINC. 
Specific 

BRASS. 

Specific 
wt.  =  1.052. 

GUN 
METAL. 

Specific 
wt.  =  1.092. 

LEAD. 

Specific 

inch. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

i/l6 

2.50 

2-34 

2-55 

2.89 

2.41 

2.28 

2.63 

2-73 

3-71 

j£ 

5.00 

4.69 

5.10 

5-79 

4.81 

4-55 

5.26 

5.46 

7.41 

3/i6 

7.50 

IO.O 

7-03 
9.38 

7.65 
10.2 

8.68 
ii.  6 

7.22 
9-63 

6.83 
9.10 

7.89 
10.5 

8.19 
10.9 

II.  I 

14.8 

5/i6 

12.5 

ii.  7 

12.8 

14-5 

I2.O 

11.4 

13.2 

13-7 

I8.5 

M 

15.0 

14.1 

15-3 

17.4 

14.4 

13-7 

15.8 

16.4 

22.2 

7/i  6 

16.4 

17.9 

20.3 

16.8 

15-9 

18.4 

I9.I 

25-9 

# 

20.0 

18.7 

20.4 

23-2 

19-3 

18.2 

21.  1 

21.9 

29.7 

9/i6 

22.5 

21.  I 

23.0 

26.0 

21.7 

20.5 

23-7 

24.6 

33-4 

^ 

25.0 

23-5 

25-5 

28.9 

24.1 

22.8 

26.3 

27-3 

27-5 

25.8 

28.1 

31.8 

26.5 

25.O 

28.9 

30.0 

40.8 

» 

30.0 

28.1 

30.6 

34-7 

28.9 

27-3 

31-6 

32-8 

44-5 

J3/i6 

32.5 

3°-5 

33-2 

37-6 

3i-3 

29.6 

34-2 

35-0 

48.2 

I5/i6 

35-o 
37-5 

32.8 
35-2 

35-7 
38.3 

40-5 
43-4 

33-7 
36.1 

31-9 

34-1 

36.8 
39-5 

38.2 
4I.O 

51-9 
55-6 

I 

40.0 

37-5 

40.8 

46.3 

38.5 

36.4 

42.1 

43-7 

59-3 

Thickness  advancing  by  Twentieths  of  an  Inch. 


inch. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

•05 

2.00 

1.88 

2.04 

2.32 

1-93 

1.82 

2.  II 

2.19 

2.96 

.IO 

4.OO 

3-75 

4.08 

4-63 

3-85 

3.64 

4.21 

4-37 

5-93 

•15 

6.00 

5.63 

6.12 

6.95 

5-78 

5-46 

6.32 

6.56 

8.90 

.20 

8.00 

7-50 

8.16 

9.26 

7.70 

7.28 

8.42 

8-74 

II.O 

•25 

IO.O 

9-38 

IO.2 

ii.  6 

9-63 

9-IO. 

10.5 

10.9 

14.8 

•30 

12.0 

n-3 

12.2 

13-9 

1  1.6 

IO-9 

12.6 

i3-i 

17.8 

•35 

14.0 

i3-i 

14-3 

16.2 

13-5 

12.7 

14.7 

15.3 

20.8 

.40 

16.0 

15.0 

16-3 

18.5 

i5-4 

14.6 

16.8 

17.5 

23-7 

•45 

18.0 

16.9 

18.4 

20.8 

17-3 

16.4 

18.9 

19.7 

26.7 

•5° 

20.  o 

18.8 

20.4 

23.2 

19-3 

18.2 

21.  I 

21.9 

29.7 

•55 

22.0 

20.6 

22.4 

25-5 

21.2 

20.0 

23.2 

24.0 

32.7 

.60 

24.0 

22.5 

24-5 

27.8 

23.1 

21.8 

25.3 

26.2 

35-6 

.65 

26.O 

24.4 

26.5 

30.1 

25.0 

23-7 

27.4 

28.4 

38.6 

.70 

28.0 

26.3 

28.6 

32.4 

27.0 

25-5 

29-5 

30.6 

4i-5 

•75 

3O.O 

28.1 

3O.6 

34.7 

28.9 

27-3 

31-6 

32.8 

44-5 

.80 

32.0 

30.0 

32.6 

37-o 

30.8 

29.1 

33-7 

35-o 

47-5 

•85 

34-o 

31.9 

34-7 

39-4 

32.7 

30-9 

35-8 

37-2 

50.4 

.90 

36.0 

33-8 

36.7 

41.7 

34-7 

32-8 

37-9 

39-3 

53-4 

•95 

38.0 

35-6 

38.8 

44.0 

36.6 

34-6 

40.0 

41-5 

56.3 

1.  00 

40.0 

37-5 

40.8 

46.3 

38-5 

36.4 

42.1 

43-.7 

59-3 

Note  to  Table  73,  next  page.  — To  find  the  weight  of  I  lineal  foot  or  I  lineal  yard  of 
hammered  iron,  copper,  tin,  zinc,  or  lead,  multiply  the  tabular  weight  for  rolled  wrought 
iron  of  the  given  dimensions  by  the  following  multipliers,  respectively : — 

EXACT.          APPROXIMATE. 

Hammered  Iron 1.008 i.oi  equivalent  to  I  per  cent.  more. 

Copper 1.158 1.16  ,,          16        ,,          more. 

Tin 962 96  ,,  4        ,,          less. 

Zinc 91 91  ,,  9       „          less. 

Lead 1-483 1.48  ,,         48        ,,          more. 


WEIGHT   OF   METALS   OF   A   GIVEN    SECTIONAL  AREA.         221 


Table  No.  73. — WEIGHT  OF  METALS,  OF  A  GIVEN  SECTIONAL  AREA, 
PER  LINEAL  FOOT  AND  PER  LINEAL  YARD. 


SECT. 
AREA. 

ROLLED 
WROUGHT  IRON. 
Sp.  Weight=i. 

CAST  IRON. 
Sp.Weight=.937S- 

STEEL. 
Sp.  Weight=i.o2. 

BRASS. 
Sp.  Weight—  1.052. 

GUN  METAL. 
Sp.Weight=i.o92. 

i  Foot. 

i  Yard. 

i  Foot. 

i  Yard. 

i  Foot. 

i  Yard. 

i  Foot. 

i  Yard. 

i  Foot. 

i  Yard. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

.1 

-333 

1.  00 

.313 

.938 

•340 

1.02 

•351 

1.05 

.364 

1.09 

.2 

•3 

.667 
1.  00 

2.OO 
3.00 

.625 
.938 

1.88 
2.8l 

.680 
1.02 

2.04 
3-06 

.701 

2.IO 
3-16 

.728 
1.09 

2.18 
3-28 

•4 

1-33 

4.OO 

1.25 

3-75 

1.36 

4.08 

1-43 

4.21 

1.46 

4-37 

1.67 

5-00 

I.56 

4.69 

1.70 

5.10 

1-75 

5.26 

1.82 

.6 

2.OO 

6.00 

1.88 

5-63 

2.O4 

6.12 

2.  1  1 

6.3I 

2.18 

6-ls 

-7 

2-33 

7.00 

2.19 

6.56 

2.38 

7.14 

2.46 

7.36 

2-55 

7.64 

.8 

2.67 

8.00 

2.50 

7-5° 

2.72 

8.16 

2.81 

8.42 

2.91 

8.74 

•9 

3.00 

9.00 

2.81 

8-44 

3-06 

9.18 

3.16 

9-47 

3-28 

9-83 

I.O 

3-33 

10.0 

3-15 

9-38 

3-40 

10.2 

3.51 

10.5 

3-64 

10.9 

.1 

3-67 

II.  0 

3-44 

10.3 

3-74 

II.  2 

3.86 

11.6 

4.00 

12.0 

.2 

4.OO 

12.0 

3-75 

11.3 

4.08 

12.2 

4.21 

12.6 

4-37 

I3-I 

•3 

4-33 

13.0 

4.06 

12.2 

4.42 

13-3 

4-56 

13.7 

4-73 

14.2 

•4 

4.67 

14.0 

4-38 

I3«  i 

4.76 

14-3 

4.91 

14.7 

5.10 

I5>3 

•5 

5.00 

15.0 

4.69 

14.1 

5.26 

15.8 

5-46 

16.4 

.6 

5-33 

16.0 

5.00 

15.0 

5-44 

16.3 

5.61 

16.8 

5.82 

17-5 

•  7 

5.67 

17.0 

5-31 

15-9 

5-78 

17-3 

5-96 

17.9 

6.19 

18.6 

.8 

6.00 

18.0 

5.63 

16.9 

6.12 

I8.4 

6.31 

18.9 

6.55 

19.7 

•9 

2.0 

6-33 
6.67 

19.0 

20.0 

5-94 
6.25 

17.8 
18.8 

6.46 
6.80 

19.4 
20.4 

6.66 
7.01 

20.0 
21.  0 

6.92 
7.28 

20.8 
21.8 

2.1 

7.00 

21.0 

6.56 

19.7 

7.14 

21.4 

7.36 

22.1 

7.64 

22.9 

2.2 

7-33 

22.0 

6.88 

20.6 

7.48 

22.4 

7-72 

23.1 

8.01 

24.0 

2-3 

7.67 

23.O 

7.19 

21.6 

7.82 

23-5 

8.07 

24.2 

8-37 

25.1 

2.4 

8.00 

24.0 

7-5° 

22.5 

8.16 

24-5 

8.42 

25-3 

8-74 

26.2 

2-5 

8-33 

25.0 

23.4 

8.50 

25-5 

8-77 

26.3 

9.10 

27.3 

2.6 

8.67 

26.0 

8.13 

24.4 

8.84 

26.5 

9.12 

27.4 

9.46 

28.4 

2-7 

9.00 

27.0 

8.44 

25.3 

9.18 

27-5 

9-47 

28.4 

9.83 

29-5 

2,8 

9-33 

28.0 

8-75 

26.3 

9-52 

28.6 

9.82 

29-5 

10.2 

30.6 

2.9 

9.67 

29.0 

9.06 

27.2 

9.86 

29.6 

10.2 

30-5 

10.6 

31.7 

3-0 

10.0 

30.0 

9.38 

28.1 

IO.2 

3O.6 

10.5 

31.6 

10.9 

32.8 

3.1 

10.3 

31.0 

9.69 

29.1 

10.5 

31.6 

IO.9 

32.6 

"•3 

33.9 

3-2 

10.7 

32.0 

10.  0 

30.0 

10.9 

32.6 

II.  2 

33-7 

11.7 

34.9 

3-3 

II.  O 

33-0 

10.3 

30.9 

II.  2 

33-7 

n.6 

34.7 

12.0 

36.0 

3-4 

"•3 

34-o 

10.6 

31.9 

ii.  6 

34-7 

11.9 

35-8 

12.4 

37.1 

3-5 

11.7 

35-0 

10.9 

32.8 

11.9 

35-7 

12.3 

36.8 

12.7 

38.2 

3.6 

12.0 

36.0 

IJ-3 

33-8 

12.2 

36.7 

12.6 

37-9 

I3.I 

39.3 

3-7 
3.8 

12.3 
12.7 

3&o 

ii.  6 
11.9 

34-7 
35-6 

12.6 
12.9 

37-7 
38.8 

13.0 
13.3 

38.9 
40.0 

13-8 

40.4 
41.5 

3-9 

13-0 

39-0 

12.2 

36.6 

13-3 

39-8 

13.7 

41.0 

I4.2 

42.6 

4.0 

13-3 

40.0 

12.5 

37-5 

13-6 

40.8 

14.0 

42.1 

14.6 

43.7 

4.1 

13-7 

41.0 

12.8 

38.4 

13-9 

41.8 

14.4 

43-1 

14.9 

44-8 

4.2 

I4.O 

42.0 

I3-  * 

39-4 

14-3 

42.8 

14.7 

44.2 

15.3 

45-9 

4-3 

14-3 

43-  ° 

13.4 

40.3 

14.6 

43-9 

15.1 

45-2 

15.7 

46.9 

4.4 

14.7 

44.0 

13.8 

4L3 

15.0 

44-9 

15.4 

46.3 

16.0 

48.0 

4-5 

15.0 

14.1 

42.2 

15.3 

45-9 

15.8 

47-3 

16.4 

49.1 

4.6 

15-3 

46.0 

14.4 

43-  * 

15-6 

46.9 

16.  i 

48.4 

16.7 

50.2 

4-7 

15.7 

47.0 

14.7 

44.1 

16.0 

47-9 

16.5 

49-4 

17.1 

51.3 

4.8 

16.0 

48.0 

15.0 

45-o 

16.3 

49.0 

16.8 

50-5 

17.5 

52.4 

4-9 

16.3 

49.0 

15.3 

45-9 

16.7 

50.0 

17-2 

51.6 

17.8 

53-5 

5-0 

16.7 

50.0 

15.6 

46.9 

17.0 

51.0 

17-5 

52.6 

18.2 

54-6 

222 


WEIGHT   OF   METALS. 
Table  No  73  (continued}. 


SECT. 
AREA. 

ROLLED 
WROUGHT  IRON. 
Sp.  Weight^  i. 

CAST  IRON. 
Sp.Weight=.937S. 

STEEL. 
Sp.  Weight=i.o2. 

BRASS. 
Sp.  \Veight=  i  .  052. 

GUN  METAL. 
Sp.Weight=i.o92. 

i  Foot. 

i  Yard. 

i  Foot. 

i  Yard. 

i  Foot. 

i  Yard. 

i  Foot. 

i  Yard. 

i  Foot. 

i  Yard. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

5-1 

17.0 

51.0 

15-9 

47-8 

17-3 

52.0 

17.9 

53.7 

18.6 

55-7 

5-2 

17-3 

52.0 

16.3 

48.8 

17.7 

53-0 

18.2 

54-7 

18.9 

56.8 

5-3 

17.7 

53-o 

16.6 

49-7 

18.0 

54- 

1  8.6 

55-8 

19-3 

57-9 

5-4 

iS.o 

54-0 

16.9 

50.6 

18.4 

55- 

18.9 

56.8 

19.7 

58.9 

5-5 

18.3 

55-o 

17.2 

51.6 

18.7 

56. 

i9-3 

57-9 

20.  o 

60.0 

5.6 

18.7 

56.0 

J7-5 

52.5 

19.0 

57- 

19.6 

58.9 

20.4 

61.1 

5-7 

19.0 

57.0 

17.8 

53-4 

19.4 

58. 

20.0 

60.0 

20.8 

62.2 

5.8 

19-3 

58.0 

18.  i 

54-4 

19.7 

59-2 

20.3 

61.0 

21.1 

63.3 

5-9 
6.0 

19.7 
20.  o 

59-0 
60.0 

18.4 
18.8 

55-3 
56.3 

20.1 

20.4 

60.2 
61.2 

20.7 
21.0 

62.1 
63-1 

21-5 
21.8 

64.4 
65-5 

6.1 

20.3 

61.0 

19.1 

57-2 

20.7 

62.2 

21.4 

64.2 

22.2 

66.6 

6.2 

20.7 

62.0 

19.4 

58.1 

21.  1 

63.2 

21.7 

65.2 

22.6 

67.7 

6-3 

21.0 

63.0 

19.7 

59-1 

21.4 

64.3 

22.1 

66.3 

22.9 

68.8 

6.4 

21.3 

64.0 

20.  o 

60.0 

21.8 

65-3 

22.4 

67-3 

23-3 

69.9 

6.5 

21.7 

65.0 

20.3 

60.9 

22.1 

66.3 

22.8 

68.4 

23-7 

70.9 

6.6 

22.  0 

66.0 

20.  6 

61.9 

22.4 

67-3 

23.1 

69.4 

24.0 

72.0 

6.7 

22.3 

67.0 

20.9 

62.8 

22.8 

68.3 

23-5 

70.5 

24.4 

73-i 

6.8 

22.7 

68.0 

21.3 

63.8 

23.1 

69.4 

23-9 

71-5 

24.8 

74-2 

6.9 

23.0 

69.0 

21.6 

64.7 

23-5 

70.4 

24.2 

72.6 

25-1 

75-3 

7.0 

23-3 

70.0 

21.9 

65.6 

23.8 

71-4 

24.6 

73-6 

25-5 

76.4 

7-i 

23-7 

71.0 

22.2 

66.6 

24.1 

72.4 

24.9 

74-7 

25.8 

77-5 

7.2 

24.0 

72.0 

22.5 

67.5 

24.5 

73-4 

25-3 

75-7 

26.2 

78.6 

7-3 
7-4 

24-3 
24-7 

73-o 
74.0 

22.8 
23.1 

68.4 
69.4 

24.8 
25-2 

74-5 
75-5 

25.6 
26.O 

76.8 
77-9 

26.6 
26.9 

79-7 
80.8 

7-5 
7.6 

25.0 
25-3 

75-o 
76.0 

23-4 
23-8 

70.3 
7i.3 

25-5 
25.9 

76.5 
77-5 

26.3 
26.7 

78.9 
80.0 

27.3 

27-7 

81.9 
83-0 

7-7 

25-7 

77.0 

24.1 

72.2 

26.2 

78.5 

27.0 

81.0 

28.0 

84.1 

7.8 

26.0 

78.0 

24.4 

73«i 

26.5 

79.6 

27.4 

82.1 

28.4 

85.2 

7-9 

26.3 

79.0 

24.7 

74.1 

26.9 

80.6 

27.7 

83-1 

28.8 

86.3 

8.0 

26.7 

80.0 

25.0 

75-o 

27.2 

81.6 

28.1 

84.2 

29.I 

87.4 

8.1 

27.0 

81.0 

25-3 

75-9 

27-5 

82.6 

28.4 

85.2 

29-5 

88.5 

8.2 

27-3 

82.0 

25.6 

76.9 

27-9 

83.6 

28.8 

86.3 

29.9 

89.5 

8-3 

27-7 

83.0 

25-9 

77-8 

28.2 

84.7 

29.1 

87-3 

30.2 

90.6 

8.4 

28.0 

84.0 

26.3 

78.8 

28.6 

85.7 

29.5 

88.4 

3O.6 

91.7 

8-5 

28.3 

85.0 

26.6 

79-7 

28.9 

86.7 

29.8 

89.4 

30-9 

92.8 

8.6 

28.7 

86.0 

26.9 

80.6 

29.2 

87.7 

30.2 

90.5 

3L3 

93-9 

8.7 

29.0 

87.0 

27.2 

81.6 

29.6 

88.7 

30-5 

9i.S 

31-7 

95-0 

8.8 

29-3 

88.0 

27-5 

82.5 

29.9 

89.8 

3°-9 

92.6 

32.0 

96.1 

8.9 

29.7 

89.0 

27.8 

83-4 

3°-3 

90.8 

31.2 

93-6 

32.4 

97.2 

9.0 

30.0 

90.0 

28.1 

84-4 

30.6 

91.8 

31-6 

94-7 

32.8 

98.3 

9-1 

30-3 

91.0 

28.4 

85-3 

30-9 

92.8 

3i-9 

95-7 

33-1 

99-4 

9.2 

30.7 

92.0 

28.8 

86.3 

31-3 

93-8 

32.3 

96.8 

33-5 

100.5 

9-3 

31.0 

93-° 

29.1 

87.2 

31.6 

94-9 

32.6 

97-8 

33-9 

101.6 

9-4 

31-3 

94.0 

29.4 

88.1 

32.0 

95-9 

33-o 

98.9 

34-2 

102.7 

9-5 

31-7 

•95-0 

29.7 

89.1 

32-3 

96.9 

33-3 

99-9 

34-6 

103.7 

9.6 

32.0 

96.0 

30.0 

90.0 

32.6 

97-9 

33-7 

IOI.O 

34-9 

104.8 

9-7 

32.3 

97.0 

30-3 

90.9 

33-0 

98.9 

34-0 

102.0 

35-3 

105.9 

9-8 

32.7 

98.0 

30.6 

91.9 

33-3 

IOO.O 

34-4 

103.1 

35-7 

107.0 

9-9 

33-o 

99.0 

30-9 

92.8 

33-7 

IOI.O 

34-7 

104.2 

36.0 

108.1 

10.0 

33-3 

100.0 

31-3 

93-8 

34-o 

102.0 

35-i 

105.2 

36.4 

109.2 

See  note  at  foot  of  page  220. 


RULES   FOR   WEIGHT.  223 

RULES  FOR  THE  WEIGHT  OF  IRON  AND  STEEL. 

The  following  rules  for  finding  the  weight  of  wrought  iron,  cast  iron, 
and  steel,  are  based  on  the  data  contained  in  Tables  No.  70  and  71. 

RULE  i. — To  FIND  THE  WEIGHT  OF  IRON  OR  STEEL,  when  the  volume 
in  cubic  feet  is  given.  Multiply  the  volume  by 

4.29  for  wrought  iron, 
4.02  for  cast  iron, 
4.37  for  steel. 
The  product  is  the  weight  in  hundredweights. 

RULE  2. —  When  the  volume  in  cubic  inches  is  given,  multiply  the  volume 
by 

.278  (or  .28)  for  wrought  iron, 
.26  for  cast  iron, 
.283  for  steel. 

The  product  is  the  weight  in  pounds. 

RULE  3. —  When  the  quantity  is  reduced  to  square  feet,  one  inch  in  thickness, 
multiply  the  area  by 

40  for  wrought  iron, 

37^  for  cast  iron, 

40.8  (or  41)  for  steel. 
The  product  is  the  weight  in  pounds. 
Or,  multiply  the  area  by 

•357  f°r  wrought  iron, 
.335  for  cast  iron, 
.364  for  steel. 
The  product  is  the  weight  in  hundredweights. 

RULE  4. —  When  the  sectional  area  in  square  inches,  and  the  length  in  feet, 
of  a  bar  or  prism  are  given,  multiply  the  sectional  area  by  the  length, 
and  by 

3  x/s  for  wrought  iron, 
3^6  for  cast  iron, 
3.4  for  steel. 

The  product  is  the  weight  in  pounds. 

For  large  masses,  multiply  the  sectional  area  by  the  length,  and  divide 
the  product  by 

672  for  wrought  iron, 
717  for  cast  iron, 
659  for  steel. 
The  quotient  is  the  weight  in  tons. 

RULE  5. —  When  the  sectional  area  in  square  inches,  and  the  length  in  yards, 
of  a  bar  or  prism,  are  given,  multiply  the  sectional  area  by  the  length,  and  by 

10  for  wrought  iron, 
9-375  f°r  cast  iron, 
10.2  for  steel. 

The  product  is  the  weight  in  pounds. 


224  WEIGHT   OF   METALS. 

RULE  6. — To  FIND  THE  SECTIONAL  AREA  OF  'A  BAR  OR  PRISM  OF  IRON 
OR  STEEL,  when  the  length  and  the  total  weight  are  given.  Divide  the  weight 
in  pounds  by  the  length  in  feet,  and  by 

3  x/3  for  wrought  iron, 
3^6  for  cast  iron, 
3.4  for  steel. 

The  quotient  is  the  sectional  area  in  square  inches. 

RULE  7. — To  FIND  THE  LENGTH  OF  A  BAR,  PRISM,  OR  OTHER  PIECE 
OF  UNIFORM  SECTION  OF  IRON  OR  STEEL,  when  the  total  weight  and  the 
sectional  area  are  given.  Divide  the  weight  in  pounds  by  the  sectional  area 
in  square  inches,  and  by 

3  T/3  for  wrought  iron, 
3  J$  for  cast  iron, 
3.4  for  steel. 

The  quotient  is  the  length  in  feet. 

In  applying  the  last  rule  to  calculate  the  length  of  wire  of  a  given  size, 
for  a  given  weight,  say  i  cwt.  of  wire,  the  sectional  area  of  the  wire  is 
found,  in  the  usual  way,  by  multiplying  the  square  of  the  thickness  or 
diameter,  d,  by  .7854.  Then,  by  the  rule,  the  length  in  feet  of  i  cwt.  of 
iron  wire  is  equal  to 

i£2 _  42^78 

</2X.  7854x3    1/3  d* 

In  the  same  way,  the  dividends  of  the  fractions  to  express  the  length  of 
i  cwt.  of  other  metals  may  be  found,  and  the  following  is  a  special  rule 
for  wire : — 

RULE  8. — To  FIND  THE  LENGTH  OF  ONE  HUNDREDWEIGHT  OF  WIRE 
OF  A  GIVEN  THICKNESS.  Divide  the  following  numbers  by  the  square  of 
the  diameter  or  thickness,  in  parts  of  an  inch : — 

42.78  for  wrought  iron, 
42  for  steel, 
37.43  for  copper, 
38.54  for  brass, 
31.34  for  silver, 
17.12  for  gold, 
15.28  for  platinum. 

The  quotient  is  the  length  in  feet. 

Note. — This  rule  may  be  used  for  finding  the  weight  of  round  bar  iron. 

2.  It  is  known  that  the  density  of  wire  is  not  perfectly  constant,  but 
that  there  is  some  degree  of  variation,  according  to  the  size.  It  is  generally 
understood  that  the  density  is  reduced  as  the  wire  is  drawn  smaller,  but 
it  appears  from  the  table  of  the  weight  of  Warrington  wire,  that  the  density 
is  greater  for  the  smallest  sizes.  The  same  inference  is  to  be  drawn  from 
tabular  statements  of  the  length  of  one  kilogramme  of  wire  according  to 
the  French  gauge  (Table  No.  Ji,  page  148).  One  of  these  statements  is 
given  on  the  next  page,  from  which  it  is  apparent  that  the  length  of  iron 


RULES   FOR  WEIGHT. 


225 


required  to  weigh  a  kilogramme  decreases  more  rapidly  than  the  sectional 
area  increases.     For  example,  the  diameter  being 

6,       12,       24,       30  tenths  of  a  millimetre, 
the  squares  of  which,  or  the  relative  volumes  of  a  given  length,  are  as 

i,         4,       16,       25; 
the  lengths  of  a  kilogramme  are 

405,     115,       30,       20  metres, 
which  are  inversely  as 


3-5, 


20.2. 


Showing  that  a  shorter  length  is  required  in  proportion  to  the  volume,  as 
the  diameter  of  the  wire  is  reduced,  and  that  the  density  of  the  smaller 
wire  must  therefore  be  the  greater. 

Table  No.  730.  —  WEIGHT  OF  GALVANIZED  IRON  WIRE  (French). 


No.  of  Gauge. 

Diameter. 

Length  of 
i  Kilogramme. 

N*o.  of  Gauge. 

Diameter. 

Length  of 
i  Kilogramme. 

millimetres. 

metres. 

millimetres. 

metres. 

I 

0.6 

405 

13 

2.0 

40 

2 

0.7 

370 

14 

2.2 

35 

3 

0.8 

260 

*5 

2-4 

3° 

4 

0.9 

215 

16 

2.7 

25 

5 

I.O 

J75 

i7 

3-° 

20 

6 

i.i 

140 

18 

3-4 

15 

7 

1.2 

"5 

i9 

3-9 

10 

8 

i-3 

103 

20 

4-4 

9 

9 

1.4 

82 

21 

4.9 

6 

10 

i-5 

70 

22 

5-4 

5 

ii 

1.6 

65 

23 

5-9 

4 

12 

1.8 

5° 

3.  The  densities  of  metals  assumed  in  the  foregoing  rules  are  those  which 
are  tabulated  in  Table  No.  65. 

4.  In  estimating  the  weight  of  cast  iron  from  plans,  the  weight  is  fre- 
quently calculated  at  the  same  rate  as  for  wrought  iron,  which  is  heavier 
than  cast  iron,  with  the  object  of  providing  an  allowance,  by  way  of  com- 
pensation,  for  occasional  swellings  or  enlargements  of  castings  in  excess  of 
the  exact  dimensions  of  patterns. 

The  following  tables  of  the  weight  of  metals  in  various  forms  have  been 
calculated  by  means  of  the  preceding  rules.  The  sectional  areas  of  bars 
and  other  pieces  of  uniform  section  are,  in  some  tables,  added  for  each 
scantling.  The  length  of  bar,  and  the  area  of  plates  and  sheets,  required 
to  weigh  i  cwt,  or  i  ton,  are  given. 

15 


226  WEIGHT  OF  METALS. 

LIST   OF   TABLES   OF  THE  WEIGHT    OF  WROUGHT    IRON, 
IN  BARS,  PLATES,  SHEETS,  HOOP-IRON,  WIRE,  AND  TUBES. 

TABLE  No.  74. — Weight  of  Flat  Bar  Iron;  width,  i  to  n  inches;  thick- 
ness, 7'j6  to  i  inch;  length,  i  to  9  feet. 

TABLE  No.  75. — Weight  of  Square  Iron;  ^  to  6  inches  square;  length, 
i  to  9  feet. 

TABLE  No.  76. — Weight  of  Round  Iron,  ^  to  24  inches  in  diameter; 
length,  i  to  9  feet. 

TABLE  No.  77. — Weight  of  Angle-Iron  and  Tee-Iron;  sum  of  the  width 
and  depth,  i^  to  20  inches;  thickness,  ^  to  i  inch;  length,  i  foot. 

In  the  composition  of  this  table,  it  has  been  assumed  that  the  base  and 
the  web  or  flange  are  of  equal  thicknesses;  and  that  the  reduction  of  area 
of  section  by  rounding  off  the  edges,  is  compensated  by  the  filling  in  at 
the  root  of  the  flange. 

TABLE  No.  78. — Weight  of  Wrought-iron  Plates;  area,  i  to  9  square 
feet;  thickness,  ^  to  15  inches. 

TABLE  No.  79. — Weight  of  Sheet  Iron,  according  to  wire-gauge  used  by 
South  Staffordshire  sheet-rollers;  area,  i  to  9  square  feet;  thickness,  No.  i 
to  No.  32  wire-gauge. 

TABLE  No.  80. — Weight  of  Black  and  Galvanized  Iron  Sheets  (Morton's 
Table). 

TABLE  No.  81. — Weight  of  Hoop  Iron;  width,  fa  to  3  inches;  thickness, 
No.  4  to  No.  21  wire-gauge;  length,  i  foot. 

TABLE  No.  82. — Weight  and  Strength  of  Warrington  Iron  Wire. 

TABLE  No.  83. — Weight  of  Wrought-iron  Tubes,  by  internal  diameter ; 
diameter,  J/s  to  36  inches;  thickness,  ^  inch  to  No.  18  wire-gauge;  length, 
i  foot. 

TABLE  No.  84. — Weight  of  Wrought-iron  Tubes,  by  external  diameter; 
diameter,  i  to  10  inches;  thickness,  No.  15  wire-gauge  to  s/i6  inch;  length, 
i  foot. 


Multipliers,  derived  from  table  No.  70,  are  subjoined,  by  which  the 
tabulated  weights  of  wrought  iron  may  be  multiplied,  in  order  to  find  from 
these  tables  the  weight  of  bars,  plates,  or  sheets  of  other  metal. — 

Multipliers. 

Hammered  Iron i.oi 

Cast  Iron 94 

Steel 1.02 

Sheet  Copper 1.14 

Hammered  Copper 1.16 

Lead 1.48 

Cast  Brass 1.05 

Brass  Wire 1. 1 1 

Gun  Metal 1.09 


FLAT  BAR   IRON. 


227 


Table  No.  74- — WEIGHT   OF   FLAT    BAR   IRON. 

i  INCH  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

.  feet. 

X 

5/i6 

X 

.250 
•313 

•375 

.833 
1.04 
I.2.S 

2.08 
2.50 

2.50 
3.12 

3-75 

3-33 
4.16 
5-00 

4.17 
5.20 
6.25 

5.00 
6.24 
7-50 

5.83 
7.28 

8-75 

6.67 
8.32 
IO.O 

7-50 
9.36 
"•3 

134-4 

"7-5 

89.6 

7/i6 

.43» 

1.46 

2.92 

4.38 

5-84 

7.29 

8.76 

10.2 

11.7 

I3-I 

76.8 

H 

.500 

1.67 

3-33 

5-00 

6.67 

8-33 

IO.O 

II.7 

13.3 

15.0 

67.2 

9/16 

»/x6 

.563 
.625 
.688 

1.88 
2.08 
2.29 

3-75 
4.16 
4.58 

5.62 
6.25 
6.87 

7.50 
8.33 
9.17 

9-37 
10.4 
II.4 

II-3 
12.5 
13.8 

I3i 

14.6 
16.0 

15.0 
16.6 
18.3 

16.9 
18.8 

20.6 

59.7 
48.9 

X 

•750 

2.50 

5.00 

7-50 

IO.O 

12-5 

15-0 

17-5 

20.0 

22.5 

44.8 

*3/i6 

.813 

2.71 

5-42 

8.12 

10.8 

13-5 

16.3 

19.0 

21.7 

24.4 

41.4 

% 

.87S 

2.92 

5-84 

8.76 

11.  7 

I4.6 

17-5 

20.4 

23-4 

26.3 

S8.4 

z5/i6 
I 

•938 

1.  00 

3-13 

3-33 

6.25 
6.67 

9.38 

IO.O 

12.5 
13.3 

IS.6 
I6.7 

18.8 
20.  o 

21.9 
23-3 

25.0 
26.7 

28.1 
30.0 

35-8 
33-6 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

l/i 

.281 

.938 

1.88 

2.81 

3-75 

4.69 

5.63 

6.56 

7.50 

8.44 

II9-5 

5/i6 

.352 

.422 

1.17 
1.41 

ilt 

3-52 
4.22 

4.68 
5-62 

5-86 
7-03 

7-03 

8-44 

8.20 

9.84 

9.37 

"•3 

10.6 
12.7 

95-6 
79.6 

7/x6 

•492 

3.28 

4.92 

6.56 

8.20 

9-84 

ii-5 

14.8 

68.3 

/* 

.563 

1.88 

3.75 

5.62 

7.5° 

9.38 

ii.3 

15-0 

16.9 

59-7 

9/i6 

•633 

2.  1  1 

422 

6-33 

8-44 

10.6 

12.7 

14.8 

16.9 

19.0 

53-i 

•fa 

.703 

2.34 

4.69 

7.03 

9.38 

11.7 

I3-I 

16.4 

18.8 

21.  1 

47-8 

ii/jS 

•773 

2.58 

5.16 

7-73 

10.3 

12.9 

18.0 

20.  6 

23.2 

43-4 

3/ 

/4 

•844 

2.81 

5.63 

8.44 

14.0 

16.9 

19.7 

22.5 

25-3 

39-8 

J3/i6 

•914 

3-05 

6.09 

9-14 

12.2 

15-2 

18.3 

21.3 

24.4 

27.4 

36.8 

* 

.984 

3.28 

6.56 

9.84 

13.  1 

16.4 

19.7 

23.0 

26.3 

29-5 

34.1 

1.06 

3-52 

7.03 

10.6 

I4.I 

17-6 

21.  1 

24.6 

28.1 

31.6 

3L9 

I 

1-13 

3-75 

7.5° 

II-3 

15.0 

18.8 

22.5 

26.3 

30.0 

33-8 

29-9 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

14 

.313 

1.04 

2.08 

3-12 

4.17 

5-21 

6.25 

7.29 

8.33 

9-37 

107.5 

5/i6 

•391 

I.30 

2.60 

3-91 

5-21 

6.51 

7.82 

9.II 

10.4 

u.  7 

94.0 

H 

.469 

1.56 

3.13 

4.69 

6.25 

7.81 

9.38 

10-9 

12.5 

14.1 

71.7 

7/16 

•  547 

1.82 

3.65 

5-47 

7-29 

9.12 

10.9 

12.8 

14.6 

16.4 

61.2 

% 

.625 

2.08 

4.17 

6.25 

8-33 

10.4 

12.5 

14.6 

16.7 

18.8 

53-8 

9/i  6 

.703 

2-34 

4.69 

7-03 

9.38 

11.7 

14.1 

16.4 

18.8 

14.1 

47.8 

H 

.781 

2.60 

5-21 

7.8l 

10.4 

13.0 

15.6 

18.2 

20.8 

23-4 

43-0 

n/jg 

.859 

2.86 

5-73 

8-59 

n-5 

14.3 

17.2 

20.1 

22.9 

39.1 

M 

.938 

3-i3 

6.25 

9.38 

12.5 

15.6 

18.8 

21.9 

25.0 

28.1 

35-8 

J3/i6 

1.02 

3-39 

6.77 

10.2 

'3-5 

16.9 

20.3 

23.7 

27.1 

30.5 

33-1 

H 

I.  II 

3-65 

7.29 

IO.9 

14.6 

18.2 

21.9 

25-5 

29.2 

32-8 

30.7 

15/16 

1.  17 

3-9i 

7.81 

ii.  7 

15.6 

19-5 

23-4 

27-3 

31.2 

35-i 

28.7 

i 

1.25 

4.17 

8-33 

12.5 

16.7 

20.8 

25.0 

29.2 

33-3 

37-5 

26.9 

228 


WEIGHT   OF   METALS, 


WEIGHT  OF  FLAT  BAR  IRON. 
1 34  INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

•344 

I-I5 

2.29 

3-44 

4.58 

5-73 

6.87 

8.02 

9.17 

10.3 

97-7 

5/i  6 

•430 

i-43 

2,86 

4-30 

5.73 

7.16 

8.59 

IO.O 

"•5 

12.9 

78.2 

H 

.516 

1.72 

3-44 

5.16 

6.87 

8-59 

10.3 

12.  0 

15-5 

65.6 

7/i6 

.602 

2.01 

4.01 

6.  02 

8.02 

IO.O 

12.0 

14.0 

16.0 

18.0 

55-9 

.688 

2.29 

4-58 

6.87 

9.17 

"•5 

13-8 

16.0 

18.3 

20.6 

48.9 

9/i6 
"6 

•773 
.859 
•945 

lit 

3-15 

5.16 
5-73 
6.31 

7-73 
8-59 
9-45 

10.3 

"•5 

12.6 

12.9 

14-3 
15-8 

15-5 
17.2 
I8.9 

18.0 
20. 

22. 

20.6 

22.9 

25.2 

23.2 

25.8 

28.4 

43-4 
39-1 

35-5 

^ 

1.03 

3-44 

6.88 

10.3 

13.8 

17.2 

20.  6 

24. 

27.5 

30-9 

32.6 

T3/i6 

1.  12 

3-72 

7-45 

II.  2 

14.9 

18.6 

22.3 

26. 

29.8 

33.5 

30.1 

% 

1.20 

4.01 

8.02 

12.0 

16.0 

20.0 

24.1 

28. 

32.1 

36.1 

27.9 

IS/!  6 

1.29 

4.30 

8.59 

12.9 

17.2 

21-5 

25.8 

30. 

34.4 

38.7 

26.1 

I 

1.38 

4.58 

9.17 

13-8 

18-3 

22.9 

27-5 

32. 

36.7 

41.3 

24.4 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

•275 

1.25 

2.50 

3-75 

5.00 

6.25 

7.50 

8.75 

IO.O 

»«3 

89.6 

5/i6 

.469 

I.56 

3-13 

4.69 

6.25 

7.82 

9.38 

10.9 

12.5 

14.1 

78.3 

H 

.563 

1.88 

3-75 

5-63 

7-50 

9-38 

"•3 

13.1 

15.0 

16.9 

59-7 

7/16 

.656 

2.19 

4-38 

6.56 

8-75 

10.9 

i3-i 

15-3 

J7-5 

19.7 

51-2 

% 

•750 

2.50 

5-00 

7.5o 

IO.O 

12.5 

15.0 

17.5 

20.0 

22.5 

44.8 

9/i6 

.844 

2.81 

5-63 

8.44 

"•3 

I4.I 

16.9 

19.7 

22.5 

25-3 

39-8 

X 

.938 

3-13 

6.25 

9-38 

I2-5 

I5.6 

18.8 

21.9 

25.0 

28.1 

35-8 

"/i6 

•03 

3-44 

6.88 

10.3 

13-8 

17.2 

20.6 

24.1 

27-5 

30.9 

32.6 

* 

•13 

3-75 

7-50 

"•3 

15-0 

18.8 

22.5 

26.3 

30.0 

33-8 

29.9 

I3/i6 

.22 

4.06 

8.13 

12.2 

16.3 

20.3 

24.4 

28.4 

32.5 

36.6 

27.6 

H 

•31 

4-38 

8-75 

I3-I 

21.9 

26.3 

30.6 

35-o 

39-4 

25.6 

*5/x6 

.41 

4.69 

9-38 

I4.I 

18.8 

23-4 

28.1 

32.8 

37-5 

42.2 

23-9 

I 

•50 

5.00 

IO.O 

15-0 

20.0 

25-0 

30.0 

35-o 

40.0 

45-° 

22.4 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs: 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.406 

i-35 

2.71 

4.06 

5-41 

6.8 

8.10 

9.48 

10.8 

12.2 

82.7 

5/rf 

.508 

1.69 

3-39 

5-07 

6.77 

8-5 

10.2 

u.8 

13-5 

15.2 

66.2 

H 

.609 

2.03 

4.06 

6.09 

8.12 

IO.2 

12.2 

14.2 

16.2 

18.3 

55-i 

7/l6 

.711 

2.37 

4-74 

7.II 

9.48 

n.8 

14.2 

16.6 

19.0 

21.3 

47-3 

X 

•8l3 

2.71 

5-42 

8.12 

10.8 

13.5 

16.2 

19.0 

21.6 

24.4 

41-3 

9/i6 

.914 

3-05 

6.09 

9.14 

12.2 

15.2 

18-3 

21.3 

24.4 

27.4 

36.8 

H 

.02 

3-39 

6.77 

IO.2 

13-5 

16.9 

20.3 

23-7 

27.1 

30.5 

33-i 

™/i6 

.12 

3-72 

7-45 

II.  2 

14.9 

18.6 

22.3 

26.1 

29.8 

33-5 

30.1 

% 

.22 

4.06 

8.13 

12.2 

16.3 

20.3 

24.4 

28.4 

32.5 

36.6 

27.6 

'3/16 

•32 

4.40 

8.80 

13.2 

I7.6 

22.0 

26.4 

30.8 

35-2 

39-6 

25-4 

K 

•43 

4-74 

9.48 

14.2 

19.0 

23-7 

28.4 

33-2 

37-9 

42.7 

23-6 

15/16 

•53 

5.08 

10.2 

15.2 

20.3 

25-4 

30.5 

35-5 

40.6 

45-7 

22.1 

i 

•63 

5-42 

10.8 

I6.3 

21.7 

27.1 

32-5 

37-9 

43-3 

48.8 

21.2 

FLAT  BAR   IRON. 


229 


WEIGHT  OF  FLAT  BAR  IRON. 
i      INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.638 

1.46 

2.92 

4-37 

5.83 

7-29 

8.74 

IO.2 

11.7 

I3.I 

76.8 

5/i6 

•547 

1.82 

5-47 

7.29 

9.II 

10.9 

12.8 

14.6 

16.4 

61.4 

^ 

.656 

2.19 

4.38 

6.56 

8.75 

10.9 

13-1 

15-3 

17.5 

19.7 

51.2 

7/i6 

.766 

2-55 

5-10 

7.66 

10.2 

12.8 

15-3 

17.9 

20.4 

23.0 

43.9 

/^ 

.875 

2.92 

5.83 

8.75 

ii.  7 

14.6 

17-5 

2O.4 

23-3 

26.2 

38.4 

9/i6 

.984 

3-28 

6.56 

9.84 

13.1 

16.4 

19.7 

23.0 

26.2 

29.5 

34-1 

M 

.09 

3-65 

7-29 

10.9 

14.6 

19.2 

21.9 

25.5 

29.2 

32-8 

30.7 

.20 

4.01 

8.02 

12.0 

16.0 

20.0 

24.1 

28.1 

32.1 

36.1 

27.9 

^/l6 

.31 

4.38 

8-75 

13-  ! 

17-5 

21.9 

26.3 

30.6 

35-0 

39-4 

25.6 

13/16 

.42 

4-74 

9-48 

14.2 

19.0 

23-7 

28.4 

33-2 

37-9 

43-2 

23.7 

y% 

•53 

5.io 

10.2 

15-3 

20.4 

25-5 

30.6 

35-7 

40.8 

45-9 

21.9 

J5/i6 
I 

.64 

•75 

5-47 
5-83 

10.9 
II.7 

16.4 
17-5 

21.9 
23-3 

27-3 
29.2 

32.8 
35-0 

38.3 
40.8 

43.7 
46.7 

49-2 
52.5 

20.5 
19.2 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.469 

1.56 

3.13 

4.69 

6.25 

7.81 

9.38 

10.9 

12.5 

14.1 

71.7 

5/i  6 
H 

.586 
.703 

1-95 

2-34 

3-91 
4.69 

5.86 
7-03 

7.8l 

9-37 

9.66 
11.7 

11.7 
14.1 

13.7 
16.4 

TQ  o 

17.6 

21.  1 

57.3 

47-8 

7/i6 

.820 

2-73 

5-47 

8.20 

10.9 

13-7 

16.4 

19.1 

21.9 

24.6 

41.0 

% 

.938 

3-13 

6.25 

9.38 

12.5 

15.6 

18.8 

21.9 

25.0 

28.1 

35-8 

9/i6 

.06 

3-52 

7.03 

10.5 

14.1 

17.6 

21.  1 

24.6 

28.1 

31-6 

31.8 

H 

•17 

3-91 

7-8i 

11.7 

14.6 

i9>5 

23-4 

27-3 

31.2 

35-2 

28.7 

^fie 

.29 

4-30 

8.59 

12.9 

17.2 

21.5 

25.8 

30.1 

34-4 

38.7 

26.1 

tf 

.41 

4.69 

9.38 

14.1 

18.8 

23-4 

28.1 

32-8 

37-5 

42.2 

23.9 

J3/i6 

•52 

5.08 

10.2 

15-2 

20.3 

25-4 

30.5 

35-5 

40.6 

45-7 

22.1 

7/B 

.64 

5.47 

IO.9 

16.4 

21.9 

27.3 

32-8 

43-9 

49.4 

20.5 

W/x6 

.76, 

5.86 

ii.  7 

17.6 

23-4 

29-3 

35-i 

41.0 

46.9 

52.7 

I9.I 

I 

.88 

6.25 

12.5 

18.8 

25.0 

3i-3 

37-5 

43-8 

50.0 

56.2 

17.9 

2   INCHES    WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet 

X 

.500 

1.67 

3.33 

5.00 

6.67 

8-33 

IO.O 

11.7 

13-3 

15.0 

67.2 

5/i6 

.625 

2.08 

4.17 

6.25 

8.33 

10.4 

12.5 

14.6 

I6.7 

18.8 

53-8 

H 

•750 

2.50 

5-00 

7.50 

IO.O 

12-5 

15.0 

!7-5 

2O.  O 

22.5 

44-8 

7/i6 

•875 

2.92 

5-83 

8-75 

11.7 

I4.6 

17-5 

20.4 

23-3 

26.3 

38.4 

X 

1.  00 

3-33 

6.67 

10.0 

13-3 

I6.7 

20.0 

23-3 

26.7 

30.0 

33-6 

9/i  6 

•!3 

3-75 

7.50 

"•3 

15.0 

18.8 

22.5 

26.3 

30.0 

33-8 

29.9 

H 

.25 

4.17 

8-33 

12.5 

16.7 

20.8 

25.0 

29.2 

33-3 

37-5 

26.9 

n/ie 

.38 

4o8 

9.16 

13-8 

18.3 

22.9 

27-5 

32.1 

36.7 

41.2 

24.4 

* 

.50 

5.00 

10.  0 

15.0 

20.0 

25.0 

30.0 

35-o 

40.0 

45-° 

22.4 

J3/i6 

.63 

5-42 

10.8 

16.3 

21.7 

27.2 

32.5 

37-9 

43-3 

48.8 

20.7 

H 

•75 

5-83 

11.7 

23-3 

29.2 

35.0 

40.8 

46.7 

52.5 

19.2 

*5/*fi 

.88 

6.25 

12.5 

18.8 

25.0 

3i-3 

37-5 

43-8 

50.0 

56.3 

17.9 

I 

2.00 

6.67 

13-3 

20.  o 

26.7 

33-3 

40.0 

46.7 

53-4 

60.0 

16.8 

230 


WEIGHT   OF   METALS. 


WEIGHT  OF  FLAT  BAR  IRON. 

2l/%    INCHES   WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

1" 

'Ml 

•797 

1.77 
2.21 

2.66 

3-54 

4-43 
5-31 

6.64 
7-97 

7.08 

8.85 

10.6 

8.85 

11.7 

13-3 

10.6 
13-3 
15-9 

12.4 
15-5 

18.6 

14.2 
17.7 

21.2 

15-9 
19.9 

23-9 

63.2 
50.6 
42.2 

7/i6 

•93° 

3-10 

6.20 

9-3° 

12.4 

15-5 

18.6 

21.7 

24.8 

27.9 

36.1 

X 

i.  06 

3-54 

7.08 

10.6 

14.2 

17.7 

21.3 

24.8 

28.3 

31-9 

31-6 

9/i6 

.20 

3-98 

7-97 

12.0 

15-9 

20.  o 

23-9 

27.9 

31-9 

35-8 

28.1 

H 

•33 

4-43 

8.85 

13-3 

17.7 

22.1 

26.6 

31.0 

35-4 

39-8 

25-3 

J1/i6 

.46 

4.87 

9-74 

14.6 

19-5 

24.4 

29.2 

34-i 

39-o 

43-8 

23.0 

X 

•59 

5-31 

10.6 

15-9 

21.2 

26.6 

31-9 

37-2 

42.5 

47-8 

21.  1 

*3/i6 

•74 

5-76 

"•5 

17.3 

23.0 

28.8 

34-5 

40-3 

46.0 

51.8 

I9.8 

7/* 

.86 

6.  20 

12.4 

24.8 

31.0 

37-2 

43-4 

49.6 

55-8 

18.  i 

*5/i6 

.98 

6.64 

13-3 

19.9 

26.6 

33-2 

39-8 

46.5 

53-i 

59-8 

16.9 

1 

2.13 

7.08 

14.2 

21.3 

28.3 

35-4 

42-5 

49.6 

56.7 

63.8 

15-8 

INCHES    WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.563 

1.88 

3.75 

5.63 

7-50 

9-4 

11.3 

13.1 

15.0 

16.9 

59-7 

5/i6 

.703 

2.34 

4.69 

7.03 

9.38 

ii.  7 

14.1 

16.4 

18.8 

21.  1 

47.8 

M 

.844 

2.  8  1 

8.44    11.3 

14.1 

16.9 

19.7 

22.5 

25-3 

39-8 

7/i6 

.984 

3.28 

6*56 

9.84 

13.1 

16.4 

19.7 

23.0 

26.3 

29-5 

34-1 

/z 

3-75 

7.50 

"'3 

15.0 

18.8 

22.5 

26.3 

30.0 

33-8 

29.9 

9/i6 

1.27 

4.22 

8.44 

12.7 

I6.9 

21.  1 

25.3 

29-5 

33-8 

38.0 

26.5 

$6 

I.4I 

4.69 

9.38 

14.1 

18.8 

23-4 

28.1 

32.8 

37-5 

42.2 

23-9 

T 

'•55 
1.69 

5.16 
5.63 

10.3 
"•3 

16.9 

20.6 

22.5 

25.8 
28.1 

3°-9 
33-8 

36.1 

39-4 

41-3 
45-o 

46.4 
50.6 

21.7 
19.9 

J3/i6 

1.83 

6.09 

12.2 

18.3 

24.4 

30.5 

36-6 

42.7 

48.8 

54-9 

18.4 

% 

1.97 

6.56 

I3-  I 

19.7 

26.3 

32.8 

39-4 

45-9 

52.5 

59-1 

17.1 

'5/16 

2.  1  1 

7-03 

14.1 

21.  1 

28.1 

35-2 

42.2 

49-2 

56-3 

63.3 

15-9 

I 

2.25 

7.50 

15-0 

22.5 

30.0 

37-5 

45-° 

52-5 

60.0 

67.5 

14.9 

INCHES   WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

•594 

1.98 

3-96 

5-94 

7.92 

9.90 

11.9 

13-9 

I5.8 

I7.8 

56.6 

5/i6 

.742 
.891 

2.47 
2.97 

4-95 
5-94 

7.42 
8.91 

9.90 

ii.  9 

12.4 
14.8 

14.8 
17.8 

17-3 
20.8 

19.8 
23-8 

22.3 
26.7 

45-3 
37-7 

7/i6 

1.04 

3-46 

6-93 

10.4 

13-9 

17-3 

20.8 

24.2 

27.7 

31.2 

32.3 

% 

1.19 

3-96 

7.92 

11.9 

15.8 

I9.8 

23.8 

27.7 

31-7 

35-6 

28.3 

9/i6 

1-34 

4.45 

8.91 

13-4 

17.8 

22.3 

26.7 

31.2 

35-6 

40.1 

25.2 

H 

1.48 

4-95 

9.90 

14.8 

19.8 

24.7 

29.7 

34-6 

39-6 

44-5 

22.6 

1.67 

5-44 

10.9 

16.3 

21.8 

27.2 

32.7 

38.1 

43-5 

49.0 

20.6 

X 

1.78 

5-94 

11.9 

17.8 

23.8 

29.7 

35.6 

41.6 

47-5 

53-4 

18.9 

*3/i6 

i-93 

6.43 

12.9 

19-3 

25.7 

32.2 

38.6 

45-° 

51.5 

57-9 

17.4 

A 

2.08 

6-93 

13-9 

20.8 

27.7 

34-6 

41.6 

48.5 

55-4 

62.3 

16.2 

*5/i6 

2.23 

7.42 

14.8 

22.3 

29.7 

37-1 

44-5 

Si-9 

59-4 

66.8 

15-1 

I 

2.38 

7.92  i   15.8 

23.8 

31-7 

39-6 

47-5 

55-4 

63-3 

71-3 

14.2 

FLAT   BAR   IRON. 


231 


WEIGHT  OF  FLAT  BAR  IRON. 
zz  INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
o  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.625 

2.08 

4.17 

6.25 

8-33 

10.4 

12.5 

14.6 

I6.7 

18.8 

53-8 

5/i6 

.781 

2.60 

5.21 

7.8l 

10.4 

13.0 

15.6 

18.2 

20.8 

23-4 

43-° 

H 

7/i6 

.938 
.09 

3-13 
3-65 

6.25 
7.29 

9.38 
10.9 

I2-5 
14.6 

15.6 

18.2 

_0    0 

21.9 

21.9 

25.5 

25.0 
29.2 

32'-  8 

35-8 
30-7 

/* 

•25 

4.17 

8.33 

12.5 

16.7 

20.8 

25.0 

29.2 

33-3 

37-5 

26.9 

K 

.41 
•56 

4.69 
5.21 

9.38 
10.4 

I4.I 
I5.6 

18.8 

20.8 

23.4 

26.0 

28.1 
31.3 

32.8 
36.5 

37-5 
41.7 

42.2 
46.9 

23-9 
21.5 

.72 

5-73 

ii-5 

17.2 

22.9 

28.6 

34.4 

40.1 

45-8 

51.6 

19.6 

% 

.88 

6.25 

12.5 

18.6 

25.0 

31.3 

37.5 

43.8 

50.0 

56.3 

18.0 

I3/i6 

2.03 

6-77 

13.5 

20.3 

27.1 

33-8 

40.6 

47.4 

54-2 

60.9 

16.5 

£°6 

2.19 

7.29 

14.6 

21.9 

29.2 

36.5 

43-7 

51.0 

58.3 

65.7 

15-4 

2-34 

7.81 

15.6 

23-4 

31.3 

39-0 

46.9 

54.7 

62.5 

70.3 

14-3 

I 

2.50 

8.33 

16.7 

25.0 

33-3 

41.7 

50.0 

58.3 

66.7 

75-0 

13-4 

INCHES   WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.656 

2.19 

4.38 

6.56 

8.75 

10.9 

I3-I 

15.3 

17-5 

19.7 

51.2 

5/i6 

.820 

2.73 

5.47 

8.20 

10.9 

13-7 

16.4 

I9.I 

21.9 

24.6 

4I.O 

7/i6 

.984 
•15 

3.28 
3.83 

6.56 

7.66 

9.84 

ii-S 

i3-i 
15-3 

16.4 
I9.I 

19.7 
23.0 

23.0 
26.8 

26.2 
30.6 

29-5 
34-4 

34-2 
29-3 

^ 

•31 

4.38 

8-75 

13-1 

17-5 

21.9 

26.3 

30.6 

35-0 

39-4 

25.6 

9/16 

.48 

4.92 

9.84 

14.8 

19.7 

24.6 

29-5 

34-5 

39-4 

44-3 

22.8 

* 

.64 

5-47 

10.9 

16.4 

21.9 

27-3 

32.8 

38.3 

43-8 

49.2 

2O.5 

"/i6 

.81 

6.02 

12.0 

18.1 

24.1 

30.2 

36.1 

42.1 

48.1 

54-1 

18.6 

% 

•97 

6.56 

I3-I 

19.7 

26.3 

32.8 

39-4 

45-9 

52.5 

59-1 

17.1 

I3/i6 

2.13 

7.11 

14.2 

21.3 

28.4 

35-5 

42.7 

49.8 

56.9 

64.0 

15.8 

ft 

2.30 

7.66 

15.3 

23.0 

30.6 

38.3 

45-9 

53«6 

61.3 

68.9 

14.7 

iS/i6 

2.46 

8.20 

16.4 

24.6 

32-8 

41.0 

49-2 

57-4 

65.6 

73-8 

13.7 

I 

2.63 

8.75 

17.5 

26.3 

35-0 

43-8 

52.5 

61.3 

70.0 

78.8 

12.8 

INCHES   WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.688 

2.29 

4.58 

6.87 

9.17 

"•5 

13.8 

16.1 

I8.3 

20.6 

48.9 

5/i6 

.859 

2.86 

5.73 

8-59 

II-5 

14-3 

17.2 

20. 

22.9 

25.8 

39-1 

H 

.03 

3-44 

6.88 

10.3 

13-8 

17.2 

20.6 

24. 

27-5 

30-9 

32.8 

7/,6 

.20 

4.01 

8.02 

12.0 

16.0 

20.1 

24.1 

28. 

32.1 

36.1 

27.9 

X 

.38 

4.58 

9.17 

13-8 

18.3 

22.9 

27.5 

32. 

36.7 

41-3 

24-4 

9/i6 

•55 

5.16 

10.3 

15-5 

20.6 

25.8 

30.9 

36. 

41-3 

46.4 

21.7 

H 

•72 

5-73 

ii-S 

17.2 

22.9 

28.6 

34-4 

40. 

45-8 

51.6 

19.5 

"/i6 

.89 

6.30 

12.6 

18.9 

25.2 

31-5 

37-8 

44. 

50.4 

56.7 

17.8 

% 

2.06 

6.88 

13.8 

20.6 

27-5 

34-4 

41.3 

48. 

55-o 

61.9 

16.3 

x3/i6 
K 

2.23 
2.41 

7-45 

8.02 

14.9 

16.0 

22.3 
24.1 

29.8 
32.1 

37-2 
40.1 

44-7 
48.1 

52.1 

56.1 

••59-6 
64.2 

67.0 

72.2 

15.0 
14.0 

*S/i6 

2.58 

8-59 

17.2 

25.8 

34.4 

43-° 

51.6 

60.  i 

68.8 

77.3 

13.0 

I 

2-75 

9.17 

18.3 

27.5 

36.7 

45-8 

55-o 

64.2 

73-3 

82.5 

12.2 

232 


WEIGHT   OF   METALS. 


WEIGHT  OF  FLAT  BAR  IRON. 

?.      INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
o  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.719 

2.40 

4-79 

7.19 

9.58 

12.0 

14.4 

16.8 

19.2 

21.6 

46.7 

5/i6 

.898 

3.00 

6.00 

9.00 

12.0 

I5.O 

18.0 

21.0 

24.0 

27.0 

37-4 

H 

.08 

3-59 

7.19 

10.8 

14.4 

18.0 

21.6 

25.2 

28.8 

32-3 

31.2 

7/i6 

.26 

4.19 

8-39 

12.6 

16.8 

21.0 

25.2- 

29.4 

33.5 

37-7 

26.7 

X  . 

•44 

4-79 

9.58 

14.4 

19.2 

24.0 

28.8 

33-5 

38.3 

43-1 

23-4 

9/i6 

.62 

5-39 

10.8 

16.2 

21.6 

27.0 

32.3 

37-7 

43-i 

48.5 

20.8 

k 

.80 

6.00 

12.0 

18.0 

24.0 

30.0 

36.0 

42.0 

48.0 

54-o 

18.7 

"Ii6 

.98 

6-59 

13.2 

19.8 

26.4 

33-0 

40.5 

46.1 

52.7 

59-3 

17.0 

H 

2.16 

7.19 

14.4 

21.6 

28.8 

36.0 

43-i 

50-3 

57-5 

6f7 

15.6 

J3/i6 

2-34 

7-79 

I5.6 

23-4 

3M 

39-o 

46.7 

54-5 

62.3 

7p."i 

14.4 

7A 

2.52 

8.39 

16.8 

25.2 

33-5 

42.0 

50-3 

58.7 

67.1 

75-5 

13-4 

'5/16 

2.70 

8.98 

18.0 

27.0 

35-9 

45-° 

53-9 

62.9 

71.9 

80.9 

12.4 

I 

2.88 

9-58 

19.2 

28.8 

38.3 

48.0 

57-5 

67.1 

76.7 

86.3 

11.7 

3  INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.750 

2.50 

5.00 

7-50 

IO.O 

12.5 

15  o 

17-5 

20.0 

22.5 

44.8 

5/i6 

.938 

3-13 

6-25 

9.38 

12.5 

16.7 

18.8 

21.9 

25.0 

28.1 

35-8 

& 

•13 

3-75 

7.50 

H3 

15.0 

18.8 

22.5 

26.3 

3O.O 

33-8 

29.9 

7/i6 

•31 

4.38 

8.75 

I3-I 

17.5 

21.9 

26.3 

30.6 

35-o 

39-4 

25.6 

y* 

•50 

5-00 

IO.O 

15.0 

20.0 

25.0 

30.0 

35-o 

40.0 

45-° 

22.4 

9/i6 

.69 

6? 

"•3 

16.9 

22.5 

28.2 

33-8 

39-4 

4S-o 

50.6 

19.9 

H 

.88 

12.5 

18.8 

25.0 

31-3 

37-5 

43-8 

50.0 

56.3 

17.9 

II/i6 

2.06 

6!  88 

13-8 

20.6 

27-5 

34-4 

41-3 

48.1 

55-o 

61.9 

16.3 

% 

2.25 

7-50 

15.0 

22.5 

30.0 

37-5 

45-o 

52.5 

60.0 

67.5 

14.9 

'3/16 

2.44 

8.13 

16.3 

24.4 

32-5 

40.7 

48.8 

56.9 

65.0 

73-i 

13-8 

% 

2.63 

8-75 

17.5 

26.3 

35-° 

43-8 

52.5 

61.3 

70.0 

78.8 

12.8 

*5/i6 

2.81 

9-38 

18.8 

28.1 

37-5 

46.9 

56-3 

65.6 

75-0 

84.4 

12.0 

I 

3.00 

10.  0 

20.0 

30.0 

40.0 

50.0 

60.0 

70.0 

80.0 

90.0 

II.  2 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.813 

2.71 

5-42 

8.13 

10.8 

13.6 

16-3 

19.0 

21.7 

24.4 

41-3 

5/i6 

1.02 

3-39 

6.77 

10.2 

13-5 

16.9 

20.3 

23.7 

27.1 

30.5 

33-1 

y$> 

1.22 

4.06 

8.13 

12.2 

16.3 

20.3 

24.4 

28.4 

32.5 

36.6 

27.5 

7/16 

1.42 

4-74 

9.48 

14.2 

19.0 

23.7 

28.4 

33-2 

37-9 

42.7 

23-6 

A- 

1.63 

5-42 

10.8 

16-3 

21.7 

27.1 

32.5 

37-9 

43-3 

48.8 

20.7 

9/i6 

1.83 

6.09 

12.2 

18.3 

24.4 

30.5 

36.6 

42.7 

48.7 

54.8 

18.4 

'/S 

2.03 

6.77 

13-5 

20.3 

27.1 

33-9 

40.6 

47-4 

54-2 

60.9 

16.5 

2.23 

7-45 

14.9 

22-3 

29.8 

37-2 

44-7 

52.1 

59-6 

67.0 

15.0 

% 

2.44 

8.13 

I6.3 

24.4 

32-5 

40.6 

48.8 

56.9 

65.0 

73-  l 

13-7 

i3/i6 

2.64 

8.80 

I7.6 

26.4 

35-2 

44.0 

52.8 

61.6 

70.4 

79.2 

12.7 

H 

2.84 

9.48 

19.0 

28.4 

37-9 

47-4 

56.9 

66.4 

75-8 

85-3 

n.8 

15/i6 

3-05 

10.2 

20.3 

30.5 

40.6 

50.8 

60.9 

71.1 

81.2 

91.4 

II.  0 

I 

3-25 

10.8 

21.7 

32.5 

43-3 

54-2 

65.0 

75-8 

86.7 

97-5 

10.3 

1 

FLAT   BAR   IRON. 


233 


WEIGHT  OF  FLAT  BAR  IRON. 
£  INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

•875 

2.92 

5.83 

8.75 

ii.  7 

14.6 

17-5 

20.4 

23-3 

26.3 

38.4 

5/i6 

1.09 

3.65 

7.29 

10.9 

14.6 

18.2 

21.9 

25.5 

29.2 

32.8 

30-7 

H 

'•Si 

4.38 

8.75 

I3-I 

17-5 

21.9 

26.3 

30.6 

35.0 

39-4 

25.6 

7/i6 

1.53 

5.10 

10.2 

15-3 

20.4 

25-5 

30.6 

35-7 

40.8 

45-9 

21.9 

X 

i-75 

5.83 

ii.  7 

17-5 

22.3 

29.2 

35-0 

40.8 

46.7 

52.5 

19.2 

9/i6 

1.97 

6.56 

13-1 

19.7 

26.3 

32.8 

39-4 

45-9 

52.5 

59-1 

I7.I 

# 

2.19 

7.29 

14.6 

21.9 

29.2 

36.5 

43-7 

51.0 

58.3 

65.6 

15-4 

»/x6 

2.41 

8.02 

16.0 

24.1 

32.1 

40.1 

48.1 

56.1 

64.2 

72.2 

14.0 

# 

2.63 

8.75 

17-5 

26.3 

35-o 

43-8 

52.5 

61.3 

70.0 

78.8 

12.8 

I3/i6 

2.84 

9.48 

19.0 

28.4 

37-9 

47-4 

56.9 

66.4 

75-8 

85.3 

II.9 

$ 

3-o6 

IO.2 

20.4 

30.6 

40.8 

51.0 

61.2 

7i-5 

81.6 

91.9 

II.  O 

'5A6 

3-28 

IO.9 

21.9 

32.8 

43-8 

54-7 

65.6 

87.5 

98.4 

10.2 

I 

3-50 

II.7 

23-3 

35-o 

46.7 

58.3 

70.0 

8i!7 

93-3 

105.0 

9.6O 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

.938 

3.13 

6.25 

9.38 

12.5 

15.6 

18.8 

21.9 

25.0 

28.1 

35-8 

5/i6 

I.I7 

3.91 

7-8l 

II.7 

15.6 

19.5 

23.4 

27.3 

31.3 

35-2 

28.7 

3/£ 

I.4I 

4.69 

9.38 

I4.I 

18.8 

23.4 

28.1 

32.8 

37-5 

42.2 

23-9 

7/x6 

1.64 

5-47 

10.9 

16.4 

21.9 

27.3 

32.8 

38.3 

43-7 

49-2 

20.5 

K 

1.88 

6.25 

12.5 

18.8 

25.0 

31.3 

37-5 

43-8 

50.0 

56.3 

17.9 

9/i  6 

2.  1  1 

7.03 

I4.I 

21.  1 

28.1 

35.3 

42.2 

49.2 

56.3 

63.3 

15-9 

S^ 

2-34 

7.81 

I5.6 

23.4 

31.2 

39-1 

46.9 

54-7 

62.5 

70.3 

14-3 

n/I6 

2.58 

8.59 

17.2 

34-4 

43-0 

51.6 

60.2 

68.8 

77-3 

13-0 

X 

2.81 

•9.38 

18.8 

28.1 

37-5 

46.9 

56-3 

65.6 

75-° 

84-4 

I2.O 

J3/i6 

3.05 

10.2 

20.3 

30-5 

40.6 

50.8 

60.9 

71.1 

81.3 

91.4 

II.  0 

£/x6 

3.28 
3-52 

IO.9 
II-7 

21.9 
23-4 

32-8 
35-2 

43-8 
46.9 

%l 

65.6 
70.3 

76.6 
82.0 

87.5 
93-7 

98.4 
105.5 

10.2 
9-56 

I 

3-75 

12-5 

25-0 

37-5 

50.0 

62.5 

75-0 

87.5 

IOO.O 

112.5 

8.96 

4  INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

lbs.1 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

I.OO 

3-33 

6.67 

10.  0 

13-3 

I6.7 

20.0 

23.3 

26.7 

30.0 

33.6 

5/i6 

1-25 

4.17 

8-33 

12.5 

20.8 

25.0 

29.2 

33-3 

37.5 

26.9 

H 

1.50 

5.00 

10.0 

15.0 

20.0 

25.0 

30.0 

35.0 

40.0 

45-° 

22.4 

7/i6 

1-75 

5-83 

ii.  7 

17.5 

23.3 

29.2 

35-0 

40.8 

46.7 

52.5 

I9.2 

X 

2.OO 

6.67 

13.3 

20.0 

26.7 

33.3 

40.0 

46.7 

53-3 

60.0 

16.8 

9/i6 

2.25 

7-50 

15.0 

22.5 

3O.O 

37.5 

45-0 

52.5 

60.0 

67.5 

14.9 

"As 

2.50 
2-75 

8-33 
9.17 

3 

25.0 

27-5 

33-3 
36.7 

41.7 

45-8 

50.0 
55-o 

64.2 

66.7 
73-3 

75-o 
82.5 

13.4 

12.2 

X 

3-00 

10.0 

20.0 

3O.O 

40.0 

50.0 

60.0 

70.0 

80.0 

90.0 

II.  2 

x3/x6 

3.25 

10.8 

21.7 

32-5 

43-3 

54-2 

65.0 

75.8 

86.7 

97-5 

10.3 

% 

3-50 

ii.  7 

23.3 

35-0 

46.7 

58.4 

70.0 

81.7 

93-3 

105.0 

9.60 

'5/i6 

3-75 

12.5 

25.O 

37-5 

50.0 

62.5 

75.o 

87.5 

IOO.O 

112.5 

8.96 

I 

4.00 

13-3 

26.7 

40.0 

53-3 

66.7 

80.0 

93-3 

106.7 

120.0 

8.40 

234 


WEIGHT   OF   METALS. 


WEIGHT  OF  FLAT  BAR  IRON. 
4"  INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

1.  06 

3-54 

7.08 

10.6 

14.2 

17.7 

21.3 

24.8 

28.3 

31-9 

31-6 

5/i6 

i-33 

4-43 

8.85 

13-3 

17.7 

22.1 

26.6 

31.0 

35-4 

39-8 

25.3 

H 

i-59 

5-3i 

10.6 

15-9 

21.3     26.6 

31-9 

37-2 

42.5 

47.8 

21.  1 

7/16 

1.85 

6.20 

12.4 

24.8      31.0 

37.2 

43-4 

49.6 

55-8 

18.1 

l/2 

2.13 

7.08 

14.2 

21.3 

28.3 

35-4 

42.5 

49.6 

56.7 

63-8 

15.8 

t 

2-39 
2.66 

7-97 
8.85 

15-9 
17.7 

23-9 
26.6 

31.9 

35.4 

39-8 
44-3 

47.8 

53-1 

55-8 
62.0 

63.7 
70.8 

71.7 

79-7 

14.1 
12.7 

«/i6 

2.92 

9-74 

19-5 

29.2 

39.0 

48.7 

58.4 

68.2 

77-9 

87.7 

11.5 

H 

3-19 

10.6 

21.3 

3i-9 

42.5 

53-1 

63.8 

74-4 

85.0 

95-6 

10.5 

*3/i6 

3-45 

H.5 

23.0 

34-5 

46.0 

57.6 

69.1 

80.6 

92.1 

103.6 

9-9 

ft 

3-72 

12.4 

24.8 

37-2 

49.6 

62.0 

74-4 

86.8 

99.2 

in.  6 

9.0 

*S/x6 

3.98 

13-3 

26.6 

39-8 

53.1 

66.4 

79-7 

93-0 

106.2 

"9-5 

8.4 

I 

4-25 

14.2 

28.3 

42.5 

56.7 

70.8 

85.0 

99-2 

"3-3 

127-5 

7-9 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

$.13 

3-75 

7.5 

II-3 

15.0 

18.8 

22.5 

26.3 

30.0 

33-8 

29.9 

5/i6 

I.4I 

4.69 

9.38 

I4.I 

18.8 

23-4 

28.1 

32.8 

37-5 

42.2 

23.9 

H 

1.69 

5-63 

"•3 

16.9 

22.5 

28.1 

33.8 

39-4 

45-° 

50.6 

19.9 

7/i6 

1.97 

6.56 

13-1 

19.7 

26.3 

32.8 

39-4 

45-9 

52.5 

59-1 

I7.I 

A. 

2.25 

7.50 

15.0 

22.5 

30.0 

37-5 

45-° 

52.5 

60.0 

67.5 

14.9 

9/i6 

2-53 

8-44 

16.9 

25-3 

33-8 

42.2 

50.6 

59  I 

67.5 

75-9 

13-3 

H 

2.8l 

9.38 

18.8 

28.1 

37-5 

46.9 

56-3 

65.6 

75>° 

84.4 

12.0 

3.09 

10.3 

20.6 

30.9 

41-3 

51.6 

61.9 

72.2 

82.5 

92.8 

10.9 

X*6 

3.38 

II-3 

22.5 

33-8 

45-° 

56-3 

67.5 

78.8 

90.0 

101.3 

9-95 

13/16 

3-66 

12.2 

24.4 

36.6 

48.8 

60.9 

73-1 

85.3 

97-5 

109.7 

9.19 

% 

3-94 

'3-  I 

26.3 

39-4 

52-5 

65.6 

78.8 

91.9 

105.0 

118.1 

8-53 

J5/i6 

4.22 

14.1 

28.1 

42.2 

56.3 

70.3 

84.4 

98.4 

112.5 

126.6 

7.96 

I 

4.50 

15.0 

30.0 

45-° 

60.0 

75-0 

90.0 

105.0 

120.0 

i35-o 

7.46 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

5/i6 

I.I9 

1.48 

3.96 

4-95 

7.92 
9.90 

II.9 
14.8 

15.8 

19.8 

19.8 
24.8 

23.8 

29.7 

27.7 

34.6 

31.7 

39.6 

35.6 

44.4 

28.3 
22.6 

X 

1.78 

5-94 

11.9 

I7.8 

23.8 

29.7 

35-6 

41.6 

47.5 

53-4 

18.9 

7/i6 

2.08 

6-93 

13-9 

20.8 

27.7 

34-7 

41.6 

48.5 

55-4 

62.3 

16.2 

X 

2.38 

7.92 

15.8 

23.8 

31-7 

39-6 

47-5 

55.4 

63.3 

7L3 

14.2 

9/i6 

2.67 

8.91 

17.8 

26.7 

35-6 

44.6 

53-4 

62.3 

71-3 

80.2 

12.6 

H 

2.97 

9.90 

19.8 

29.7 

39-6 

49-5 

59-4 

69.3 

79.2 

89.1 

II.3 

«/i6 

3-27 

10.9 

21.8 

32.7 

4-3.5 

54-5 

65-3 

76.2 

87.1 

98.0 

10.3 

* 

3-56 

11.9 

23-8 

35-6 

47-5 

59-4 

7i-3 

83.1 

95-  o 

106.9 

9.4 

13/16 

3-86 

12.9 

25-7 

38.6 

5L5 

64-3 

77.2 

90.1 

102.9 

115.8 

8.7 

% 

4.16 

13-9 

27.7 

41.6 

55-4 

69-3 

97.0 

uo.8 

124.7 

8.1 

'5/16 

4-45 

14.8 

29.7 

44-5 

59-4 

74.2 

89/1 

103.9 

118.8 

133.6 

7-5 

1 

4-75 

15.8 

31.7 

47.5 

63.3 

79-2 

95.° 

1  10.  8 

126.7 

142.5 

7-i 

FLAT   BAR   IRON. 


235 


WEIGHT  OF  FLAT  BAR  IRON. 
5  INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

1.25 

4.17 

8-33 

12.5 

16.7 

20.9 

25.0 

29.2 

33-3 

37-5 

26.9 

5/i6 

1.56 

5.21 

10.4 

IS.6 

20.8 

26.1 

31-3 

36.5 

41.7 

46.9 

21-5 

ft 

1.88 

6.25 

12-5 

18.8 

25.0 

31-3 

37-5 

43-8 

50.0 

56.3 

17.9 

7/16 
£ 

2.19 

2.50 

7.29 
8-33 

14.6 
I6.7 

21.9 

25.0 

29.2 
33-3 

36.5 
41.7 

43-8 
50.0 

51.0 

58.3 

58,3 
66.7 

65.6 
75-0 

15-4 
13-4 

9/i6 

2.81 

9.38 

18.8 

28.1 

37-5 

46.9 

56.3 

65.6 

75-o 

84.4 

12.0 

X 

3-13 

10.4 

20.8 

3i-3 

41.7 

52.1 

62.5 

72.9 

83-3 

93-8 

10.8 

"/i6 

3-44 

"•3 

22.9 

34-4 

45-8 

57-3 

68.8 

80.2 

91.7 

103.1 

9-77 

K 

3-75 

12.5 

25.0 

37-5 

50.0 

62.5 

75-0 

87.5 

IOO.O 

112.5 

8.96 

*3/i6 

4.06 

13-5 

27.1 

40.6 

54-2 

67.7 

81.3 

94-8 

108.3 

121.9 

8.27 

% 

4-38 

14.6 

29.2 

43-8 

58.3 

72.9 

87-5 

102.  1 

116.7 

J3I-3 

7.68 

*S/*6 

4.69 

15.6 

31.3 

46.9 

62.5 

78.i 

93-8 

109.4 

125.0 

140.6 

7.17 

I 

5.00 

16.7 

33.3 

50.0 

66.7 

83-3 

100.0 

II6.7 

133-3 

150.0 

6.72 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

5/i6 

1.64 

4.38 
5-47 

8.75 
10.9 

If-1 

16.4 

17-5 
21.9 

21.9 

27.3 

26.3 
32-8 

30.6 
38-3 

35-o 
43-8 

39-4 

49.2 

25.6 

20.5 

# 

1.97 

6.56 

13-1 

19.7 

26.3 

32.8 

39-4 

45-9 

52-5 

59-  * 

I7-I 

7/i6 
# 

2.JO 
2.63 

7.66 

8.75 

iS-3 
i7.5 

23.0 

26.3 

30.6 
35-o 

38.3 
43.8 

45-9 
52-5 

53-6 

61.3 
70.0 

fs.l 

14.6 
12.8 

9/i6 

2-95 

9-84 

19.7 

29.5 

39-4 

49.2 

59-1 

68.9 

78.8 

88.6 

II.4 

X 

3.28 

10.9 

21.9 

32.8 

54-7 

65.6 

76.6 

87.5 

98.4 

10.3 

"/i6 

3-61 

12.0 

24.1 

36.1 

48.1 

60.2 

72.2 

84.2 

96.3 

108.3 

9-31 

X 

3-94 

!3-! 

26.3 

39-4 

52.5 

65.6 

78.8 

91.9 

105.0 

118.1 

8.55 

*3/i6 

4.27 

14.2 

28.4 

42.7 

56.9 

71.1 

85.3 

99-5 

"3-7 

128.0 

7.88 

h 

4-59 

'5-3 

30.6 

45-9 

61.3 

76.6 

91.9 

107.2 

122.5 

137.8 

7-31 

"5/16 

4.92 

16.4 

32.8 

49.2 

65.6 

82.0 

98.4 

114.8 

W-3 

147-7 

6.83 

I 

5-25 

'7-5 

35-o 

52.5 

70.0 

87.5 

105.0 

122.5 

140.0 

157-5 

6.40 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

1.38 

4.58 

9.17 

13.8 

18-3 

22.9 

27-5 

32.1 

36.7 

41.3 

24-5 

5/i6 

1.72 

5-73 

II.5 

17.2 

22.9 

28.6 

34-4 

4O.I 

45-8 

51.6 

19-5 

3 

7/i6 

2.06 
2.41 

8.02 

16.0 

20.6 

24.1 

27.5 

34-4 
40.1 

41-3 
48.1 

48.1 

64.2 

61.9 
72.2 

16.4 
14.0 

X 

2-75 

9.17 

18.3 

27.5 

36-7 

45-8 

55-o 

64.2 

73-3 

82.5 

12.2 

9/i6 

3-09 

10.3 

20.6 

30.9 

41-3 

51.6 

61.9 

72.2 

82.5 

92.8 

10.9 

H 

3-44 

11.5 

22.9 

34-4 

45-8 

57-3 

68.8 

80.2 

91.7 

103.1 

9-77 

^ 

3-78 
4-13 

12.6 

13.8 

25.2 

27.5 

37-8 
4i-3 

50.4 
55-o 

63.0 
68.8 

i$2.5 

88.2 
96.3 

100.8 

IIO.O 

"3-4 
123.8 

8.14 

*3/i6 

4-47 

14.9 

29.8 

44-7 

59-6 

74-5 

89.4 

104.3 

119.2 

I34.I 

7-52 

H 

4.81 

16.0 

32.1 

48.1 

64.2 

80.2 

96.3 

112.3 

128.3 

144.4 

6.98 

*S/i6 

5.16 

17.2 

34-4 

51.6 

68.8 

85.9 

103.1 

120.3 

137-5 

154-7 

I 

5-50 

18.3 

36.7 

55-o 

73-3 

91.6 

no.o    128.4 

146.7 

165.0 

6  ii 

236 


WEIGHT   OF   METALS. 


WEIGHT  OF  FLAT  BAR  IRON. 
INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

# 

1.44 

4-79 

9.58 

14.4 

19.2 

24.0 

28.8 

33-5 

38.3 

43-  i 

23-4 

5/i6 

X 

1.  80 
2.16 

5-99 
7.19 

12.0 
14.4 

18.0 

21.6 

24.0 
28.8 

30.0 
35-9 

35-9 
43-i 

41.9 
50-3 

47.9 
57.5 

64.7 

18.7 
15.6 

7/i6 

X 

2.52 

2.88 

9^58 

16.8 
19.2 

25.2 
28.8 

33-5 
38.3 

41.9 
47-9 

5°-3 
57-5 

ft! 

67.1 
76.7 

£! 

13.4 
II.7 

9/i6 

3-23 

10.8 

21.6 

32-3 

43-1 

53-9 

64.7 

75-5 

86.2 

97.0 

IO.4 

H 

3-59 

12.0 

24.0 

36.0 

48.0 

60.0 

71.9 

83-9 

95-8 

107.8 

9-35 

"/i6 

3-95 

13.2 

26.4 

39-5 

52.7 

65-9 

79.1 

92.2 

105.4 

118.6 

8.50 

1? 

4-31 

14.4 

28.8 

43-i 

57-5 

71.9 

86.3 

100.6 

115.0 

129.4 

7-79 

J3/i6 

4.67 

15.6 

31.2 

46.7 

62.3 

77-9 

93-4 

109.0 

124.6 

140.2 

7.19 

% 

5-°3 

16.8 

33-5 

50.3   |  67.0 

83-9 

100.7 

117.4 

134.2 

150.9 

6.68 

JS/i6 

5-39 

18.0 

35-9 

53-9 

71.9 

89.8 

107.8 

125.8 

143-7 

161.7 

6.22 

1 

5-75 

19.2 

38-3 

57-5 

76.7 

95-8 

115.0 

134-2 

153-3 

172.5 

5-83 

6  INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs.     • 

Ibs. 

Ibs. 

.  Ibs. 

Ibs. 

feet. 

X 

1.50 

5.00 

IO.O 

15.0 

2O.  O 

25.0 

30.0 

35-0 

40.0 

45-0 

22.4 

5/i6 

1.88 

6.25 

12-5 

_  O    O 

25.0 

31-8 

37.5 

43.8 

5O.O 

56.3 

17.9 

H 

2.25 

7-50 

15.0 

22^5 

3O.O 

37-5 

45.° 

52.5 

60.0 

67-5 

14.9 

7/i6 

2.63 

8.75 

17-5 

26.3 

35-0 

43-8 

52.5 

61.3 

7O.O 

78.8 

12.8 

k 

3.00 

10.0 

20.0 

30.0 

40.0 

50.0 

60.0 

70.0 

80.0 

90.0 

II.  2 

9/i6 

3-38 

"•3 

22.5 

33-8 

45-0 

56-3 

67.5 

78.8 

9O.O 

IOI.3 

IO.O 

H 

3-75 

12.5 

25.0 

37-5 

50.0 

62.5 

75-o 

87.5 

1  00.0 

II2.5 

8.96 

n/i6 

4.13 

13-8 

27-5 

41-3 

55-0 

68.8 

82.5 

96.3 

I  IO.O 

123.7 

8.15 

X 

4-5° 

15.0 

30.0 

45-° 

60.0 

75-o 

90.0 

105.0 

120.0 

135-0 

7-47 

13/16 

4.88 

16.3 

32.5 

48.8 

65.0 

81.3 

97-5 

113.7 

130.0 

146.3 

6.90 

ft 

5.25 

17-5 

35-0 

52.5 

70.0 

87-5 

105.0 

122.5 

140.0 

157.5 

6.40 

15/16 
i 

5.63 
6.00 

18.8 

20.0 

37-5 
40.0 

56-3 

60.0 

80!  o 

93-8 

100.0 

112.5 

120.0 

131.3 

140.0 

150.0 
1  60.0 

168.7 
l8o.O 

5-97 
5-6o 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

1.63 

5-42 

10.8 

16.3 

21.7 

27.2 

32.5 

37-9 

43-3 

49.0 

20.7 

5/i6 

2.03 

6.77 

13-5 

20.3 

27.1 

33-9 

40.6 

47-4 

54-2 

60.9 

16.5 

3A 

2.44 

8.13 

16.3 

24.4 

32.5 

40.6 

48.8 

56.9 

65.0 

73-  I 

I3.8 

7/i6 

2.84 

9-47 

18.9 

28.4 

37-9 

47-4 

56.8 

66.3 

75-8 

85-2 

14.8 

# 

3-25 

10.8 

21.7 

32.5 

43-3 

54-2 

65.0 

75-8 

86.7 

97-5 

10.3 

9/i6 

3.66 

12.2 

24.4 

36.6 

48.8 

60.9 

73-1 

85-3 

97-5 

109.7 

9-2O 

X 

4.06 

13-5 

27.1 

40.6 

54-2 

67.7 

94-8 

108.3 

121.9 

8.27 

"/t6 

4-47 

14.9 

29.8 

44-7 

59-6 

74-5 

89*4 

104.3 

119.2 

i34-i 

7-52 

H 

4.98 

I6.3 

32-5 

48.8 

65.0 

81.3 

97-5 

113-8 

130.0 

146.3 

6.89 

J3/i6 

5.28 

I7.6 

35-2 

52.8 

70.4 

88.0 

105.6 

123.2 

140.8 

158.4 

6.36 

% 

5.68 

19.0 

37-9 

56.9 

75-8 

94.8 

113-8 

132.7 

151-7 

170.6 

5-91 

15/X6 

6.09 

2O.3 

40.6 

60.9 

81.3 

101.6 

121.9 

142.8 

162.5 

182.8 

5.51 

I 

6.50 

21.7 

43-3 

65.0 

86.7 

108.3 

130.0 

I5L7 

173-3 

195.0 

5-29 

FLAT   BAR    IRON, 


237 


WEIGHT  OF  FLAT  BAR  IRON. 
7  INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

i-75 

5-83 

ii.  7 

17.5 

23.3 

29.2 

35-o 

40.8 

46.7 

52-5 

19.2 

5/i6 

2.19 

7.29 

14.6 

21.9 

29.2 

36.5 

43-8 

51.0 

58-3 

65.6 

!5-4 

H 

2.63 

8.75 

17-5 

26.3 

35-o 

43-8 

52.5 

61.3 

70.0 

78.8 

12.8 

7/i6 

3-06 

10.2 

20.4 

30.6 

40.8 

51-0 

61.3 

71.5 

81.7 

91.9 

II.  O 

*/2 

3-50 

11.7 

23-3 

35-0 

46.7 

58.3 

70.0 

81.7 

93-3 

105.0 

9.60 

9/i6 

3-94 

13-1 

26.3 

39-4 

52.5 

65.6 

78.8 

91.9 

105.0 

118.1 

8-53 

"/i6 

4.38 
4.81 

14.6 
16.0 

29.2 
32.1 

43-8 
48.1 

58-3 
64.2 

72.9 
80.2 

87.5 
96.3 

102.  1 
II2.3 

116.7 

128.3 

I3L3 

144.4 

7-68 
6.98 

X 

5.25 

17-5 

35-0 

52.5 

70.0 

87.5 

105.0 

122-5 

140.0 

157.5 

6.40 

J3/i6 

5.69 

19.0 

37-9 

56.9 

75.8 

95-0 

113.8 

132.7 

I5I.7 

170.6 

5-91 

^ 

6.13 

20.4 

40.8 

61.3 

81.7 

102.  1 

122.5 

142.9 

163-3 

183.8 

5-49 

i5/i6 

6.56 

21.9 

43-8 

65.6 

87.5 

109.4 

131-3 

I53-I 

175.0 

196.9 

5.12 

I 

7.00 

23-3 

46.7 

70.0 

93-3 

II6.7 

140.0 

163.3 

186.7 

2IO.O 

4.80 

INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

1.88 

6.25 

12.5 

18.8 

25.0 

31-3 

37-5 

43-8 

50.0 

56-3 

17.9 

5/16 

2-34 

7.81 

15.6 

23.4 

31-3 

39.1 

46.9 

54-7 

62.5 

70.3 

14-3 

7/16 

2.81 
3-28 

9.38 

10.9 

18.8 
21.9 

28.1 
32.8 

37.5 
43.8 

46.9 

54-7 

56-3 
65-6 

76.'6 

75-0 
87-5 

84.4 
98.4 

II.9 

10.2 

X 

3-75 

12.5 

25.0 

37-5 

50.0 

62.5 

75-0 

87-5 

100.0 

112.5 

8.96 

r 

4.22 
4-69 

14.1 
15.6 

28.1 
31.3 

42.2 
46.9 

56.3 

62.5 

70.3 
78.1 

84-4 
93-8 

98.4 
109.4 

112.5 
125.0 

126.6 
140.6 

7.96 
7.17 

"/i6 

5.16 

17.2 

34-4 

51.6 

68.8 

85.9 

103.1 

120.3 

137.5 

154.7 

6.52 

K 

5.63 

18.8 

37-5 

56.3 

75-0 

93-8 

112.5 

131-3 

150.0 

168.8 

5-97 

J3/i6 

6.09 

20.3 

40.6 

60.9 

8i-3 

101.6 

121.9 

142.2 

162.5 

182.8 

5-51 

7/s 

6.56 

21.9 

43-8 

65.6 

87.5 

109.4 

131.3 

I53-I 

175.0 

196.9 

5-12 

*S/x6 

7-03 

23-4 

46.9 

70.3 

93-8 

117.2 

140.6 

164.1 

187.5 

2IO.9 

4.78 

I 

7-50 

25.0 

50.0 

75-0 

100.0 

125.0 

150.0 

175.0 

200.0 

225.0 

4.48 

8  INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

5/i6 

2.00 
2.50 

6.67 

8-33 

13.3 
16.7 

20.0 
25.0 

26.7 

33-3 

33-3 

41.7 

40.0 
50.0 

46.7 
58.3 

a? 

60.0 

75-o 

16.8 
13-4 

H 

3.00 

IO.O 

2O.  O 

30.0 

40.0 

50.0 

60.0 

70.0 

80.0 

90.0 

II.  2 

7/i6 

/2 

3.50 
4.OO 

II.7 
13-3 

23.3 
26.7 

35-0 
40.0 

46.7 
53.3 

58.3 
66.7 

70.0 

80.0 

8l.7 
93-3 

93-3 
106.7 

105.0 

I2O.O 

9.60 
8.40 

9/i6 

4-5° 

15.0 

30.0 

45-o 

60.0 

75-0 

90.0 

105.0 

120.0 

135-0 

7-47 

H 

5-00 

16.7 

33-3 

50.0 

66.7 

83.3 

100.0 

II6.7 

133-3 

150.0 

6.72 

t 

5-50 
6.00 

I8.3 

20.  o 

36.7 
40.0 

55.0 
60.0 

73.3 
80.0 

91.7 

100.0 

I  IO.O 
I2O.O 

128.3 
140.0 

146.7 
1  60.0 

165.0 
ISO.O 

6.ii 
5.60 

'3/i6 

6.50 

21.7 

43-3 

65.0 

86.7 

108.3 

130.0 

I5L7 

173-3 

195.0 

5-J7 

H 

7.00 

23.3 

46.7 

70.0 

93-3 

116.7 

I4O.O 

163.3 

186.7 

2  IO.O 

4.80 

x5/i6 

7.50 

25.0 

50.0 

75-0 

100.0 

125.0 

150.0 

175.0 

200.0 

225.0 

4.48 

1 

8.00 

26.7 

53-3 

80.0 

106.7 

133.3 

160.0 

186.7 

213.3 

24O.O 

4.20 

238 


WEIGHT   OF   METALS. 


WEIGHT  OF  FLAT  BAR  IRON. 
9  INCHES  WIDE. 


THICK- 
NESS. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

2.25 

7.50 

15.0 

22.5 

30.0 

37.5 

45-o 

52.5 

60.0 

67-5 

14.9 

S/i6 

2.8l 

9.38 

18.8 

28.1 

37-5 

46.9  i 

56.3 

65.6 

75-0 

84.4 

II.9 

X 

3.38 

u-3 

22.5 

33-8 

45-0 

56.3 

67.5 

78.8 

9O.O 

101.3 

10.0 

7/i6 

3-94 

26.3 

39-4 

52.5 

65.6 

78.8 

91.9 

105.0 

118.1 

8-53 

A. 

4-5° 

15.0 

30.0 

45-0 

60.0 

75-0 

90.0 

105.0 

I2O.O 

I35-o 

7-47 

9/i6 

5-06 

16.9 

33-8 

50.6 

67.5 

84.4 

101.3 

118.1 

135-0 

151.9 

6.64 

X 

5.63 

1  8.  8 

37.5 

56-3 

75-0 

93-8 

112.5 

131.3 

150.0 

168.8 

5-97 

6.19 

20.6 

41.3 

61.9 

82.5 

103.1 

123-8 

144.4 

165.0 

185.6 

5-43 

% 

6.75 

22.5 

45-o 

67.5 

90.0 

112.5 

157.5 

iSo.O 

202.5 

4.98 

J3/i6 

77:& 

24.4 
26.3 

48.8 
52.5 

7!  8 

97-5 
105.0 

121.9 
I3I-3 

146.3 
104.5 

170.6 
183.8 

195-0 
210.0 

219.4 
236.3 

4-59 

4.26 

J5/i6 

8.44 

28.1 

56-3 

84.4 

112.5 

140.6 

168.8 

196.9 

225.01253.1 

3.98 

I 

9.00 

30.0 

60.0 

90.0 

I2O.O 

150.0 

180.0 

210.0 

24O.O     27O.O 

3.73 

10  INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

2.50 

8-33 

16.7 

25.0 

33-3 

41.7 

50.0 

58.3 

66.7 

75-0 

13.4 

5/i6 

3-13 

10.4 

20.8 

31.3 

-  4L7 

52.1 

62.5 

72.9 

83.3 

93-8 

10.7 

X 

3-75 

12.5 

25.0 

37.5 

50.0 

62.5 

75.0 

87.5 

IOO.O 

112.5 

8.96 

7/i6 

4-38 

14.6 

29.2 

43.8 

58.3 

72.9 

87.5 

IO2.I 

116.7 

131.3 

7.68 

# 

5.00 

I6.7 

33-3 

50.0 

66.7 

83.3 

IOO.O 

II6.7 

133.3 

150.0 

6.72 

t 

5-63- 
6.25 

18.8 

20.8 

37-5 
41.7 

56.3 

75-0 
83-3 

93-8 
104.2 

II2.5 
125.0 

145.8 

150.0 

166.7 

168.8 
187.5 

I'll 

«/l6 

6.88 

22.9 

45.8 

68^8 

91.7 

114.6 

137.5 

160.4 

183.3 

206.3 

4.89 

H 

7-50 

25-0 

50.0 

75-0 

IOO.O 

125.0 

150.0 

175-0 

200.0 

225.0 

4.48 

J3/i6 

8.13 

27.1 

54-2 

81.3 

108.3 

135.4 

162.5 

189.6 

216.7 

243-8 

4.14 

H 

8-75 

29.2 

58.3 

87.5 

116.7 

145.8 

175.0 

204.2 

233-3 

262.5 

3.84 

z5/i6 

9-40 

31.3 

62.5 

93-8 

125.0 

156.3 

187.5 

218.8 

25O.O 

281.3 

3-58 

I 

IO.O 

33-3 

66.7 

loo.o    133.3 

166.7 

2OO.O 

233-3 

266.7 

300.0 

3-36 

ii  INCHES  WIDE. 


inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

X 

2-75 

9.17 

18.3 

27-5 

36.7 

45-8 

55.0 

64.2 

73-3 

82.5 

12.2 

5/i6 

3-44 

II.5 

22.9 

34-4 

45-8 

57-3 

68.8 

80.2 

91.7 

103.1 

9-77 

H 

4-13 

13-8 

27.5 

4L3 

55-0 

68.8 

82.5 

96.3 

IIO.O 

123.8 

8.15 

7/i6 

4.81 

16.0 

32.1 

48.1 

64.2 

80.2 

96.3 

II2.3 

128.3 

144.4 

6.98 

X 

5.50 

18.3 

36.7 

55-0 

73-3 

91.7 

IIO.O 

128.3 

146.7 

165.0 

6.ii 

9/16 

6.19 

20.6 

41.3 

61.9 

82.5 

103.1 

123.8 

144.4 

165-0 

185.6 

5-43 

X 

6.88 

22.9 

45-8 

68.8 

91.8 

114.6 

137-5 

160.4 

183-3 

206.3 

4.89 

I1/i6 

7.56 

25.2 

50.4 

75-6 

100.8 

126.0 

151.3 

176.5 

201.7 

226.9 

4-44 

X 

8.25 

27-5 

55-0 

82.5 

IIO.O 

I37-5 

165.0 

192.5 

220.0 

247.5 

4.07 

13/16 

8-94 

29.8 

59-6 

89.4 

119.2 

149.0 

178.8 

208.5 

238.3 

268.1 

3.76 

n 

9-63 

32.1 

64.2 

96.3 

128.3 

160.4 

192.5 

224.6 

256.7 

288.8 

3-49 

'5/i6 

10.4 

34-4 

68.8 

103.1 

137-5 

171.9 

206.3 

240.6 

275.0 

309.4 

3.26 

I 

II.  0 

36.7 

73-3 

IIO.O 

146.7 

183.3 

220.0 

256.7 

293-3 

330.0 

3.06 

SQUARE  IRON. 


239 


Table  No.  75.— WEIGHT    OF   SQUARE    IRON. 


SIDE. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

incMes. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

/4 

.0156 

•  052 

.104 

.156 

.208 

.260 

•313 

•365 

.417 

.469 

2154 

3/i6 

.0351 
.0625 

.117 

.208 

•234 
.417 

:gj 

.468 
-833 

.584 
1.04 

.701 
1.25 

.818 
1.46 

•935 
1.67 

1.05 

960.0 
537.6 

5/i6 

.0977 

.326 

.651 

•977 

1.68 

1-95 

2.28 

2.60 

2-93 

343-8 

3^ 

.141 

.469 

.938 

1.41 

1.88 

2.34 

2.81 

3.28 

3-75 

4.22 

238-3 

7/i6 

.191 

.638 

1.28 

1.91 

2.55 

3-19 

3.83 

4.46 

5.10 

5.74 

176.0 

•25 

.833 

1.67 

2.50 

3-33 

4.17 

5.00 

5.83 

6.67 

7-50 

J34-4 

9/x6 

.316 

1.  06 

2.  II 

3.16 

4.22 

5-27 

6.33 

7.38 

8.44 

9.49 

106.3 

•391 

1.30 

2.60 

3.91 

5.21 

6.51 

7.81 

9.11 

10.4 

II.7 

85.9 

u/l6 

.473 

I.58 

3.15 

4-73 

6.30 

7.88 

9.45 

II.  O 

12.6 

14.2 

71.0 

^ 

.563 

1.88 

3-75 

5-63 

7-50 

9.38 

11.3 

I3.I 

15.0 

16.9 

59.7 

I3/i6 

.661 

2.20 

4.40 

6.61 

8.80 

II.  O 

13.2 

15-4 

1  6.  6 

I9.8 

50.8 

^8 

.766 

2-55 

5.10 

7.66 

10.2 

12.8 

15.3 

17.9 

20.4 

23.0 

43-9 

T5/i6 

.879 

2-93 

5.86 

8-79 

ii.  7 

14.7 

17.6 

20.5 

23-4 

26.4 

38.2 

I 

1.  00 

3-33 

6.67 

10.  0 

13.3 

16.7 

20.  o 

23.3 

26.7 

30.0 

33-6 

I  «/i6 
Iff 

.13 

.27 

3.76 
4.22 

7-53 
8-44 

"•3 
12.7 

15.1 

16.9 

18.8 

21.  1 

22.6 
25.3 

26.3 
29-5 

30.1 

33.8 

38.0 

29.7 
26.5 

I  3/i6 

.41 

4.70 

9.40 

14.1 

18.8 

23-5 

28.2 

32.9 

37-6 

42.3 

23.8 

'X 

.56 

5.21 

10.4 

J5-6 

20.8 

26.0 

31-3 

36.5 

41-7 

46.9 

21.5 

I  5/i6 

.72 

5-74 

ii.  5 

23.0 

28.7 

34-4 

4O.2 

45-9 

51-7 

19-5 

I^g 

.89 

12.6 

18.9 

25.2 

31-5 

37-8 

44.1 

50.4 

56.7 

17.8 

I  7/i6 

2.07 

6.89 

13-8 

20.7 

27.6 

34-5 

4L3 

48.2 

55-i 

62.0 

16.2 

JK 

2.25 

7-50 

15.0 

22.5 

30.0 

37-5 

45-0 

52.5 

60.0 

67-5 

14.9 

I  9/i6 

2.44 

8.14 

16.3 

24.4 

32.6 

40.7 

48.8 

57-0 

65-1 

73.2 

13.8 

I/^ 

2.64 

8.80 

17.6 

26.4 

35-2 

44-0 

52.8 

61.6 

70.4 

79.2 

12.7 

IIJ/i6 

2.88 

9.60 

19.2 

28.8 

38.4 

48.0 

57-6 

67.2 

76.8 

86.4 

11.7 

I* 

3.06 

10.2 

20.4 

30.6 

40.8 

51.0 

61.3 

71.4 

81.6 

91.9 

II.  0 

I  «3/i6 

3-29 

II.  O 

21.9 

32.9 

43-8 

54-8 

65-7 

76.7 

87.6 

98.6 

10.2 

1% 

3.52 

11.7 

23-4 

35-2 

46.9 

58.6 

70.3 

82.0 

93-8 

105.5 

9.56 

II5/i6 

3.75 

12.5 

25.0 

37-5 

50.1 

62.6 

75-1 

87.6 

100.  1 

II2.6 

8-95 

2 

4.00 

13.3 

26.7 

40.0 

53-3 

66.7 

80.0 

93-3 

106.7 

120.0 

8.40 

21A 

4-52 

15.1 

30.1 

45-2 

60.2 

75-3 

90-3 

105.4 

120.0 

135.5 

7-43 

2# 

5.06 

16.9 

33-8 

50.6 

67.1 

84.4 

101.3 

118.1 

135-0 

I5L9 

6.64 

5.64 

18.8 

37-6 

56.4 

75-2 

94-o 

II2.8 

131.6 

150.4 

169.2 

5.96 

2^ 

6.25 

20.8 

41.7 

62.5 

83-3 

10.4 

125.0 

145.8 

166.6 

187.5 

5.38 

2/6 

6.89 

23.0 

45-9 

68.9 

91-9 

114.9 

137-8 

160.8 

183.9 

206.7 

4.99 

2/^ 

7.56 

25.2 

50.4 

75-6 

100.8 

126.1 

151.3 

176.5 

201.7 

226.9 

4-44 

2^ 

8.27 

27.6 

55.1 

82.7 

no.  2 

137-8 

165-3 

192.9 

22O.4 

248.0 

4.06 

3 

9.00 

30.0 

60.0 

90.0 

120.0 

150.0 

180.0 

210.0 

240.0 

27O.O 

3-73 

3/£ 

10.6 

35-2 

70.4 

105.6 

140.8 

176.0 

211.3 

246.5 

281.7 

316.9 

3-17 

3^s 

12.3 

40.8 

81.7 

122.5 

163.3 

204.2 

245.0 

285.8 

326.7 

367.5 

2-73 

32^ 

14.1 

46.9 

93-8 

140.6 

187.5 

234.4 

281.3 

328.1 

375-o 

42L9 

2.38 

4 

16.0 

53-3 

106.7 

1  60.0 

213.3 

266.7 

320.0 

373-0 

426.0 

480.0 

2.10 

4X 

18.1 

60.2 

120.4 

180.6 

240.8 

301.1 

361.2 

421.5 

481.7 

541.9 

.86 

4^ 

20.3 

67-5 

135.0 

202.5 

270.0 

337-5 

405.0 

472.5 

540.0 

607.5 

.66 

4^ 

22.6 

75-2 

150.4 

225.6 

300.8 

376.1 

451-3 

526.5 

601.7 

676.9 

•49 

5 

25.0 

83-3 

166.7 

250.0 

333-3 

416.7 

500.0 

583.3 

666.7 

750.0 

•34 

5X 

27.6 

91.9 

183.8 

275.6 

367.5 

459-4 

551.3 

643-1 

735-o 

826.9 

.21 

5/^ 

30.3 

100.8 

201.7 

302.5 

403.3 

504.2 

605.0 

705.8 

806.7 

907.5 

.11 

5/^ 

no.  2 

220.4 

330.6 

440.8 

*\  ^  I.O 

661.3 

77L5 

881.7 

991.8 

.02 

6 

36.0 

120.0 

240.0 

360.0 

480.0 

600.0 

720.0 

840.0 

960.0 

1080 

•933 

240 


WEIGHT   OF   METALS. 


Table  No.  76.— WEIGHT    OF    ROUND    IRON. 


DlAM. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 

I  CWt. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

1A 

.0123 

.041 

.082 

.123 

.164 

.205 

•245 

.286 

.327 

-368 

2738 

3/i6 

.0276 

.092 

.184 

.276 

.368 

.460 

.552 

.644 

.736 

.828 

1217 

X 

.0491 

.164 

.327 

.491 

.655 

.818 

.982 

1.15 

1.31 

1.47 

684.4 

5/i6 

.0767 

.256 

•S" 

.767 

1.  02 

1.28 

i-53 

1.79 

2.04 

2.30 

438.1 

X 

.110 

•  368 

.736 

1.  10 

1.47 

1.84 

2.21 

2.58 

2.94 

3-31 

305.4 

7/i6 

.150 

.501 

1.  00 

1.50 

2.00 

2.51 

3.01 

3.51 

4.01 

4-51 

224.0 

>/2 

.196 

•654 

1.  21 

1.96 

2.62 

3-27 

3-93 

4.58 

5-23 

5.89 

I7I.4 

9/z6 

.248 

.828 

1.66 

2.49 

3-31 

4.14 

4-97 

5.80 

6.63 

7.46 

135-5 

H 

•3°7 

I.  O2 

2.05 

3-°7 

4.09 

5-II 

6.14 

7.16 

8.18 

9.20 

109.5 

«/i6 

•371 

1.24 

2.48 

3-71 

4-95 

6.19 

7.42 

8.66 

9.90 

II.  I 

90.6 

y 

.442 

1.47 

2-94 

4.42 

5.89 

7.36 

8.83 

10.3 

11.8 

13.3 

76.0 

J3/i6 

.518 

1-73 

3-46 

5-19 

6.91 

8.64 

10.4 

12.  1 

13-8 

70.5 

# 

.601 

2.OO 

4.01 

6.01 

8.02 

10.  0 

12.0 

14.0 

16.0 

18.0 

55-9 

*5/i6 

.690 

2-30 

4.60 

6.90 

9.20 

n-5 

13-8 

16.1 

18.4 

20.7 

48.7 

I 

.785 

2.62 

5-24 

7-85 

10.5 

13.1 

15-7 

18.3 

20.9 

23-6 

42.8 

I  Vi6 

.887 

2.96 

5-91 

8.87 

11.8 

14.8 

17.7 

20.7 

23.6 

26.6 

37-9 

'# 

•994 

3-31 

6.63 

9-94 

13-3 

16.6 

19.9 

23-2 

26.5 

29.8 

33-8 

I  3/i6 

.11 

3.69 

7.38 

ii.  i 

14.8 

18.5 

22.2 

25.8 

29-5 

33-2 

30.3 

i* 

•23 

4.09 

8.18 

12.3 

16.4 

20.5 

24-5 

28.6 

32.7 

36.8 

27-3 

I  5/i6 

•35 

4-51 

9.02 

13-5 

18.0 

22.6 

27.1 

31-6 

36-1 

40.6 

24.9 

IN 

.48 

4-95 

9.90 

14.9 

19.8 

24.8 

29.7 

34-6 

39-6 

46.6 

22.7 

I  7/i6 

.62 

5.08 

IO.2 

16.2 

20.3 

25-9 

32.5 

35-5 

40.6 

48.7 

20.7 

'# 

•77 

5.89 

ii.  8 

17.7 

23.6 

29-5 

35-3 

41.2 

47.1 

53-0 

19.0 

I  9/i6 

1.92 

6-39 

12.8 

19.2 

25.6 

32.0 

38.4 

44-7 

5I-I 

57-5 

17-5 

I# 

2.07 

6.91 

13.8 

20.7 

27.7 

34-6 

41.5 

48.4 

55-3 

62.9 

16.2 

I«/i6 

2.24 

7.46 

14.9 

22.4 

29.8 

37-3 

44-7 

52.2 

59-6 

67.1 

15-0 

MC 

2.41 

8.02 

16.0 

24.1 

32-1 

4O.I 

48.1 

56.1 

64.1, 

72.2 

13-9 

1*3/16 

2.58 

8.60 

17.2 

25.8 

34-4 

43-0 

51-6 

60.2 

68.8 

77-4 

13.0 

1$ 

2.76 

9.20 

18.4 

27.6 

36.8 

46.0 

55-2 

64.4 

73-6 

82.8 

12.2 

1*5/16 

2-95 

9.83 

19.7 

29-5 

39-3 

49.1 

59-0 

68.8 

79.6 

88.4 

II.4 

2 

3-14 

10.5 

20.9 

31-4 

41.9 

52.4 

62.8 

73-3 

83.8 

94-3 

10.7 

2^ 

3-55 

1  1.8 

23-6 

35-5 

47-3 

59-1 

70.9 

82.8 

94.6 

106.4 

9-47 

2# 

3-98 

J3-3 

26.5 

39-8 

53-0 

66.3 

79-5 

92.8 

1  06.0 

"9-3 

8.44 

2^ 

4-43 

14.8 

29-5 

44-3 

59-i 

73-8 

88.6 

103-3 

118.1 

132.9 

7-59 

2^ 

4.91 

16.4 

32.7 

49.1 

65.5 

81.8 

98.2 

II4-5 

130.9 

147-3 

6.84 

2^ 

5-4i 

18.0 

36-1 

54-1 

72.2 

90.2 

108.2 

126.2 

144.3 

162.3 

6.21 

«# 

5-94 

19.8 

39-6 

59-4 

79-2 

QQ.O 

118.8 

138.5 

158.4 

178.2 

5.66 

2% 

6.49 

21.6 

43-3 

64.9 

86.6 

108.2 

129.8 

151-5 

I73-I 

194.8 

5.18 

3 

7.07 

23-6 

47.1 

70.7 

94-3 

117.8 

141.4 

164.9 

188.5 

212.  1 

4-75 

3X 
3K 

8.30 
9.62 

27.7 
32.1 

64.1 

83.0 
96.2 

110.4 
128.3 

138.3 
160.4 

165-9 

192.4 

193.6 

224.5 

221.2 
256.6 

248.9 
288.6 

4-05 
3-49 

3^ 

II.  0 

33-5 

73-6 

110.4 

147-3 

164.1 

220.9 

257-7 

294-5 

331-3 

3-04 

4 

12.6 

41.9 

83-8 

125-7 

167.6 

209.4 

251-3 

293.2 

335-0 

377-0 

2.67 

4# 

14.2 

47-3 

94.6 

141.9 

189.1 

236.4 

283.7 

33i.o 

378.3 

425-6 

2-37 

4^ 

15.9 

53-o 

106.0 

159.0 

212.  1 

265.1 

319-1 

37I-I 

424.1 

477-1 

2.  II 

4^ 

17.7 

59-i 

118.1 

177.2 

236.3 

295-3 

354-4 

413.5 

472.5 

531.6 

.90 

5 

19.6 

65-5 

130.9 

196.4 

261.8 

327.3 

392-7 

458.2 

523.6 

589.1 

•71 

5# 

21.7 

72.2 

144-3 

216.5 

288.6 

360.8 

432.9 

505.1 

577-3 

649.4 

•55 

S</2 

23.8 

79-2 

158.4 

237.6 

316.7 

396.0 

475-2 

554-3 

633-6 

712.7 

.41 

$% 

26.0 

86.6 

173-  i 

259-7 

346.2 

432.8 

5!9-3 

605.9 

692.4 

779-0 

-29 

28.3 

94.2 

188.5 

282.7 

377-0 

471.2 

565.5 

659.7 

754-0 

848.2 

-19 

ROUND  IRON. 
WEIGHT  OF  ROUND  IRON. 


241 


DlAM. 

SECT. 
AREA. 

LENGTH  IN  FEET. 

Length 
to  weigh 
i  ton. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Inches. 

sq.  in. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

feet. 

6^ 

33-2 

.9876 

1.975 

2.963 

3.950 

4.938 

5.926 

6.613 

7.901 

8.888 

20.2 

7 

38.5 

1.145 

2.291 

3.436 

4.582 

5-727 

6.872 

8.018 

9.163 

10.31 

!7-5 

r/2 

44.2 

•3r5 

2.629 

3-944 

5.258 

6.573 

7.887 

9.202 

10.52 

11.84 

15.2 

8 

50-3 

.496 

2.992 

4.448 

5.984 

7.480 

8.976 

io.47> 

11.97 

13.46 

13-4 

8/2 

56.7 

.689 

3.378 

5.067 

6.756 

8.444 

10.13 

11.82' 

13.50 

15.20 

n,8 

9 

63.6 

.893 

3.786 

5.680 

7.572 

9-46 

11.36 

13-25 

15.14 

17.04 

10.6 

91A 

70.9 

2.  1  10 

4.220 

6.329 

8.440 

IO-55 

12.66 

14-77 

16.88 

18.99 

9.48 

10 

78.5 

2.338 

4.676 

7.012 

9-352 

11.69 

14.03 

16.37 

18.70 

21.04 

8.56 

10^ 

86.6 

2-577 

4-754 

7.731 

10.31 

12.89 

15.46 

18.04 

19.02 

23.19 

7.76 

II 

95-° 

2.828 

5.656 

8.485 

11.31 

14.14 

16.97 

19.80 

22.62 

25-46 

7-07 

II# 

103.9 

3.088 

6.176 

9-265 

12.35 

J5-44 

18.53 

21.62 

24.70 

27.80 

6-47 

12 

113.1 

3.366 

6.732 

10.10 

13.46 

16.83 

20.20 

23-56 

26.93 

30.29 

5-94 

12^ 

122.7 

3.656 

7.312 

10.96 

14.62 

18.28 

21.91 

25-59 

29.25 

32.90 

5.48 

13 

132.7 

3-95° 

7.900 

11.85 

15.80 

19-75 

23.70 

27.65 

31.60 

35-15 

5-o6 

'3# 

I43-I 

4.260 

8.520 

12.78 

17.04 

21.30 

25-56 

29.82 

34.08 

38.34 

4-70 

H 

153-9 

4.581 

9.162 

13-74 

18.32 

22.90 

26.49 

32.07 

36.65 

41.23 

4-37 

H^ 

165.1 

4.915 

9.830 

14.74 

19.66 

24-58 

28.49 

34.41 

39.32 

44-24 

4.07 

15 

176.7 

5-259 

10.52 

15.78 

21.04 

26.30 

31.46 

36.81 

42.08 

47-33 

3-8o 

i5K 

188.7 

5.616 

11.23 

16.85 

22.46 

28.08 

32.70 

39-31 

44.92 

50.54 

3-56 

16 

201.  1 

5-984 

11.97 

17-95 

23.93 

29.92 

35-90 

41.89 

47-88 

53-86 

3-34 

i6# 

213.8 

6.364 

12.73 

19.09 

25.46 

31.82 

38.18 

44-55 

50.92 

57-28 

3-14 

17 

227.0 

6-755 

13-51 

20.27 

27.02 

33-78 

40.53 

47-29 

54.04 

60.80 

2.96 

i7# 

240.5 

7-159 

14.32 

21.48 

28.64 

35-80 

42.95 

50.11 

57.28 

64-43 

2-79 

18 

254-5 

7-573 

15-15 

22.72 

30.29 

37-86 

45-44 

53-01 

60.60 

68.16 

2.64 

19 

283.5 

8.438 

16.88 

25-32 

33-75 

42.19 

50.63 

59-03 

67-52 

75-94 

2-37 

20 

314.2 

9-35° 

18.70 

28.05 

37-40 

46.75 

56.10 

65-45 

74-80 

84.15 

2.14 

21 

346.4 

10.31 

20.62 

30.93 

41.23 

51-54 

61.85 

72.16 

82.47 

92.78 

1.94 

22 

380.1 

11.31 

22.63 

33-94 

45-25 

56.57 

67.88 

79.19 

90.51 

101.8 

1.77 

23 

4I5-5 

12.37 

24-73 

37.10 

49.46 

61.83 

74-19 

86.56 

93-92 

111.3 

1.62 

24 

452-4 

13.46 

26.93 

40-39 

53.86 

67.32 

80.78 

94-25 

107.7 

121.3 

1.49 

16 


242 


WEIGHT   OF   METALS. 


Table  No.  77.— WEIGHT    OF   ANGLE-IRON    AND   TEE-IRON, 
i  FOOT  IN  LENGTH. 

NOTE. — When  the  base  or  the  web  tapers  in  section,  the  mean  thickness  is  to  be  measured. 


THICK- 
NESS. 

SUM  OF  THE  WIDTH  AND  DEPTH  IN  INCHES. 

'X 

*H 

*Jf 

1% 

2 

2/8 

2# 

2/8 

2/2 

2^ 

2% 

inches. 

yf 

3/i6 
5/i6 

lbs.i 

•57 
.81 
1.04 
1.24 

Ibs. 

.62 

.89 
1.15 

i-37 

Ibs. 

.68 
•  97 
1-25 
1.50 

Ibs. 

•73 

3 

1.63 

Ibs. 
.78 

I-I3 
I.46 
I.76 

Ibs. 

.83 
1.  21 

l'.89 

Ibs. 

.88 
1.29 
1.67 

2.  02 

Ibs. 
.94 

1.37 
1.77 
2.15 

Ibs. 

.99 
1.45 

1.88 
2.28 

Ibs. 
1.04 

1:11 

2.41 

Ibs. 

1.09 
1.  60 
2.08 
2-54 

2^ 

3 

3/8 

3# 

3/8 

3/2 

3/8 

3^ 

3^8 

4 

4# 

3/i6 

5/i6 

H 

7/1  6 

I.I4 

1.68 
2.19 
2.67 
3-13 

3-57 

i.  20 
1.76 

2.29 
2.80 
3-28 
3-75 

1-25 

1.84 
2.40 
2-93 
3-44 
3-93 

1.30 
1.91 

2.50 
3-06 

3-59 
4.11 

1-45 
1.99 
2.60 
3-19 
3-75 
4.29 

1.41 

2.07 
2.71 
3.32 
3.91 

4.48 

1.46 
2.15 

2.81 

3-45 
4.06 
4.66 

i.5i 
2.23 

2.92 
3.58 
4.22 
4.84 

1.56 

2.30 

3-02 

3.71 
4.38 
5.02 

1.62 
2.38 
3-J3 
3-84 
4-53 
5.20 

1.72 

2-54 
3-33 
4.10 
4.84 
5-56 

4^ 

4^ 

5 

5X 

5/2 

5K 

6 

6X 

6/2 

6% 

7 

3/i6 
X 
5/i6 

H 

7/i  6 

* 

9/i6 

2.70 
3-54 
4-36 
5-i6 
5-92 

7.38 

2.85 

3-75 
4.62 

5-47 
6.29 
7.08 
7.85 

3.01 

1  78 
6.65 

7-50 
8.32 

3-16 
4.17 

I'14 

6.09 

7.02 
7.92 
8.79 

3-32 
4-38 
5-40 
6.41 
7.38 
8.33 
9.26 

3-48 
4-58 
5.66 
6.72 
7-75 
8.75 
9.73 

3.63 
4.79 
5.92 
7-03 
8.  ii 
9.17 

IO.2O 

3-79 
5.00 
6.18 

7-34 
8.48 

9.58 
10.66 

3-95 

5-21 

6-45 
7.66 
8.84 

IO.OO 

11.13 

4.10 
5-42 
6.71 

7-97 
9.21 
10.42 
ii.  60 

4.26 

5.63 
6.97 

8.28! 

9-57 
10.83 
12.07 

7X 

7/2 

7X 

8 

8K 

8/2 

S% 

9 

9# 

9/2 

9^ 

X 
5/i6 
H 
7/i6 
K 
9/i  6 
# 

5-83 
7-23 
8-59 
9-93 
11.25 
12.54 
13.80 

6.04 

7-49 
8.91 
10.30 
11.67 
13.01 
14.32 

6.25 

7-75 
9.22 
10.66 
12.08 
13.48 
14.84 

6.46 
8.01 

9-53 
11.03 
12.50 
I3.94 
15-36 

6.67 
8.27 
9-84 

"•39 
12.92 
14.41 
15-89 

6.88 

8-53 
10.  16 
11.76 

13-33 
14.88 
16.41 

7.08 
8.79 
10.47 

12.12 
13-75 
15.35 
16.93 

7.29 
9-05 
10.78 
12.49 
14.17 
15.82 
17-45 

7.50 
9.31 
11.09 
12.85 
14.58 
16.29 
17.97 

7.71 

9-57 
11.41 
13.22 
15.00 
16.76 
18.49 

7.92 

9.83 
11.72 

13.58 
15-42 
17.23 
19.01 

10 

ioX 

ii 

n/2 

12 

12^ 

13 

13/2 

14 

14^ 

15 

N 

7/i6 

9/i6 
# 
# 

12.03 
13-95 

15-83 
17.70 

19.53 

23-13 

12.66 
14.67 
16.67 
18.63 
20.57 
24.38 

13.28 
15.40 
17-50 
19.57 
2I.OI 

25-63 

13-91 
16.13 

18.33 
20.51 

22.66 

26.88 

H-53 
16.86 
19.17 
21.44 
23.70 
28.13 

17-59 
2O.OO 
22.38 
24-74 
29-37 

18.31 
20.84 

23.3I 
25.78 
30.63 

19.04 
21.67 
24.25 
26.83 
31.88 

19.77 
22.50 

25-I9 
27.87 

33.13 

20.50 

23-34 
26.12 
28.91 
34.38 

21.22 

24.17 
27.06 
29-95 
35.63 

12 

"# 

13 

I3K 

14 

15 

16 

17 

18 

19 

2O 

^ 

I 

I 

23.70 
28.13 

32.45 
36.67 

24.74 
29-37 
33.91 
38.33 

25.78 
30.63 
35.36 
40.00 

26.83 
31.88 
36.82 
41.67 

27.87 

33-13 

38.28 

43-33 

29-95 
35.63 
41.19 
46.67 

32.03 
38.13 
44.12 
50.00 

34.12 
40.63 
47.02 
53-33 

36.20 
4LI3 
49.95 
56.67 

38.28 
43-63 
52.87 
60.00 

40.36 
46.13 
55-78 
63.33 

WROUGHT-IRON   PLATES.  243 

Table  No.  78. — WEIGHT   OF  WROUGHT-IRON    PLATES. 


SECT. 

AREA  IN  SQUARE  FEET. 

Number 

THICK- 
NESS. 

AREA, 
when 
i  foot 

of  sq.  ft. 
in 

wide. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

i  ton. 

inches. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

sq.  feet. 

X 

3-oo 

10.  0 

20.0 

30.0 

40.0 

50.0 

60.0 

70.0 

80.0 

9O.O 

224.0 

5/i6 

3-75 

12.5 

25.0 

37-5 

50.0 

62.5 

75.0 

87.5 

IOO.O 

II2.5 

179.2 

X 

4-5° 

15.0 

3°.0 

45.o 

60.0 

75-0 

90.0 

105.0 

I2O.O 

135-0 

149.3 

7/i6 

5.20 
6.00 

17-5 
20.0 

35-0 
40.0 

60.0 

70.0 

80.0 

87.5 

IOO.O 

105.0 

120.0 

122.5 
140.0 

140.0 
1  6O.O 

157-5 
iSo.O 

128.0 
II2.O 

9/i6 

6.75 

22.5 

45-o 

67.5 

90.0 

112.5 

135-0 

150.0 

ISO.O 

202.5 

99-67 

i& 

7-5° 

25.0 

50.0 

75-0 

IOO.O 

125.0 

I5O.O 

175-0 

2OO.O 

225.0 

89.60 

XI/i6 

8.25 

27.5 

55-0 

82.5 

IIO.O 

137.5 

165.0 

192.5 

220.0 

247-5 

81.45 

3/ 

9.00 

30.0 

60.0 

90.0 

I2O.O 

150.0 

iSo.O 

2IO.O 

24O.O 

270.0 

74.67 

13/16 

9-75 

32-5 

65.0 

97-5 

I3O.O 

162.5 

195-0 

227.5 

260.0 

292.5 

68.92 

^ 

11.50 

35-o 

70.0 

105.0 

140.0 

175-0 

2IO.O 

245.0 

280.0 

3I5.O 

64.00 

I5/i6 

11.25 

37-5 

75-o 

112.5 

I5O.O 

187.5 

225.0 

262.5 

300.0 

337-5 

59-73 

I 

I2.OO 

40.0 

80.0 

120.0 

1  6O.O 

2OO.O 

24O.O 

280.0 

32O.O 

360.0 

56.00 

I  J/i6 

12-75 

42.5 

85.0 

127.5 

I7O.O 

212.5 

255.0 

297-5 

340.0 

382.5 

52-71 

I/^ 

13.50 

45-o 

90.0 

135-0 

1  80.0 

225.0 

270.0 

3I5"° 

360.0 

405.0 

49.78 

I  3/i6 

14.25 

47-5 

95-o 

142.5 

I9O.O 

237.5 

285.0 

332.5 

380.0 

427.5 

47.16 

IX 

I5.O 

50.0 

IOO.O 

150.0 

200.0 

250.0 

300.0 

350.0 

400.0 

450.0 

44.80 

l|^ 

16.5 

55-o 

IIO.O 

165.0 

220.0 

275.0 

330.0 

385-0 

440.0 

495-0 

40.73 

i/4 

18.0 

60.0 

I2O.O 

iSo.O 

24O.O 

300.0 

360.0 

420.0 

480.0 

540.0 

37-33 

i|/ 

21.0 

70.0 

140.0 

210.0 

280.0 

350.0 

42O.O 

490.0 

560.0 

630.0 

32.00 

2 

24.0 

80.0 

1  6O.O 

24O.O 

32O.O 

4OO.O 

480.0 

560.0 

640.0 

720.0 

28.00 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

2*A 

30 

.893 

1.79 

2.68 

3-57 

4.46 

5.36 

6.25 

7.14 

8.04 

23-40 

3 

36 

.07 

2.14 

3-21 

4-29 

5.36 

6.64 

7.50 

8.57 

9.64 

18.67 

42 

.25 

2.50 

3-75 

5-oo 

6.25 

7.50 

8.75 

10.00 

11.25 

16.00 

4 

48 

•43 

2.86 

4-29 

5-7i 

7.14 

8.57 

IO.OO 

11.43 

12.86 

14.00 

54 

.61 

3.21 

4.82 

6-43 

8.04 

9.64 

11.25 

12.86 

14.46 

12.44 

5  2 

60 

•79 

3-57 

5-36 

7.14 

8.93 

10.71 

12.50 

14.29 

16.07 

11.20 

5>£ 

66 

1.96 

3-93 

5-89 

7.86 

9.82 

11.79 

13.75 

I5.7I 

17.68 

IO.I8 

6 

72 

2.14 

4.29 

6-43 

8.57 

10.71 

12.86 

15.00 

17.14 

19.29 

9-33 

7 

84 

2.50 

5.00 

7-50 

10.00 

12.50 

15.00 

17.50 

20.00 

22.50 

8.00 

g 

96 

2.86 

8.57 

11.43 

10.29 

17.14 

20.00 

22.86 

25-71 

7.00 

9 

108 

3.21 

6.43 

9.64 

12.86 

16.07 

19.29 

22.50 

25.7I 

28.93 

6.22 

10 

120 

3-57 

7.14 

10.71 

14.29 

12.86 

21.43 

25.00 

28.56 

32-14 

5.60 

ii 

I32 

3-93 

7.86 

11.79 

I5-7I 

19.64 

23.57 

27.50 

31-43 

35.36 

5.09 

12 

144 

4.29 

8-57 

12.86 

17.14 

21-43 

25.7I 

30.00 

34-29 

38.57 

4.67 

13 

I56 

4.64 

9.29 

13-93 

23.21 

27.86 

32.50 

37-14 

41.79 

4.31 

1  68 

5.00 

10.00 

15.00 

20.00 

25.OO 

3O.OO 

35-oo 

4O.OO 

45.00 

4.00 

15 

180 

5.36 

10.71 

16.07 

21.43 

26.79 

32.14 

42.86 

48.21 

3-73 

244 


WEIGHT  OF   METALS. 


Table  No.  79.— WEIGHT   OF   SHEET    IRON. 

AT  480  LBS.    PER   CUBIC   FOOT. 

According  to  Wire-gauge  used  in  South  Staffordshire  (Table  No.  17). 


THICKNESS. 

AREA  IN  SQUARE  FEET. 

Number 
of  sq.  ft. 
in  i  ton. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

B.W.G. 

inch. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

sq.  ft. 

32 

.OI25 

.500 

.00 

1.50 

2.00 

2.50 

3.00 

3.5° 

4.00 

4-5° 

4480 

31 

.0141 

.562 

•13 

1.69 

2.25 

2.8l 

3.38 

3.94 

4.50 

5.06 

3986 

30 

.0156 

.625 

.25 

1.88 

2.50 

3-13 

3-75 

4.38 

5.00 

5-63 

3584 

29 

.0172 

.688 

.38 

2.06 

2-75 

3-44 

4-13 

4.81 

5.50 

6.19 

3256 

.28 
27 

.0188 
.0203 

.750 
.813 

.50 
.63 

2.25 
2.44 

3.00 
3.25 

4.06 

Jjg 

4.00 

5.25 
5.69 

6.00 
6.50 

6.75 
7-31 

2987 
2755 

26 

.O2I9 

.875 

•75 

2.63 

3-50 

4.38 

5-25 

6.13 

7.00 

7.88 

2560 

25 

.0234 

.938 

.88 

2.81 

3-75 

4.69 

5-63 

6.56 

7.50 

8-44 

2388 

24 

.0250 

.00 

2.00 

3.00 

4.00 

5.00 

6.00 

7.00 

8.00 

9.00 

2240 

23 

.0281 

•13 

2.25 

3.38 

4-50 

5.63 

6.75 

7.88 

9.00 

10.  1 

1982 

22 

.0313 

•25 

2.5O 

3-75 

5.00 

6.25 

li° 

8-75 

IO.O 

n-3 

1792 

21 

•0344 

.38 

2-75 

4.13 

6.88 

9-63 

II.  O 

12.4 

1623 

2O 

•°375 

•50 

3.00 

4-5° 

6.00 

7.50 

9.00 

10.5 

12.0 

i3-5 

1493 

19 

.0438 

•75 

5-25 

7.00 

8.75 

10.5 

12.3 

I4.O 

1280 

18 

.0500 

2.OO 

4.OO 

6.00 

8.00 

IO.O 

I2.O 

14.0 

16.0 

iS.'o 

1  120 

17 

•0563 

2.25 

4-5° 

6.75 

9.00 

II-3 

13-5 

15.8 

iS.o 

20.3 

996 

16 

.0625 

2.50 

5.00 

7-5° 

IO.O 

12.5 

I5.O 

17-5 

20.0 

22.5 

896 

15 

.0750 

3.00 

6.00 

9.00 

I2.O 

15.0 

18.0 

21.0 

24.0 

27.0 

747 

.0875 

3-50 

7.00 

10.5 

14.0 

17.5 

21.  0 

24-5 

28.0 

31.5 

640 

13 

.1000 

4-OO 

8.00 

12.  0 

16.0 

20.0 

24.0 

28.0 

32.0 

36.0 

560 

12 

.1125 

4-50 

9.00 

13-5 

18.0 

22.5 

27.0 

31-5 

36.0 

40.5 

498 

II 

.1250 

5.00 

IO.O 

15.0 

20.  o 

25.0 

30.0 

35-0 

40.0 

45-o 

448 

10 

.1406 

5.63 

H'3 

16.9 

22.5 

28.1 

33-8 

49-4 

45'° 

50.6 

398 

9 

•1563 

6.25 

12.5 

16.8 

25.0 

31-3 

37-5 

43-8 

50.0 

56.3 

358 

8 

.1719 

6.88 

13-8 

20.6 

27-5 

34-4 

41.3 

48.1 

55-0 

61.9 

326 

7 

•1875 

7-50 

15.0 

22.5 

30.0 

37-5 

52.5 

60.0 

67.5 

299 

6 

.2031 

8.13 

16.3 

24.4 

32.5 

40.6 

4&8 

56.9 

65.0 

72.1 

276 

5 

.2188 

8.75 

17-5 

26.3 

35-0 

43-8 

52.5 

61.3 

70.0 

78.8 

256 

4 

•  2344 

9-38 

18.8 

28.1 

37-5 

46.9 

56-3 

65.6 

75.0 

84-4 

239 

3 

.2500 

IO.O 

20.  o 

30.0 

40.0 

50.0 

60.0 

70.0 

80.0 

90.0 

224 

2 

.2813 

11.25 

22.5 

33-8 

45-0 

56-3 

67.5 

78.8 

90.0 

101.3 

199 

I 

•3125 

12.5 

25.0 

37-5 

50.0 

62.5 

87.5 

100.0 

112.5 

179 

IRON   SHEETS. 


245 


Table  No.  80. — WEIGHT   OF   BLACK  AND   GALVANIZED 
IRON    SHEETS. 

(MORTON'S  TABLE,  FOUNDED  UPON  SIR  JOSEPH  WHITWORTH  &  Co.'s  STANDARD 
BIRMINGHAM  WIRE-GAUGE.) 

NOTE. — The  numbers  on  Holtzapffel's  wire-gauge  are  applied  to  the  thicknesses 
on  Whitworth's  gauge. 


Gauge  of  Black  Sheets. 

Approximate  number  of 
square  feet  in  i  ton. 

Gauge  of  Black  Sheets. 

Approximate  number  of 
square  feet  in  i  ton. 

Wire- 
Gauge. 

Thickness. 

Black 
Sheets. 

Galvanized 
Sheets. 

Wire- 
Gauge. 

Thickness. 

Black 
Sheets. 

Galvanized 
Sheets. 

No. 

inch. 

square  feet. 

square  feet. 

No. 

inch. 

square  feet. 

square  feet. 

I 

.300 

187 

I85 

17 

.060 

933 

876 

2 

.280 

200 

197 

18 

.050 

1  120 

1038 

3 

.260 

215 

212 

19 

.040 

I4OO 

1274 

4 

.240 

233 

229 

20 

.036 

1556 

1403 

5 

.220 

254 

250 

21 

.032 

1750 

1558 

6 

.200 

280 

275 

22 

.028 

2000 

1753 

7 

.180 

311 

3°4 

23 

.024 

2333 

2004 

8 

.I65 

339 

33i 

24 

.022 

2545 

2159 

9 

.150 

373 

363 

25 

.020 

2800 

2339 

10 

•135 

415 

403 

26 

.018 

3III 

2553 

ii 

.120 

467 

452 

27 

.016 

35°° 

2808 

12 

.110 

509 

491 

28 

.014 

4000 

3I22 

r3 

.095 

589 

566 

29 

.013 

4308 

3306 

H 

.085 

659 

630 

30 

.012 

4667 

3513 

15 

.070 

800 

757 

31 

.OIO 

5600 

4017 

16 

.065 

862 

813 

32 

.009 

6222 

4327 

246 


WEIGHT   OF   METALS. 


Table  No.  81.— WEIGHT    OF   HOOP    IRON. 

I    FOOT   IN   LENGTH. 

According  to  Wire -gauge  used  in  South  Staffordshire. 


WIDTH  IN  INCHES. 

X 

X 

% 

i 

x# 

x# 

x* 

'# 

B.  W.  G. 

inches. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

21 

•0344 

.0716 

.0861 

.100 

•"5 

.129 

.144 

.158 

.172 

20 

•0375 

.0781 

.0938 

.109 

.125 

.141 

.156 

.172 

.188 

19 

.0438 

.0911 

.109 

.128 

.146 

.164 

.182 

.2OO 

.219 

18 

.0500 

.104 

.125 

.146 

.167 

.188 

.208 

.229 

.250 

17 

•0563 

.117 

.141 

.164 

.188 

.211 

•234 

.258 

.281 

16 

.0625 

.130 

.156 

.182 

.208 

•234 

.260 

.286 

•313 

15 

.0750 

.156 

.188 

.219 

.250 

.281 

•313 

•344 

•375 

.0875 

•  I83 

.219 

.256 

•293 

•329 

•366 

.402 

.438 

13 

.1000 

.208 

.250 

.292 

•333 

•375 

.416 

•458 

.500 

12 

.1125 

•234 

.281 

.328 

•375 

.422 

.469 

.516 

•563 

II 

.1250 

.260 

•313 

•365 

.417 

.469 

.521 

•573 

.625 

10 

.1406 

•293 

•352 

.410 

.469 

•527 

.586 

•645 

•703 

9 

.1563 

•  326 

•391 

•456 

.522 

•587 

.652 

.717 

•  783 

8 

.1719 

•358 

•430 

.501 

•573 

.644 

.716 

.788 

•859 

7 

.1875 

•391 

.469 

•547 

.625 

•703 

.781 

.859 

.938 

6 

.2031 

•423 

.508 

•593 

•677 

.762 

.836 

•931 

1.02 

5 

.2188 

.456 

•547 

.638 

•  729 

.820 

.912 

1O.O 

1.09 

4 

•2344 

.488 

.586 

•683 

.781 

•879 

•977 

10.7 

I.I7 

WIDTH  IN  INCHES. 

rp            „„ 

X# 

x# 

i# 

2 

*x 

*y* 

2^ 

3 

B.W.G. 

inches. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

21 

•0344 

.197 

.201 

.215 

.229 

.258 

.287 

•315 

•344 

20 

•0375 

.203 

.219 

.224 

.250 

.281 

.313 

•344 

•375 

19 

.0438 

•  238 

•257 

•274 

.292 

.328 

•365 

.400 

•437 

18 

.0500 

.271 

.292 

.312 

•333 

•375 

.417 

.458 

.500 

17 

•°563 

•305 

•328 

•351 

•375 

.422 

.469 

.516 

•563 

16 

.0625 

•339 

.365 

•391 

.417 

.469 

.521 

•573 

.625 

15 

.0750 

•307 

.438 

.469 

.500 

.562 

.625 

.687 

•750 

14 

.0875 

•475 

.512 

•  549 

.658 

.804 

•875 

13 

.1000 

•543 

.584 

.626 

.667 

•75° 

^833 

.917 

1.  00 

12 

.1125 

.609 

.656 

.703 

•750 

.842 

.938 

•°3 

•13 

II 

.1250 

.677 

•729 

.781 

•833 

•937 

.04 

•  *5 

.25 

IO 

.1406 

.762 

.820 

•879 

.938 

i.  06 

•  17 

.29 

.16 

9 

•1563 

.848 

.913 

•978 

.04 

•17 

•30 

•43 

.56 

8 

.1719 

•931 

.OO 

.07 

•15 

.29 

•43 

•58 

.72 

7 

.1875 

1.02 

.09 

•17 

•25 

.41 

•56 

•72 

.88 

6 

.2031 

1.  10 

.19 

•27 

•35 

•52 

.69 

1.86 

2.03 

5 

.2188 

I.I9 

.28 

•37 

.46 

.64 

.82 

2.00 

2.19 

4 

•2344 

1.27 

•37 

.46 

.56 

.76 

•95 

2.15 

2-35 

WARRINGTON   IRON   WIRE. 


247 


Table  No.  82. — WEIGHT  AND   STRENGTH   OF  WARRINGTON 

IRON   WIRE. 

TABLE  OF  WIRE  MANUFACTURED  BY  RYLANDS  BROTHERS. 
NOTE.— The  Wire-Gauge  is  that  of  Rylands  Brothers. 


Size  on 
Wire- 
Gauge. 

Diameter. 

Weig 
100  Yds. 

htof 
i  Mile. 

Leng 

i  Bundle 
of  63  Ibs. 

thof 
iCwt. 

Breakin 

An- 
nealed. 

g  Strain. 
Bright. 

Specific 
Density, 
the  aver- 
age den- 
sity of  iron 
=i. 

. 

average 

inch. 

milli- 
metres. 

Ibs. 

Ibs. 

yards. 

yards. 

Ibs. 

Ibs. 

iron 
=  i. 

7/o 

# 

12.7 

193-4 

3404 

33 

58 

10470 

15700 

.9852 

6/o 

15/32 

II.9 

170.0 

2991 

37 

66 

92OO 

13810 

S/o 

7/i6 

II.  I 

148.1 

2606 

43 

76 

8O2O 

12000 

4/o 

13/32 

10.3 

127.6 

2247 

49 

88 

6910 

10370 

3/c 

r8 

9-5 

108.8 

1915 

58 

103 

5890 

8835 

.9852 

»/o 

n/32 

8.7 

91.4 

1609 

69 

123 

4960 

7420 

O 

.326 

8-3 

82.1 

1447 

77 

136 

4450 

6678 

I 

.300 

7.6 

69.6 

1227 

9o 

161 

3770 

5655 

2 

.274 

7.0 

58.1 

1022 

108 

193 

3HO 

4717 

3 

.250  (i) 

6.4 

48.4 

85I 

130 

232 

26l8 

3927 

.9852 

4 
5 

.229 
.209 

5-8 
5-3 

40.6 
33-8 

714 

595 

III 

276 
332 

2197 
1830 

3295 
2740 

6 

.191 

4-9 

28.2 

495 

223 

397 

1528 

2290 

7 

.174 

4.4 

23-4 

412 

269 

479 

1268 

I9OO 

8 

.159 

4.0 

19.6 

344 

322 

573 

1060 

1558 

9 

.i46 

3-7 

16.5 

290 

382 

680 

893 

1340 

10 

•J33 

3-4 

13-7 

241 

460 

819 

741 

1  1  10 

10/2 

•I25(i) 

3-2 

12.  1 

213 

52i 

927 

654 

980 

.9852 

II 

.117 

3-o 

10.6 

186 

595 

1059 

573 

860 

12 

.ioo(TV) 

2.6 

8.0 

142 

783 

1393 

436 

650 

13 

.090 

2-3 

6.3 

no 

1006 

1790 

339 

509 

14 

.079 

2.0 

4.8 

85 

1305 

2322 

261 

390 

15 

.069 

.8 

3-7 

65 

1715 

3052 

199 

299 

16 

.0625  (^) 

•  5 

2.9 

5i 

2188 

3894 

156 

233 

•9378 

17 

.053 

•  3 

2.2 

38 

2900 

5160 

118 

176 

18 

.047 

.2 

1-7 

30 

3687 

6560 

93 

138 

19 

.041 

.O 

i-3 

23 

4847 

8620 

70 

105 

20 

.036 

•9 

I.O 

18 

5985 

1  1  120 

54 

Si 

21 

.  03125^) 

.8 

.8 

14 

!    7574 

I4I52 

43 

64 

1.0843 

22 

.028 

•7 

.6 

n 

9893 

18486 

33 

49 

Mem.  This  Table  of  the  weight  and  strength  of  Warrington  wire  is  given  by  permission 
of  Messrs.  Rylands  Brothers;  and  it  is  said  to  be  based  on  very  accurate  measurements  of 
sizes  and  weights.  The  last  column  is  added  by  the  author,  to  show  that  the  density  of 
the  wire  is  stationary  for  diameters  of  from  ^  inch  to  %  inch,  and  probably  somewhat 
smaller  diameters ;  but  that,  contrary  to  current  opinions  of  the  density  of  wire,  the 
density  becomes  greater  when  the  diameter  is  reduced  to  x/32  inch. 


248 


WEIGHT   OF   METALS. 


Table  No.  83. — WEIGHT    OF   WROUGHT-IRON    TUBES, 

BY  INTERNAL  DIAMETER. 
LENGTH,  I  FOOT.     Thickness  by  Holtzapffel's  Wire-Gauge. 


THICK- 
NESS. 
W.  G. 

4 

5 

6 

7 

-238 

.220 

.203 

.180 

INCH. 

# 

9/i6 

K 

7/i6 

H 

5/i6 

X 

i5/64/ 

7/32/ 

13/64 

3/i6& 

INT. 

DIAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

DM. 

inches. 

% 

4.91 

4-05 

3.27 

2.58 

1.96 

1-43 

.982 

.905 

-795 

.698 

•575 

X 

5-73 

4-79 

3-93 

3-15 

2-45 

1.84 

1.31 

1.22 

i.  08 

•963 

.811 

X 

6.54 

5-52 

4-58 

3-72 

2-95 

2.25 

1.64 

1.53 

i-37 

1.23 

1.05 

7.36 

6.26 

5-24 

4-3° 

3-44 

2.66 

1.96 

1.84 

1.66 

1.50 

1.28 

X 

9.00 

7-73 

6-55 

5-44 

4-40 

3-48 

2.62 

2.46 

2.24 

2.03 

1-75 

I 

10.6 

9.20 

7.86 

6-59 

5.40 

4-30 

3.27 

3-09 

2.81 

2.56 

2.23 

JX 

12.3 

10.7 

9.17 

7-73 

6.38 

5.11 

3-93 

3-71 

3-39 

3-09 

2.70 

i/5 

13-9 

12.2 

10.5 

8.88 

7.36 

5-93 

4.58 

4-33 

3.96 

3-62 

3-17 

i# 

15.6 

13-6 

n.8 

IO.O 

8-34 

6.75 

5-24 

4.96 

4-54 

4.15 

3-64 

2 

17.2 

I5-1 

*3-* 

II.  2 

9-33 

7-57 

5.89 

5.58 

5.12 

4.68 

4.11 

2X 

18.8 

16.6 

14.4 

12.3 

10.3 

8.38 

6.55 

6.20 

5.69 

5-21 

4-58 

2/^j 

20.5 

18.0 

15-7 

13-5 

"•3 

9.20 

7.20 

6.83 

6.27 

5-75 

5-05 

2X 

22.1 

19-5 

17.0 

14.6 

12.3 

IO.O 

7.85 

7.45 

6.84 

6.28 

5.52 

3r 

23-7 

21.0 

18.3 

15.8 

13-3 

10.8 

8.51 

8.07 

7-42 

6.81 

6.00 

27.0 

23-9 

20.9 

18.0 

15.2 

12.5 

9.82 

9.32 

8-57 

7-87 

6.94 

4 

30.3 

26.9 

23-6 

20.3 

17.2 

14.1 

n.  i 

10.6 

9.72 

8.94 

7.88 

4>£ 

33-5 

29.8 

26.2 

22.6 

19.1 

15.8 

12.4 

ii.  8 

10.9 

IO.O 

8.82 

5 

36.8 

32.8 

28.8 

24-9 

21.  1 

17.4 

13.7 

13.1 

12.0 

ii.  i 

9-77 

5^2 

40.1 

35-7 

3i-4 

27.2 

23.1 

19.0 

15.1 

14-3 

13.2 

12.  1 

10.7 

6 

43-4 

38.7 

34-o 

29-5 

25.0 

20.7 

16.4 

15-6 

14-3 

13.2 

11.7 

6X 

46.6 

41.6 

36.7 

31-8 

27.0 

22.3 

17.7 

16.8 

14.3 

12.6 

7 

49-9 

44-6 

39-3 

34-1 

29.0 

23-9 

19.0 

18.0 

i6!6 

15.3 

13-5 

7K 

53-2 

47-5 

41.9 

36.4 

30.9 

25.6 

20.3 

19-3 

17.8 

16.4 

14-5 

8 

56.5 

50.4 

44-5 

38-7 

32.9 

27.2 

21.6 

20.5 

18.9 

17.4 

15-4 

9 

63.0 

56-3 

49-7 

43-2 

36.8 

3°-5 

24.2 

23.0 

21.2 

19.6 

17-3 

10 

69.5 

62.2 

55-o 

47-8 

40-7 

33-8 

26.8 

25-5 

23-5 

21.7 

19.2 

ii 

76.1 

68.1 

60.2 

52.4 

44-7 

37-o 

29.5 

28.0 

23.8 

21.  1 

12 

82.6 

74-0 

65.5 

57-0 

48.6 

40.3 

32.1 

30.5 

28.1 

25-9 

23.0 

13 

89.2 

80.0 

70.7 

61.6 

52.5 

43-6 

34-7 

33-0 

30.4 

28.1 

24.9 

95-7 

85.8 

75-9 

66.2 

56.5 

46.8 

37-3 

35-5 

32.7 

30.2 

26.7 

15 

102.3 

91.7 

81.2 

70.7 

60.4 

50.1 

39-9 

38-0 

35-o 

32.3 

28.6 

16 

108.8 

97-6 

86.4 

75-3 

64-3 

53-4 

42-5 

40.5 

37-3 

34-4 

30.5 

17 

II5-4 

103-5 

91.6 

79-9 

68.2 

56.7 

45-2 

43-0 

39-6 

36.6 

32.4 

18 

121.9 

109.3 

96.9 

84-5 

72.2 

59-9 

47-8 

45-5 

41.9 

38.7 

34-3 

19 

128.5 

115.2 

102.  I 

89.1 

76.1 

63-2 

5°-4 

48.0 

44-2 

40.8 

36.2 

20 

135-0 

121.  1 

107.3 

93-6 

80.0 

66.5 

53-0 

5°-4 

46.5 

42.9 

38.0 

21 

141-5 

I27.O 

112.  6 

98.2 

83-9 

69.7 

55-6 

52-9 

48.8 

45-1 

39-9 

22 

148.1 

132.9 

117.8 

102.8 

87.9 

73-o 

58.3 

55-4 

51-  ! 

47.2 

41.8 

23 

154-6 

138.8 

123.1 

107.4 

91.8 

76-3 

60.9 

57-9 

53-4 

49-3 

43-7 

24 

161.2 

144.7 

128.3 

II2.0 

95-7 

79-6 

63-5 

60.4 

55-7 

51.5 

45-6 

26 

174.3 

156.5 

138.8 

121.  1 

103.6 

86.1 

68.7 

65-4 

60.3 

55.7 

49-3 

28 

187.4 

168.3 

149.2 

130.3 

111.4 

92.7 

74-o 

70.4 

64.9 

60.0 

53-1 

30 

200.4 

ISO.O 

159.7 

139-5 

119-3 

99.2 

79-2 

75-4 

69-5 

64.2 

56.8 

32 

213-5 

I9I.8 

170.2 

148.6 

127.1 

105.7 

84-4 

80.4 

74.1 

68.5 

60.6 

34 

226.6 

203.6 

180.6 

157.8 

I35-° 

112.3 

89-7 

85-4 

78.7 

72.8 

64-4 

36 

239-7 

215.4 

191.1 

167.0 

142.9 

118.8 

94-9 

90.4 

83-4 

77.0 

68.1 

WROUGHT-IRON   TUBES. 

Table  No.  83  (continued}. 
LENGTH,  i  FOOT.     Thickness  by  Holtzapffel's  Wire-Gauge. 


249 


THICK- 
NESS. 
W.  G. 

8 

9 

10 

ii 

12 

13 

14 

15 

16 

jr 

18 

.165 

.148 

•134 

.120 

.109 

•095 

•  083 

.072 

.065 

.058 

.049 

INCH. 

11  /  646. 

9/64  / 

ft6- 

3/3a/ 

5/64^. 

Vi6  b. 

3/64/. 

INT. 

DIAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

inches. 

1A 

.501 

.423 

•364 

•  318 

.267 

.219 

.181 

.149 

.130 

.Ill 

•  0895 

1A 

.717 

.610 

•539 

.472 

.410 

•343 

.290 

•243 

•215 

.187 

•154 

Y% 

•934 

•797 

.714 

.625 

•553 

.468 

.398 

•337 

.300 

•263 

.218 

g 

1.15 

1.58 

I.OO 

.890 
1.24 

•  779 
1.09 

.841 

.507 
.718 

•431 
.620 

.385 

•555 

•339 

•49  i 

.282 
.410 

i 

2.OI 

!'.?§ 

i-59 

1.41 

1.27 

1.09 

•935 

.808 

•  725 

•643 

.538 

l/i 

2-45 

2.17 

1.94 

1.72 

i-55 

i-34 

•997 

.895 

•795 

.667 

l*/2 

2.88 

2.55 

2.29 

2.04 

1.84 

i-59 

i-37 

1.19 

1.07 

.946 

•795 

1H 

3-31 

2.94 

2.64 

2.35 

2.12 

1.84 

i-59 

i-37 

.24 

1.  10 

•923 

2 

3-74 

3-33 

3.00 

2.66 

2.41 

2.08 

1.81 

1.56 

.41 

1.25 

.05 

2X 

4.17 

3-72 

3-35 

2.98 

2.69 

2-33 

2.02 

1-75 

•58 

1.40 

.18 

i      2j^5 

4.61 

4.10 

3-29 

2.98 

2.58 

2.24 

1.94 

•75 

i-55 

.31 

2^ 

5-04 

4-49 

4-05 

3-61 

3-26 

2.83 

2.46 

2.13 

.92 

1.71 

•44 

3 

5-47 

4.88 

4.40 

3-92 

3-55 

3-o8 

2.68 

2.31 

2.09 

1.86 

•57 

3^ 

6-33 

5.65 

5.10 

4-55 

4.12 

3-58 

3-" 

2.69 

2-43 

2.16 

.82 

4 

7.20 

6-43 

5.80 

5.18 

4.69 

4.07 

3-55 

3-07 

2.77 

2.47 

2.08 

4tYz 

8.06 

7.20 

6.50 

5-8i 

5.26 

4-57 

3.98 

3-45 

3.11 

2-77 

'2.34 

5 

8-93 

7.98 

7.21 

6.44 

5-83 

5-07 

4.42 

3-83 

3-45 

3-07 

2-59 

5K 

9-79 

8-75 

7.91 

7.06 

6.40 

5-57 

4-85 

4.20 

3-79 

3.38 

2.85 

6 

10.7 

9-53 

8.61 

7.69 

6.97 

6.07 

5-29 

4-58 

4-13 

3.68 

6^ 

"•5 

10.3 

9-31 

8.32 

7-55 

6.56 

5-72 

4.96 

4-47 

3-98 

136 

7 

12.4 

ii.i 

IO.O 

8.95 

8.12 

7.06 

6.16 

5-33 

4.81 

4.29 

3.62 

13-3 

10.7 

9.58 

8.69 

7.56 

6-59 

5-15 

4-59 

3-88 

8 

14.1 

\2.6 

11.4 

10.2 

9.26 

8.06 

7-03 

6.09 

5-49 

4.90 

4-13 

9 

15-8 

14.2 

12.8 

"•5 

10.4 

9-05 

7.90 

6.84 

6.17 

5-5° 

4-65 

10 

17.6 

15-7 

14.2 

12.7 

"•5 

IO.O 

8.77 

7.60 

6.85 

6.  ii 

5-  16 

ii 

19-3 

17-3 

15.6 

14.0 

12.7 

II.  0 

9.64 

8-35 

7-53 

6.72 

5.67 

12 

21.0 

18.8 

I5  2 

13-8 

12.0 

10.5 

9.10 

8.21 

7-33 

6.19 

13 

22.7 

20.4 

18.4 

16.5 

15.0 

13-0 

11.4 

9.86 

8.89 

7-93 

6.70 

H 

24-5 

21.9 

19.8 

17.7 

16.1 

14.0 

12.2 

10.6 

9-57 

8-54 

7.22 

15 

26.2 

23-5 

21.3 

19.0 

17.2 

15.0 

*3-  * 

11.4 

10.3 

9-15 

7-73 

16 

27.9 

25.0 

22.7 

20.3 

18.4 

16.0 

14.0 

12.  1 

10.9 

9.88 

8.24 

i  17 

29.6 

26.6 

24.1 

21.5 

19-5 

17.0 

14.9 

12.9 

11.6 

10.4 

8.76 

18 

3M 

28.1 

25-5 

22.8 

20.6 

18.0 

13-6 

12.3 

II.  0 

9-27 

19 

33-i 

29.7 

26.9 

24.0 

21.8 

19.0 

i6!6 

14.4 

13.0 

n.6 

9.78 

20 

34-8 

31.2 

28.3 

25-3 

22.9 

20.0 

17-5 

I5.I 

13-7 

12.2 

10.3 

21 

36.6 

32.8 

29.7 

26.5 

24.1 

21.0 

18.3 

15-9 

14.3 

12.8 

10.8 

22 

38.3 

34-3 

27.8 

25.2 

22.0 

19.2 

16.6 

15.0 

13-4 

11.3 

23 

40.0 

35-9 

32.5 

29.1 

26.4 

23.0 

20.1 

17.4 

15-7 

I4.O 

ii.  8 

24 

41.8 

37-4 

33-9 

30-3 

27.5 

24.0 

20.9 

18.1 

16.4 

14.6 

12.6 

26 

45-2 

40.5 

36.7 

32-8 

29.8 

26.0 

22.6 

19.7 

17.7 

I5.8 

13-4 

28 

48.7 

43-6 

39-5 

35-3 

32.1 

28.0 

24-4 

21.2 

19.1 

17.0 

14.4 

30 

52.1 

46.7 

42.3 

37-8 

34-4 

3O.O 

26.1 

22.7 

20.5 

18.3 

15-4 

32 

55-5 

49-8 

45-1 

40.4 

36.7 

32.0 

27.9 

24.2 

21.8 

19-5 

16.5 

34 

59-0 

52.9 

48.0 

42.9 

39-0 

34-0 

29.7 

25.8 

23.2 

20.7 

17.5 

36 

62.4 

56.0 

50.8 

45-4 

41-3 

36.0 

31-4 

27-3 

24.6 

21.9 

18.6 

250 


WEIGHT   OF   METALS. 


Table  No.  84.— WEIGHT    OF   WROUGHT-IRON    TUBES, 
BY  EXTERNAL  DIAMETER. 

LENGTH,  i  FOOT.     Thickness  by  Holtzapffel's  Wire-Gauge. 


THICKNESS. 

W.  G. 

7 

8 

9 

IO 

II 

12 

13 

14 

15 

INCH* 

.180 

.165 

.148 

•  134 

.I2O 

.109 

•095 

•  083 

.072 

3/i6  & 

»/64  b. 

9/64  / 

9/64  b. 

/s 

b. 

7/64 

3/3*/ 

S/64/ 

S/64  b. 

EXT.  DIAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs 

Ibs. 

Ibs. 

Ibs. 

I  inch. 

1.55 

1.44 

1-32 

22 

.11 

.02 

.900 

•797 

.700 

1/^5 

I.78 

1.66 

«•$! 

39 

.26 

.16 

1.03 

.906 

1/4 

2.  02 

1.88 

1.71 

57 

.42 

•30 

1.15          .01 

.888 

\y% 

2.25 

2.09 

1.90 

74 

•58 

•45 

1.27              .12 

•983 

I/^ 

2.49 

2.31 

2.10 

, 

92 

•f> 

'3 

•59 

1.40    !      .23 

1.08 

iH 

2.72 

2.52 

2.29 

2.09 

.i 

>9 

3 

1-52          -34 

1.17 

2.96     1     2.74 

2.48 

2. 

27 

2.05 

.j 

7 

1-65    i      -45 

1.27 

ifc 

3-19 

2.96 

2.68 

2. 

45 

2.21 

2.  02 

1-77    |      -56 

1.36 

2 

3-43 

3-J7 

2.87 

2. 

62 

2.' 

;6 

2.16 

1.90 

.67 

i-45 

2/^ 

3-67 

3-39 

3-06 

2. 

80 

2.52 

2.30 

2.  02 

.78 

2% 

3-90 

3.60 

3-26 

2. 

97 

2.68 

2-44 

2.14 

.88 

1.64 

2H 

4.14 

3-82 

3-45 

3- 

15 

2.83 

2-59 

2.27 

•99 

1.74 

"2]/z 

4-37 

4.04 

3-65 

3- 

32 

2-99 

2-7 

3 

2-39 

2.  IO 

1.83 

2/4 

4.61 

4-25 

3-84 

3- 

50 

3-J5 

2.8 

7 

2.52 

2.21 

i-93 

2^ 

4.84 

4-47 

4.03 

3- 

67 

3-31 

3.02 

2.64 

2.32 

2.02 

2/S 

5.08 

4.68 

4-23 

3-85 

\6 

3.16 

2.77 

2-43 

2.  1  1 

3 

5-32 

4.90 

4.42 

4.02 

3-62 

3.30 

2.89 

2-54 

2.21 

3/4^ 

5-79 

5-33 

4.81 

4-37 

3-94 

3-5 

9 

3-H          2.75 

2.40 

3/^5 

6.26 

5-76 

5-20 

4- 

72 

4-25 

3-8 

7 

3-39        2.97 

2-59 

3^ 

6-73 

6.19 

5.58 

5-°7 

4-f 

7 

4.16 

3-64        3.19 

2-77 

4 

7.20 

6.63 

5-97 

5- 

43 

4-* 

& 

4-44 

3-89 

3-40 

2.96 

4f 

7.67 

7.06 

6.36 

5- 

78 

5-20 

4-73 

4-13 

3.62 

3-15 

8.14 

7-49    ; 

7-45 

6. 

13 

5-f 

,i 

5.01 

4.38 

3-84 

3-34 

4^ 

8.61 

7.91 

7-13 

6. 

48 

5-* 

>2 

5-30 

4-63 

4.06 

3-53 

5 

9.08 

8.35 

7-52 

6. 

83 

6.13 

5.58 

4.88 

4.27 

3-72 

9-56 

8.79 

7.91 

7- 

18 

6.44 

5-87 

5-1" 

4-49 

3-90 

1/2 

10.0 

9.22 

8.30 

7- 

53 

6.76 

6.15 

5.38 

4.71 

4.09 

10.5 

9-65 

8.68 

7- 

88 

7.07 

6.44 

5.63 

4-93 

4.28 

6 

II.  0 

10.  1 

9.07 

8. 

23 

7-39 

6-73 

5.87 

5-H 

4-47 

6/4f 

11.4 

10.5    '< 

9.46 

8. 

58 

7.70 

7.01 

6.12 

5.36 

4.66 

6^ 

11.9 

10.9    j 

9.85 

8-93 

8.02 

7-30 

6-37 

5.58 

4-85 

6^ 

12.4 

11.4 

10.2 

9- 

28 

8-33 

7.58 

6.62 

5-79 

5-°3 

7 

12.9 

ii.  8  i 

10.6 

9-63 

8.64 

7.87 

6.87 

6.01 

5.22 

13-3 

12.2     ! 

II.  0 

9-99 

8.96 

8.15 

7.12 

6.23 

5-41 

7^» 

13.8 

12.7 

11.4 

10.3 

9.27 

8.44 

7-37 

6-45 

5.60 

73^ 

14.3 

ii.  8 

10.7 

9-59 

8.72 

7.62 

6.66 

5-79 

8 

14.7 

13.5 

12.2 

II.  0 

9.90 

9.01 

7.86 

6.88 

5.98 

THICKNESS. 

W.  G. 

4 

5 

6 

7 

8 

9 

INCH. 

.3125       .281 

.238 

.220 

.203 

.180 

.165 

.148 

S/i6              9/32 

'S/64/ 

7/32 

13/64 

'3/16  b. 

"/64  b. 

9/64  / 

EXT.  DIAM. 

Ibs.               Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

7  inch. 

21.9             19.8 

16.9 

I5.6 

14.5 

12.9 

ii.  8 

10.6 

7'A 

23-5             21.3 

18.1 

1  6,8 

15-5 

13-8 

12.7 

11.4 

8 

25.2            22.7 

19-3 

17.9 

16.6 

14.7 

J3-5 

12.2 

8/^ 

26.8            24.2 

20.6 

19.1 

17.6 

15-7 

14.4 

I2.9 

9i 

28.4            25.7 

21.8 

20.2 

18.7 

16.6 

15-3 

13-7 

30.1            27.1 

23.1 

21.4 

19.8 

17.6 

16.1 

14-5 

10 

31.7            28.6 

24-3 

22.5 

20.8 

18.5 

17.0 

15-3 

LIST  OF  TABLES  OF  CAST   IRON,   STEEL,   ETC.  251 


LIST   OF   TABLES   OF   THE   WEIGHT   OF   CAST   IRON, 
STEEL,    COPPER,    BRASS,   TIN,    LEAD,   AND   ZINC. 

The  following  Tables  are  devoted  to  the  specialities  of  manufacture  in 
Cast  Iron,  Steel,  and  other  metals,  embracing  the  utmost  range  of  dimen- 
sions to  which  objects  in  the  several  metals  are  executed  in  the  ordinary 
course  of  practice. 

Thus,  whilst  it  is  customary  for  certain  classes  of  Cylinders  in  Cast  Iron — 
steam  cylinders,  for  example — to  be  constructed  according  to  given  internal 
diameters,  other  classes  are  constructed  according  to  diameters  given 
externally,  as  the  iron  piers  of  railway  bridges.  Two  distinct  tables  accord- 
ingly have  been  composed,  showing  the  weights  of  Cylinders  of  various 
thicknesses,  and  of  diameters  as  measured  internally  and  externally. 

The  weights  of  Copper  Pipes  and  Cylinders  are  only  calculated  for  in- 
ternal diameters,  as  it  is  not  the  practice  to  construct  them  to  given  external 
diameters.  Brass  Tubes,  on  the  contrary,  are  calculated  only  for  external 
diameters,  as  they  are  not  ordinarily  made  to  given  internal  diameters. 

TABLE  No.  85. — Weight  of  Cast-iron  Cylinders,  i  foot  in  length,  advanc- 
ing, by  internal  measurement,  from  i  inch  to  10  feet  in  diameter,  and  from 
Y^  inch  to  2^  inches  in  thickness. 

TABLE  No.  86. — Weight  of  Cast-iron  Cylinders,  i  foot  in  length,  advanc- 
ing, by  external  measurement,  from  3  inches  to  20  feet  in  diameter,  and 
from  3/l6  inch  to  4  inches  in  thickness. 

TABLE  No.  87. — Volume  and  weight  of  Cast-iron  Balls,  when  the 
diameter  is  given;  from  i  inch  to  32  inches  in  diameter,  with  multipliers 
for  other  metals. 

TABLE  No.  88. — Diameter  of  Cast-iron  Balls,  when  the  weight  is  given ; 
from  y?,  pound  to  40  cwts. 

TABLE  No.  89. — Weight  of  Flat  Bar  Steel,  i  foot  in  length ;  from  ^  inch 
to  i  inch  thick,  and  from  yz  inch  to  8  inches  in  width. 

TABLE  No.  90.— Weight  of  Square  Steel,  i  foot  in  length ;  from  ^  inch 
to  6  inches  square. 

TABLE  No.  91.—  Weight  of  Round  Steel,  i  foot  in  length;  from  y%  inch 
to  24  inches  in  diameter. 

TABLE  No.  92.— Weight  of  Chisel  Steel:  hexagonal  and  octagonal,  i  foot 
in  length;  from  ^  inch  to  i^  inches  diameter  across  the  sides. 
Oval-flat,  from  ^  x  ^  inch  to  i^  x  ^  inch. 

TABLE  No.  93. — Weight  of  one  square  foot  of  Sheet  Copper;  from  No.  i 
to  No.  30  wire-gauge,  as  employed  by  Williams,  Foster,  &  Co. 

TABLE  No.  94. — Weight  of  Copper  Pipes  and  Cylinders,  i  foot  in  length, 
advancing,  by  internal  measurement,  from  ^  inch  to  36  inches  in  diameter, 
and  from  No.  oooo  to  No.  20  wire-gauge  in  thickness. 


252  WEIGHT   OF   METALS. 

TABLE  No.  95. — Weight  of  Brass  Tubes,  i  foot  in  length,  advancing,  by 
external  measurement,  from  ^  inch  to  6  inches  in  diameter,  and  from 
No.  3  to  No.  25  wire-gauge  in  thickness. 

TABLE  No.  96. — Weight  of  one  square  foot  of  Sheet  Brass;  from  No.  3  to 
No.  25  wire-gauge  in  thickness. 

TABLE  No.  97. — Size  and  weight  of  Tin  Plates. 
TABLE  No.  98. — Weight  of  Tin  Pipes,  as  manufactured. 
TABLE  No.  99. — Weight  of  Lead  Pipes,  as  manufactured. 

TABLE  No.  100. — Dimensions  and  weight  of  Sheet  Zinc.  (Vielle-Mon- 
tagne.) 


CAST-IRON   CYLINDERS. 

Table  No.  85. — WEIGHT  OF  CAST-IRON  CYLINDERS. 
BY  INTERNAL  DIAMETER,     i  FOOT  LONG. 


253 


INT. 

THICKNESS  IN  INCHES. 

DlAM. 

X 

5/x6 

H 

7/i6 

l/2 

9/i6 

H 

«/* 

X 

% 

I 

inches. 

Ibs. 

Ibs.. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

I 

3-07 

4-03 

5.06 

6.17 

7.36 

8.63 

9.97 

II.4 

12.9 

16.1 

19.6 

1^ 

4.30 

5-56 

6.90 

8.32 

9.82 

11.4 

14.8 

16.6 

20.4 

24.5 

2 

5-52 

7.09 

8.74 

10.5 

12.3 

14.2 

i6ii 

18.1 

20.3 

24-7 

29.5 

2)£ 

6.75 

8.63 

10.6 

12.6 

14.7 

16.9 

19.2 

21.5 

23-9 

29.0 

34-4 

a 

7.98 

10.2 

12.4 

14.8 

17.2 

19.7 

22.2 

24.9 

27.6 

33-3 

39-3 

9.20 

II.7 

14-3 

16.9 

19.6 

22.4 

25.3 

28.3 

31-3 

37-6 

44-2 

4 

10.4 
II.7 

13.2 

14.8 

16.1 

18.0 

19.1 

22.1 

22.1 

24-5 

28.0 

28.4 
3L5 

31-6 

35-o 

38.7 

41.9 

46.2 

49.1 

54-0 

5  2 

12.9 

16.3 

19.8 

23.4 

27.0 

30.7 

34-5 

38.4 

42-3 

50.5 

58.9 

5^ 

14.1 

17.8 

21.6 

25-5 

29.5 

33-5 

37-6 

41.8 

46.0 

54-8 

63-8 

6 

15-3 

19.4 

23.5 

27.7 

32.0 

36.2 

40.7 

45-i 

49-7 

68.7 

THICKNESS  IN  INCHES. 

X 

7/i6 

Yz 

9/i6 

H 

«/i6 

X 

H 

i 

i« 

iX 

inches. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

6 

23-5 

27.7 

32.0 

36.2 

40.7 

45.1 

49-7 

59-1 

68.7 

78.7 

89.0 

6J/2 

25-3 

29.8 

34-4 

39-0 

43.7 

48.5 

53-4 

63-4 

73-6 

84.2 

95-1 

7 

27.2 

32.0 

36.8 

41.8 

46.8 

51.9 

57-i 

67.7 

78.5 

89.7 

IOI.2 

I* 

29.0 
30.8 

39-3 
41.7 

44-5 
47-3 

49-9 
52.9 

HI 

60.8 
64.4 

71.9 
76.2 

83.5 
88.4 

95-3 
100.8 

107.4 
H3.5 

S)4 

32.7 

38'.4 

44.2 

50.0 

55-9 

62.0 

68.1 

80.5 

93-3 

106.3 

II9.7 

10 

36.4 
38.2 
40.0 

40.5 

42.7 

44-8 
47.0 

46.6 
49.1 

5L5 

54-o 

52.8 
55-6 
58.3 
61.1 

59.0 
62.0 

68^2 

65.4 

68.8 
72.1 

75-5 

71.8 

75-5 
79-2 
82.8 

84.8 
89.1 
93.4 
97-7 

98.2 
103.1 
108.0 
112.9 

in.  8 

117.4 
122.9 

128.4 

125.8 

I38]  I 
144.2 

II 

41.9 

49.1 

56.5 

63-9 

71.2 

78.9 

86.5 

102.0 

117.8 

133-9 

150.3 

H# 

43-7 

5L3 

58.9 

66.6 

74.5 

82.3 

90.2 

106.3 

122.7 

139.4 

156.5 

12 

45  -6 

53-4 

61.4 

69.4 

77-5 

85.6 

93-9 

1  10.  6 

127.6 

145.0 

162.6 

13 

49-2 

57-7 

66.3 

74-9 

83-6 

92.4 

IOI.2 

119.2 

137-5 

156.0 

174.9 

14 

52.9 

62.0 

71.2 

80.4 

89.7 

99.1 

108.6 

127.8 

147-3 

167.1 

187.2 

15 

56.6 

66.3 

76.1 

85-9 

95-9 

105.9 

116.0 

136.4 

I57-I 

178.1 

199.4 

16 

60.3 

70.6 

81.0 

9L5 

IO2.O 

II2.6 

I23>3 

145.0 

166.9 

189.1 

2II.7 

17 

64.0 

74-9 

85.9 

97.0 

108.2 

119.4 

130.7 

153-6 

176.7 

200.2 

224.0 

18 

67.7 

79-2 

90.8 

102.5 

114.3 

126.  i 

138.1 

162.2 

186.5 

211.  2 

236.2 

THICKNESS  IN  INCHES. 

X 

7/i  6 

K 

H 

X 

# 

i 

1/8 

'X 

1/8 

1^ 

inches. 

cwt. 

cwt. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

18 

.604 

.707 

.811 

.02 

1.23 

•45 

1.67 

1.89 

2.  1  1 

2.34 

2.56 

19 

.637 

.746 

.855 

.08 

1.30 

•52 

2.22 

2.46 

2.70 

20 
21 

.670 
•703 

.784 
.823 

.898 

.942 

•'3 

.19 

1.36 

1-43 

.60 

.68 

I'.ll 
1.93 

2^18 

2.33 

2.44 

2.S8 
2.70 

2.96 

22 

•736 

.861 

.986 

.24 

1.49 

.76 

2.02 

2.28 

2-55 

2.82 

3.09 

23 

.769 

.900 

1.03 

.29 

1.56 

•83 

2.  IO 

2.38 

2.66 

2.94 

3.22 

24 

.802 

•939 

1.07 

•35 

1.63 

.91 

2.19 

2.48 

2-77 

3.06 

3.35 

25 

•835 

•977 

1.  12 

.40 

1.69 

•99 

2.28 

2.58 

2.88 

3-18 

3-48 

254 


WEIGHT  OF   METALS. 

Table  No.  85  (continued}. 

BY  INTERNAL  DIAMETER,     i  FOOT  LONG. 


INT. 

THICKNESS  IN  INCHES. 

DlAM. 

H 

7/i6 

1A 

H 

¥ 

H 

I 

i# 

i# 

1/8 

'X 

inches. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

26 

.868 

.02 

.16 

.46 

1.76 

2.06 

2.37 

2.68 

2.99 

3-30 

3.62 

27 

.901 

•05 

.21 

•51 

1.82 

2.14 

2.45 

2.77 

3-09 

3.42 

3-75 

28 

•934 

.09 

•25 

•57 

1.89 

2.22 

2.54 

2.87 

3-20 

3-54 

3-88 

29 

.967 

•*3 

•29 

1.96 

2.29 

2.63 

2.97 

3-31 

3-66 

4.01 

30 

.998 

.1? 

•34 

.68 

2.02 

2-37 

2.72 

3-07 

3-42 

3-78 

4.14 

32 

.06 

•25 

•43 

•79 

2.15 

2.52 

2.89 

3-27 

3-64 

4.02 

4.41 

34 

•13 

•32 

•5i 

.90 

2.29 

2.67 

3-07 

3-46 

3-86 

4.26 

4-67 

36 

.20 

.40 

.60 

2.01 

2.42 

2.83 

3.24 

3-66 

4.08 

4-5° 

4.94 

38 

.26 

•47 

.69 

2.12 

2-55 

2.98 

3.42 

3-86 

4-30 

4-75 

5.20 

40 

•33 

•55 

•77 

2.23 

2.68 

3-14 

3-59 

4-05 

4-52 

4-99 

5-47 

42 

•39 

•63 

.86 

2-34 

2.81 

3-29 

3-77 

4-25 

4-74 

5-23 

5-73 

45 

•  49 

•75 

•99 

2.50 

3.01 

3-52 

4-03 

4-55 

5-07 

5-59 

6.13 

48 

•59 

.86 

2.12 

2.66 

3.21 

3-75 

4-3° 

4-85 

5-40 

5.96 

6.52 

THICKNESS  IN  INCHES. 

H 

X 

g 

i 

1/8 

i* 

I# 

*/* 

Ig 

2 

2X 

inches. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

48 

2.66 

3.21 

3.75 

4.30 

4.85 

5-40 

5.96 

6.52 

7.63 

8.77 

9.91 

51 

2.82 

3-40 

3.98 

4.56 

5-14 

5-73 

6.32 

6.91 

8.09 

9.29 

10.5 

54 

2.99 

3.60 

4.21 

4.82 

5-44 

6.06 

6.69 

7.31 

8.55 

9.82 

II.  I 

57 

3-!5 

3.80 

4.44 

5-09 

5-73 

6.38 

7-05 

7.70 

9.01 

10.4 

11.7 

60 

3-32 

4.00 

4.67 

5-35 

6.03 

6.  7I 

7.41 

8.10 

9-47 

10.9 

12.3 

63 

3-48 

4.19 

4.90 

5.61 

6-33 

7.04 

7.78 

8-49 

9-93 

11.4 

12.9 

66 
69 

3-64 
3-8i 

4.39 

4-59 

1:11 

5.88 
6.14 

6.62 
6.92 

7-37 
7.70 

8.14 

8.51 

8.89 
9.28 

10.4 
10.9 

11.9 
I2-5 

13-5 
14.1 

72 

3-97 

4.78 

5-59 

6.40 

7.21 

8.03 

8.87 

9.67 

U-3 

13.0 

14.7 

75 

4.14 

4-98 

5.82 

6.66 

7-51 

8.36 

9.24 

10.  1 

ii.  8 

J3-5 

J5-2 

78 

4-3° 

5.18 

6.05 

6.93 

7.81 

8.69 

9.60 

10.5 

12.2 

14.0 

I5.8 

81 

4.46 

5.38 

6.28 

7.19 

8.10 

9.02 

9-97 

10.9 

12.7 

14.6 

!6.4 

84 

4-63 

5-57 

6.51 

7-45 

8.40 

9-35 

10.3 

"•3 

13.2 

J5-1 

17.0 

87 

4-79 

5-77 

6.74 

7.72 

8.69 

9.67 

10.7 

ii.  6 

13-6 

15.6 

17.6 

90 

4.96 

5-97 

6.97 

7.98 

8.99 

IO.O 

ii.  i 

12.0 

I4.I 

16.1 

18.2 

93 

5.12 

6.17 

7.20 

8.24 

9.29 

10.3 

11.4 

12.4 

14-5 

16.7 

18.8 

96 

5.28 

6.36 

7-43 

8.51 

9.58 

10.7 

ii.  8 

12.8 

15.0 

17.2 

19.4 

99 

5-45 

6.56 

7.66 

8.77 

9.88 

II.  0 

12.2 

13.2 

15-5 

17.7 

20.  o 

102 

5.61 

6.76 

7.89 

9-03 

IO.2 

"•3 

I2-5 

13.6 

20.  6 

105 

5-78 

6.95 

8.12 

9.29 

10.5 

11.7 

12.9 

14.0 

16  4 

i8.'8 

21.2 

108 

5-94 

7-15 

8.36 

9.56 

10.8 

12.0 

!3-3 

14.4 

16^8 

19-3 

21.8 

in 

6.10 

7-35 

8-59 

9.82 

ii.  i 

12.3 

13.6 

14.8 

17-3 

19.8 

22.3 

114 

6.27 

7-55 

8.82 

IO.  I 

11.4 

12.6 

14.0 

15-2 

17.8 

20.3 

22.9 

117 

6-43 

7-74 

9-05 

10.4 

11.7 

I3.0 

14-3 

15.6 

18.2 

20.9 

23-5 

120 

6-59 

7-94 

9.28 

10.6 

12.0 

13.3 

14.7 

16.0 

18.7 

21.4 

24.1 

CAST-IRON  CYLINDERS. 

Table  No.  86. — WEIGHT  OF  CAST-IRON  CYLINDERS. 
BY  EXTERNAL  DIAMETER,     i  FOOT  LONG. 


255 


EXT. 

DlAM. 

THICKNESS  IN  INCHES. 

3/i6 

X 

5/i6 

X 

7/z6 

X 

9/i6 

# 

X 

K 

i 

inches. 

Ibs. 

Ibs. 

Ibs.          Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

3 

5.18 

6.75 

8.25       9.65 

II.  0 

12.3 

13-5 

14.6 

16.6 

18.3 

19.6 

6.10 

7.98 

9.78       II.5 

13.2 

14.7 

16.2 

17.6 

20.3 

22.6 

24.5 

4x 

7.02 

9.20 

"•3      I3.3 

15.3 

17.2 

19.0 

20.7 

24.0 

26.9 

29.5 

7-94 

10.4 

12.9 

15.2 

17.5 

19.6 

21.7 

23.8 

27.7 

31-! 

34-4 

5 

8.86 

11.7 

14.4 

17.0 

19.6 

22.1 

24.5 

26.9 

3L5 

35.4 

39-3 

3% 

9-78 

12.9 

15.9 

18.9 

21.8 

24-5 

27.3 

29.9 

35-2 

39.7 

44.2 

6 

10.7 

14.1 

17.5 

20.7 

23-9 

27.0 

30.0 

33-o 

38.9 

44-o 

49.1 

6)4 

11.6 

19.0 

22.5 

26.0 

29.5 

32.8 

36-1 

42.6 

48.3 

54-0 

7 

12.5 

16.6 

20.5 

24.4 

28.2 

3L9 

35.6 

39-1 

46.4 

52.6 

58.9 

7/£ 

13-5 

17.8 

22.1 

26.2 

30.3 

34-4 

38.3 

42.2 

50.1 

56.9 

63.8 

8 

14.4 

19.0 

23-6 

28.1 

32.5 

36.8 

41.1 

45-3 

53.8 

61.2 

68.7 

&)4 

15-3 

20.3 

25.1 

29.9 

34-6 

39-3 

43-8 

48.3 

57-5 

65-5 

73.6 

9 

16.2 

21.5 

26.7 

31.8 

36.8 

41.7 

46.6 

51-4 

61.3 

69.8 

78.5 

9/2 

17.2 

22.7 

28.2 

33-6 

38.9 

44-2 

49-4 

54-5 

65.0 

74-i 

83-5 

10 

18.1 

23-9 

29.7 

35-4 

41.1 

46.6 

52.1 

57-5 

68.7 

78.4 

88.4 

ii 

19.9 

26.4 

32.8 

39-i 

45-4 

5L5 

57-6 

63.7 

76.0 

87.0 

98.2 

12 

21.8 

28.8 

35-9 

42.8 

49-7 

56.5 

63.2 

69.8 

83-4 

95-6 

108.0 

13 

23.6 

31  3 

38.9 

46.5 

54-o 

61.4 

68.7 

75-9 

90.7 

104.2 

117.8 

25-5 

33-8 

42.0 

50.2 

58.3 

66.3 

74-2 

82.1 

98.0 

II2.8 

127.6 

15 

27-3 

36.2 

45-  * 

53-8 

62.6 

71.2 

79-7 

88.2 

105.4 

121.3 

137-4 

16 

29.1 

38.7 

48.1 

57-5 

66.9 

76.1 

94-3 

112.7 

129.9 

147-3 

17 

31.0 

41.1 

51.2  i    61.2 

71.1 

81.0 

9o.'8 

100.5 

I2O.O 

138.5 

18 

32.8 

43-6 

54-3  i    64.9 

75-4 

85-9 

96.3 

106.6 

127.4 

147.1 

166.9 

19 

34-6 

46.0 

57-3      68.6 

79-7 

90.8 

101.8 

II2.8 

134-7 

155-7 

176.7 

20 

36.5 

48.5 

60.4 

72.3 

84.0 

95-7 

107.3 

118.9 

142.0 

164.3 

186.5 

21 

38-3 

50.9 

63.5 

75-9 

88.3 

100.6 

112.9 

125.0 

149.4 

172.9 

196.4 

22 

40.2 

53-4 

66.5      79.6 

92.6 

105-5 

118.4 

131.2 

156.7 

181.5 

206.2 

23 

42.0 

55-8 

69-6      83.3 

96.9 

110.5 

123.9 

137.3 

164.0 

190.1 

215.0 

24 

43-8 

58.3 

72.7      87.0 

IOI.2 

ii5-4 

129.4 

143.4 

I7I.4 

198.7 

225.8 

25 

45-7 

60.8 

75.7  !  90.7 

105.5 

120.3 

135-0 

149.6 

178.7 

207.2 

235.6 

26 

47-5 

63-2 

78.8    94.3 

109.8 

125.2 

140.5 

155.7 

I86.I 

215.8 

245-4 

27 

49-4 

65.7 

81.9    98.0 

II4.I 

130.1 

146.0 

161.8 

193.4 

224.4 

255-3 

28 

51-2 

68.1 

85.0  j  101.7 

II8.4 

135-0 

I5I-5 

1  68.0 

200.  7 

233-0 

265.1 

29 

53-o 

70.6 

88.0 

105.4 

122.7 

139-9 

I57.o 

174.1 

208.  1 

241.6 

274.9 

30 

54-9 

73-o 

91.1 

109.1 

127.0 

144.8 

162.6 

180.2 

215.4 

250.2 

284.7 

31 

56-7 

75-5 

94-2 

II2.8 

I3I-3 

149-7 

168.1 

186.4 

222.7 

258.8 

294-5 

32 

58.6 

77-9 

97.2 

116.4 

135-6 

154.6 

173.6 

192.5 

230.1 

267.4 

304.3 

33 

60.4 

80.4 

100.3 

1  20.  i 

139-9 

*59-5 

179.1 

198.7 

237.5 

276.0 

314.2 

34 

62.2 

82.8 

103.4 

123.8 

144.2 

164.5 

184.7 

204.8 

244.8 

284.6 

324.0 

35 

64.1 

85.3 

106.4 

127.5 

148.5 

169.4 

190.2 

210.9 

252.2 

293.1 

333-8 

36 

65.9 

87.8 

109.5 

131.2 

152.7 

174.3 

195.7 

217.1 

259.5 

301.7 

343-6 

38 

69.6 

92.7 

115.6 

138-5 

161.3 

184.1 

206.8 

229.3 

274.3 

318.9 

363-2 

40 

73-3 

97-6 

121.  8 

145-9 

169.9 

193-9 

217.8 

241.6 

289.0 

336.1 

382.9 

42 

77.0 

102.5 

127.9 

153-3 

178.5 

203.7 

228.8 

253-9 

303-7 

353-3 

402.5 

45 

82-5 

109.8 

137.1 

164.3 

I9I.2 

218.5 

245-4 

272.3 

325.8 

379-1 

432.0 

48 

88.0 

117.2 

146.3 

175.4 

203.8 

233-2 

262.0 

290.7 

347-9 

404.8 

461.4 

51 

93-6 

124.6 

155.5 

186.4 

216.5 

247.9 

278.6 

309.1 

370.0 

430.6 

490.9 

54 

99.1 

I3I-9 

164.7 

197.5 

229.2 

262.6 

295.1 

327.5 

392.1 

456.4 

520.3 

57 

104.6 

139-3 

173.9 

208.5 

241.8 

277.4 

3"-7 

345-9 

414.2 

482.1 

549-8 

60 

1  10.  1 

146.6 

183.1 

219.6 

254-5 

292.1 

328.3 

364-3 

436.3 

507.9 

579-3 

256 


WEIGHT   OF   METALS. 


Table  No.  86  (continued}. 
BY  EXTERNAL  DIAMETER,     i  FOOT  LONG. 


EXT. 

THICKNESS  IN  INCHES. 

DlAM. 

3/i  6 

X 

5/i  6 

K 

7/16 

# 

9/i6 

H 

X 

7/s 

i 

ft.  in. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

5  3 

•03 

•44 

1.71 

2.06 

2.39 

2-74 

3.08 

3-42 

4.09 

4-77 

5-43 

56 

.08 

•50 

1.  80 

2.16 

2.50 

2.87 

3-22 

3.58 

4.29 

5.00 

5  9 

•*3 

•55 

1.88 

2.26 

2.62 

3.00 

3-37 

3.75 

4.49 

5-23 

5*96 

60 

.18 

.61 

1.96 

2.36 

2.74 

3-14 

3-52 

3-91 

4.69 

5-46 

6.22 

63 

•23 

•  67 

2.05 

2.45 

2.85 

3-27 

3-66 

4.08 

4.88 

5.69 

6-49 

66 

.28 

•73 

2.13 

2.55 

2.97 

3-40 

3.81 

4.24 

5.08 

5-92 

6.75 

69 

•33 

.78 

2.21 

2.65 

3.09 

4.41 

5.28 

6.15 

7.01 

7  o 

•38 

.84 

2.29 

2.75 

3.20 

3.66 

4.10 

4.57 

5-47 

6.38 

7.28 

76 

•48 

•95 

2.46 

2.95 

3-43 

3-92 

4-39 

4.90 

5-87 

6.84 

7.80 

80 

•58 

2.07 

2.62 

3.15 

3-67 

4.19 

4.69 

5.23 

6.26 

7-30 

8-33 

THICKNESS  IN  INCHES. 

i% 

i* 

I# 

«# 

*# 

2 

2X 

*y2 

2^ 

3 

3/2 

4 

inches. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

6 

.481 

.520 

•557 

•592 

.652 

.701 

.740 

.761 

6l/2 

•53° 

•575 

.618 

.657 

.789 

.838 

.872 

.906 

7 

579 

•  630 

.678 

•723 

•  805 

.876 

.938 

.982 

03 

•05 

.629 

.685 

•738 

.789 

.882 

.964 

1.04 

1.09 

15 

.18 

8 

.678 

.740 

•799 

.855 

•959 

.05 

.14 

.20 

27 

•32 

1.38 

S^2 

.727 

•794 

•859 

.921 

.04 

.14 

•23 

•31 

39 

•45 

i-53 

9 

•777 

.849 

.919 

.986 

.11 

•23 

•33 

.42 

51 

•58 

1.69 

1-75 

.826 

.904 

.980 

1.05 

•19 

•31 

•43 

•53 

63 

•71 

1.84 

i-93 

10 

.875 

•959 

.04 

.12 

.27 

.40 

•53 

.64 

75 

1.84 

1.99 

2.10 

ii 

•974 

.07 

.16 

.25 

.42 

.58 

•73 

.86 

99 

2.IO 

2.30 

2.46 

12 

.07 

.18 

.28 

•38 

•57 

•75 

.92 

2.08 

2.23 

2-: 

57 

2.61 

2.81 

13 

•17 

•29 

.40 

•51 

•73 

•93 

2.12 

2.30 

2.47 

2.(. 

>3 

2.92 

3.16 

14 

•  27 

.40 

.52 

•64 

1.88 

2.10 

2.32 

2.52 

2 

71 

2.89 

3-22 

3-  Si 

15 

•37 

•65 

•78 

2.03 

2.28 

2.52 

2.74 

2 

95 

3.] 

6 

3-53 

3-86 

16 

•47 

.62 

•77 

.91 

2.19 

2-45 

2.71 

2.96 

3 

19 

3-42     3-84 

4.21 

17 

•57 

•73 

.89 

2.04 

2-34 

2.63 

2.91 

3.18 

3 

•44 

3.68  1  4.14 

4-56 

18 

1.66 

1.84 

2.01 

2.17 

2.49 

2.81 

3.H 

3-40 

3 

.68 

3-95 

4-45 

4.91 

20 

1.86 

2.06 

2.25 

2-43 

2.80 

3.16 

3-50 

3-83 

4.16 

4-47 

5.06 

I'61 

22 

2.06 

2.27 

2.49 

2.70 

3-11 

3.51 

3-90 

4.27 

4 

.64 

5.00 

5-68 

6.32 

24 

2.26 

2-49 

2-73 

2.96 

3-41 

3.86 

4.29 

4.71 

5 

.12 

5-52 

6.29 

7.01 

27 

2-55 

2.82 

3-°9 

3-35 

3-87 

4.38 

4.88 

5-37 

5 

•85 

6.31 

7.21 

8.06 

30 

9 

2.85 
3-14 
3-44 

3-48 
3-8i 

3-82 
4.18 

3-75 
4.14 

4-54 

4-33 
4-79 
5-25 

4.91 

5-44 

5-47 
6.06 
6.66 

6.03 
6.68 
7-34 

6-57 
7.29 
8.01 

7.10 

7.89 
8.68 

8.13 
9-05 
9-97 

9.12 

10.2 
II.  2 

39 
42 

3-74 
4-°3 

4.14 

4-47 

4-54 
4.90 

4-93 
5-33 

lit 

6.49 
7.01 

7-25 
7.84 

8.00 
8.66 

8.74 
9.46 

9-47 
10.3 

10.9 
ii.  8 

12.3 
13-3 

4-33 

4-79 

5-26 

5-72 

6.64 

7-54 

8-43 

9.31 

10.2 

ii 

.1 

12.7 

14.4 

48 

4.62 

5.12 

5.62 

6.12 

7.10 

8.07 

9.02 

9-98 

10.9 

ii 

.8 

13-7 

15-4 

51 

4.92 

5-45 

5.98 

6.51 

7-56 

8-59 

9.61 

10.6 

11.6 

12.6 

14.6 

16.5 

54 

1? 

5.22 
S'.sl 

6.44 

6-35 
6.71 
7.07 

6.91 

7-30 
7.70 

8.02 

8.48 
8.94 

9.12 
9.64 

10.2 

IO.2 

10.8 
11.4 

"•3 
"'§ 

12.6 

12.4 

III 

13-4 
14.2 

15-0 

15-5 
16.4 

17-3 

17-5 

18.6 
I9.6 

CAST-IRON   CYLINDERS. 


257 


Table  No.  86  (continued}. 
BY  EXTERNAL  DIAMETER,     i  FOOT  LONG. 


EXT. 

DlAM. 

THICKNESS  IN  INCHES. 

2 

ol/ 

0  I/ 

•93/ 

0  I/ 

ft.  in. 

ZA 

2>£ 

2% 

3 

3/2 

4 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

M 

6.10 
6.40 

6.77 
7.09 

7-43 

7-79 

8.09 
8.48 

9.40 
9.86 

10.7 

II.  2 

I2.O 
12.6 

13-3 

13-9 

14.5 

1S-2 

15.8 

16.6 

18.3 

59 

6.70 

7.42 

8.15 

8.88 

10.3 

II.  8 

13.2 

14.6 

15-9 

60 

7.00 

7-75 

8.51 

9.27 

10.8 

I2.3 

13-8 

15.2 

63 

7.29 

8.08 

8.88 

9.67 

II.  2 

12.8 

14.4 

15-9 

66 

7.58 

8.41 

9.24 

10.  1 

11.7 

13-3 

14.9 

16.6 

69 

7.88 

8.74 

9.60 

10.5 

12.2 

13-9 

15-5 

17.2 

70 

8.17 

9.07 

9.96 

10.9 

12.6 

14.4 

16.1 

17.9 

76 

8-77 

9.72 

10.7 

ii.  6 

13-5 

15-4 

17-3 

19.2 

80 

9-36 

10.4 

11.4 

12.4 

14-5 

16.5 

18.5 

20.5 

86 

9-95 

II.  0 

12.  1 

13.2 

15-4 

19.7 

21.8 

90 

10.5 

ii.  7 

12.9 

14.0 

I6.3 

18.6  ' 

20.8 

23.1 

96 

ii.  i 

12.3 

13-6 

14.8 

17.2 

19.6 

22.0 

24.4 

100 

11.7 

13.0 

14.3 

15.6 

18.1 

20.7 

23.2 

25.7 

10  6 

12.3 

13-7 

15-0 

16.4 

19.1 

21.7 

24.4 

27.1 

II  0 

12.9 

14-3 

15-7 

17.2 

20.0 

22.8 

25.6 

28.4 

ii  6 

13-5 

15.0 

I6.5 

17.9 

20.9 

23-8 

26.7 

29.7 

12  0 

14.1 

15.6 

17.2 

18.7 

21.8 

24.9 

27.9 

31.0 

I30 

15-3 

16.9 

18.6 

20.3 

23-7 

27.0 

30-3 

33.6 

140 

16.5 

18-3 

20.1 

21.9 

25-5 

29.1 

32-7 

36.3 

15  o 

ll'l 

19.6 

21-5 

23-5 

27-3 

31.2 

35-  ° 

38.9 

160 
17  o 
180 

20.  o 

21.2 

20.9 

22.2 
23-5 

23.0 

24.4 
25-9 

25.0 
26.6 
28.2 

29.2 
31.0 
32.9 

33-3 
35-4 
37-5 

37-4 
!39-8 
42.2 

41.5 

£1 

190 

22.4 

24.8 

27-3 

29.8 

34-7 

39-6 

44-5 

49-4 

i 

200 

23-6 

26.1 

28.8 

31-4 

36.5 

41.7 

46.9 

52.0 

17 

258 


WEIGHT   OF   METALS. 


Table  No.  87. — VOLUME  AND  WEIGHT  OF  CAST-IRON  BALLS. 
GIVEN  THE  DIAMETER. 


Diameter. 

Contents. 

Weight. 

Diameter. 

Contents. 

Weight. 

Diameter. 

Contents. 

Weight. 

inches. 

cubic 
inches. 

pounds. 

inches. 

cubic 
inches. 

pounds. 

inches. 

cubic  feet. 

cwts. 

I 

.524 

.136 

8 

268.1 

69.8 

19 

2.078 

8.35 

I^2 

1.77 

.460 

Sj/2 

321.5 

83-7 

20 

2.424 

9-74 

2 

4.19 

I.O9 

9 

38L7 

99-4 

21 

2.806 

11.28 

2/^2 

8.18 

2.13 

448.9 

116.9 

22 

3.227 

12.97 

3 

I4.I 

3.68 

r> 

10 

523-6 

136.4 

23 

3-688 

O  O 

14.82 

3/^ 

22.5 

5-85 

inches. 

cubic  feet. 

cwts. 

24 

4.188 

16.83 

4 

33-5 

8-73 

II 

.403 

1.62 

25 

4.736 

19.03 

4/^ 

47-7 

12.4 

12 

.524 

2.10 

26 

5-327 

21.40 

5 

65-5 

17.0 

13 

.666 

2.68 

27 

5-963 

23.96 

87.1 

22.7 

14 

.832 

3-34 

28 

6.651 

26.72 

6 

113.1 

29-5 

15 

1.023 

4.11 

29 

7.390 

29.69 

6^ 

143.8 

37-5 

16 

1.241 

4.99 

30 

8.181 

32.87 

7 

179.6 

46.8 

17 

1.489 

5.98 

31 

9.027 

36.27 

7^ 

220.9 

57.5 

18 

1.767 

7.10 

32 

9.930 

39-90 

Note. — To  find  the  weight  of  balls  of  other  metals,  multiply  the  weight  given  in  the 
table  by  the  following  multipliers  : — 

For  Wrought  Iron 1.067,  making  about    7  per  cent.  more. 

Steel i. 088  ,,  9  ,, 

Brass  1.12  ,,  12  „ 

Gun  Metal  1.165  ,,  i6>£          ,, 


Table  No.  88. — DIAMETER  OF  CAST-IRON  BALLS. 
GIVEN  THE  WEIGHT. 


Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

pounds. 

inches. 

pounds. 

inches. 

pounds. 

inches. 

cwts. 

inches. 

% 

1-54 

14 

4.68 

80 

8.37 

8 

18.73 

I 

1.94 

16 

4.89 

90 

8.71 

9 

19.48 

2 

2-45 

0 

18 

5-°9 

IOO 

9.O2 

10 

20.17 

3 

4 

2.OO 
3.08 

20 
25 

5-27 
5.68 

cwts. 

T 

inches. 

9f+  >7 

12 
14 

21.44 

22.57 

5 

3-32 

28 

5-9° 

l% 

•37 
10.72 

16 

23.60 

6 

3-53 

30 

6.04 

2 

11.80 

18 

24-54 

7 

3-72 

40 

6.64 

3 

i3-5i 

20 

25.42 

8 

3-89 

5° 

7.16 

4 

14.87 

25 

27.38 

9 

4.04 

56 

7-43 

5 

16.02 

30 

29.10 

10 

4.19 

60 

7.60 

6 

17.02 

35 

30.64 

12 

4-45 

70 

8.01 

7 

17.91 

40 

32.03 

WEIGHT   OF   FLAT   BAR    STEEL. 


259 


Table  No.  89. — WEIGHT  OF  FLAT  BAR  STEEL. 
i  FOOT  LONG. 


WIDTH  n 

4  INCHES. 

THICKNESS 

# 

% 

# 

ft 

I 

iX 

K# 

IM 

inches. 

Ib. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

X 
5/i  6 

.425 

•531 

-m 

.640 
.800 

•743 
.929 

•  850 
1.  06 

1.06 
i-33 

1.28 

i-59 

1.49 

1.86 

y& 

.638 

.798 

.960 

i.  ii 

1.28 

1.91 

2.23 

7/i6 

•744 

.931 

.12 

1.30 

1.49 

1.86 

2.23 

2.60 

$ 

.850 

i.  06 

.28 

1.49 

1.70 

2.13 

2-55 

2.98 

9/i6 

— 

1.20 

•44 

1.67 

I.9I 

2-39 

2.87 

3-35 

— 

i-33 

.60 

1.86 

2.12 

2.66 

3-19 

3-72 

»     6 

— 

.76 

2.04 

2-34 

2.92 

3-51 

4.09 

3^ 

— 

— 

.92 

2.23 

2-55 

3-83 

4.46 

J3/i6 

— 

— 

2.41 

2.76 

3-45 

4.14 

4-83 

?8 

— 

— 

— 

2.60 

2.98 

3-72 

4.46 

5-21 

J5/i6 

— 

— 

— 

— 

3-19 

3-98 

4-78 

5-58 

I 

— 

— 

— 

— 

3-40 

4-25 

5-10 

5-95 

WIDTH  n 

4  INCHES. 

THICKNESS 

2 

2* 

2/2 

2^ 

3 

3X 

3/2 

4 

inches. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

X 

1.70 

1.91 

2.13 

2-34 

2-55 

2.76 

2.98 

3-40 

5/i6 

2.13 

2-39 

2.66 

2.92 

3-45 

3-72 

4-25 

N 

7/x6 

2.98 

2.87 

3-35 

3-19 
3-72 

3-51 

4.09 

3-83 

4.46 

4.14 
4-83 

4.46 
5.21 

5.10 

5-95 

X 

3-40 

3-83 

4-25 

4.68 

5.10 

5-53 

5-95 

6.80 

9/i  6 

3-83 

4-30 

4.78 

5.26 

5-74 

6.22 

6.69 

7.65 

# 

4-25 

4-78 

5-31 

5.84 

6.38 

6.91 

7-44 

8.50 

4.68 

5.26 

5-84 

6-43 

7.01 

7.60 

8.18 

9-35 

^ 

5.10 

5-74 

6.38 

7.01 

7.65 

8.29 

8-93 

IO.2 

x3/i6 

5-53 

6.22 

6.91 

7.60 

8.29 

8.98 

9.67 

II.  I 

% 

6.69 

7-44 

8.18 

8-93 

9.67 

10.4 

II.9 

i5/!  6 

6.38 

7.17 

7-97 

8-77 

9-56 

10.4 

II.  2 

12.8 

I 

6.80 

7.65 

8.50 

9-35 

10.2 

II.  I 

II.9 

13.6 

WIDTH  it 

i  INCHES. 

THICKNESS 

4X 

5 

5/2 

6 

6/2 

7 

7/2 

8 

inches. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

X 

3-82 

4.26 

4.68 

5.10 

5-52 

5.96 

6.38 

6.80 

5/i6 

4.78 

5-32 

5-84 

6.38 

6.90 

7.44 

7-97 

8.50 

N 

5-74 

6.38 

7.02 

7.66 

8.28 

8.92 

9-56 

10.2 

7/i6 

6.70 

7-44 

8.18 

8.92 

9.66 

10.4 

II.  2 

II.9 

^2 

7.66 

8.50 

9-36 

10.2 

II.  I 

11.9 

12.8 

13-6 

9/i6 

8.60 

9.56 

10.5 

lI-5 

12.4 

13-4 

14-3 

15-3 

4 

9.56 

10.6 

11.7 

12.8 

13-8 

14.9 

15-9 

17.0 

10.5 

11.7 

12.9 

14.0 

15.2 

16.4 

17-5 

18.7 

^£ 

"•5 

12.8 

14.0 

15.3 

16.6 

17.9 

I9.I 

2O.4 

»3/x6 

12.4 

13.8 

15.2 

16.6 

18.0 

19-3 

20.7 

22.2 

^  6 

13-4 

14.9 

16.4 

17.9 

19.4 

20.8 

22.3 

23-8 

H-3 

15-9 

17-5 

19.1 

20.8 

22.4 

23-9 

25.6 

I 

15-3 

17.0 

18.7 

20.4 

22.1 

23.8 

25.5 

27.2 

260 


WEIGHT   OF   METALS. 


Table  No.  90. — WEIGHT  OF  SQUARE  STEEL. 
i  FOOT  IN  LENGTH. 


Size. 

Weight. 

Size. 

Weight. 

Size. 

Weight. 

Size. 

Weight. 

inches. 

pounds. 

inches. 

pounds. 

inches. 

pounds. 

inches. 

pounds. 

/8 

•053 

'V.6 

3.06 

1% 

IO.4 

3*/4 

35-9 

3A6 

.119 

I 

3-40 

II3/i6 

II.  2 

1% 

41.6 

% 

.212 

I   Vi6 

3.83 

1/8 

II.9 

31/4 

47-8 

5/i6 

•333 

1/8 

4-3° 

II5/i6 

12.8 

4 

54-4 

tf 

.478 

I   Vi6 

4-79 

2 

13-6 

4^ 

61.4 

7/x6 

.651 

'# 

5-3i 

2/8 

15-4 

4^ 

68.9 

% 

.850 

I   Vi6 

5.86 

*X 

17.2 

4& 

76.7 

9/!6 

i.  08 

iH 

6-43 

23/8 

19.2 

5 

85.0 

& 

i-33 

I  7/i6 

7-03 

*% 

21.2 

5^ 

93-7 

*/* 

1.61 

'# 

7.65 

2/8 

23-5 

5^ 

102.8 

# 

1.92 

I  9/i6 

8.30 

2% 

25-7 

5?4 

112.4 

13A6 

2.24 

Ijl 

8.98 

2/8 

28.2 

6 

122.4 

/8 

2.60 

l"/i6 

9-79 

3 

30.6 

Table  No.  91. — WEIGHT  OF  ROUND  STEEL. 
i  FOOT  IN  LENGTH. 


Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

Diameter. 

Weight. 

inches. 

pounds. 

inches. 

pounds. 

inches. 

pounds. 

inches. 

cwts. 

/ 

.0417 

7/i6 

5.l8 

4 

42.7 

12 

3-433 

3A6 

•°939 

% 

6.01 

48.3 

l2/2 

3-729 

k 

.167 

9/i6 

6.52 

4/ 

54-6 

13 

4.029 

5/x6 

.260 

y§ 

7.05 

4^4 

60.3 

J3>2 

4-345 

•375 

"/.6 

7.61 

5 

66.8 

14 

4.682 

7/i6 

•511 

8.18 

5/4 

73-6 

14^ 

S-0  1  3 

% 

.667 

13/i6 

8.77 

80.8 

15 

c.  364 

9/x6 

•845 

1^ 

9.38 

5/4 

88.3 

I5IA 

c.  7  28 

1.04 

II5/i6 

10.  0 

6 

96.1 

16 

6.103 

I/6 

1.27 
1.50 

2 

10.7 

12.0 

inches. 

cwts. 

J7 

6.471 
6.868 

<3A6 

1.76 

2/ 

I3.6 

6/^ 

7 

.007 

.168 

7.302 

7^ 

2.04 

2^/8 

I  tC.  I 

/ 

18 

7.724 

'V.6 

2-35 

2/2 

rf>.7 

8 

•341 
.526 

i9 

8.607 

I 

2.67 

2/8 

18.4 

8)^ 

•723 

20 

9-537 

I   x/i6 

3.00 

2%i 

20.2 

9 

I-931 

21 

10.52 

1^ 

3.38 

2J/Z 

22.0 

9^3 

2.152 

22 

n-54 

I  3/x6 

3.76 

3i 

24.1 

10 

2-385 

23 

12.62 

ll^ 

4.17 

28.3 

io/ 

2.629 

24 

13-73 

I  S/i6 

4.60 

3/ 

32.7 

II 

2.884 

13/8 

5-05 

3^ 

34-2 

II/ 

3-150 

WEIGHT  OF  CHISEL   STEEL. 


26l 


Table  No.  92. — WEIGHT  OF  CHISEL  STEEL — HEXAGONAL,  OCTAGONAL, 

AND  OVAL-FLAT. 
i  FOOT  IN  LENGTH. 


HEXAGONAL  SECTION. 

OCTAGONAL  SECTION. 

Diameter 

across  the 
Sides. 

Sectional  Area. 

Weight. 

Length 
to  weigh 

Sectional  Area, 

Weight. 

Length 
to  weigh 

I  CWt. 

I  CWt. 

inches. 

square  inches. 

pounds. 

feet. 

square  inches. 

pounds. 

feet. 

M 

.1217 

.414 

245 

.1164 

•396 

268 

% 

.2165 

•736 

138 

.2070 

.704 

151 

y* 

.3383 

i.J;5 

88.3 

.3236 

1.  10 

96-5 

X 

.4871 

1.66 

6l-3 

.4659 

1.58 

67 

% 

.6631 

2.25 

45 

.6342 

2.16 

49-3 

i 

.8661 

2.94 

34-5 

.8284 

2.82 

37-7 

i# 

1.096 

3-73 

27-3 

1.048 

3.56 

3° 

'# 

1-353 

4.60 

22.5 

1.294 

4.40 

24 

13/S 

1.637 

5-57 

18.3 

1.566 

5-32 

20 

i* 

1.949 

6.63 

15-3 

1.864 

6.34 

16.8 

OVAL-FLAT  SECTION. 

inch.     inch. 

24  x  3/8 

.2510 

.853 

119 

I        x  ^ 

.4463 

1-52 

67 

1%  x^ 

.6974 

2-37 

43 

Table  No.  93. — WEIGHT  OF  ONE  SQUARE  FOOT  OF  SHEET  COPPER. 

To  Wire-Gauge  employed  by  Williams,  Foster,  &  Co. 
Specific  Weight  taken  as  1.16  (Specific  Weight  of  Wrought  Iron  =  i). 


Thickness. 

Weight  of 
i  Square 
Foot. 

Thickness. 

Weight  of 
i  Square 
Foot. 

Thickness. 

Weight  of 
i  Square 
Foot. 

Wire- 

Inch 

Wire- 

Inch 

Wire- 

Inch 

Gauge. 
No. 

(approxi- 
mate). 

pounds. 

Gauge. 

No. 

(approxi- 
mate). 

pounds. 

Gauge. 

No. 

(approxi- 
mate). 

pounds. 

I 

.306 

14.0 

II 

.123 

5^5 

21 

.0338 

i-55 

2 

.284 

13.0 

12 

.109 

5.00 

22 

.0295 

i-35 

3 

.262 

12.  0 

J3 

.0983 

4-5° 

23 

.0251 

i-i5 

4 

.240 

II.  0 

14 

.0882 

4.00 

24 

.0218 

I.OO 

5 

.222 

IO.I5 

15 

.0764 

3-5° 

25 

.0194 

.89 

6 

.203 

9-30 

16 

•0655 

3.00 

26 

.0172 

•79 

7 

.186 

8.50 

17 

.0568 

2.60 

27 

•0153 

.70 

8 

.168 

7.70 

18 

.0491 

2.25 

28 

•o!35 

.62 

9 

•J53 

7.00 

!9 

•0437 

2.00 

29 

.0122 

•56 

10 

.138 

6.30 

20 

.0382 

i-75 

3° 

.OIIO 

•5o 

262 


WEIGHT   OF   METALS. 


Table  No.  94. — WEIGHT  OF  COPPER  PIPES  AND  CYLINDERS, 
BY  INTERNAL  DIAMETER. 

LENGTH,   i  FOOT.     Thickness  by  Holtzapffel's  Wire-Gauge  (Table  No.  13). 
Specific  Weight=i.i6  (Specific  Weight  of  Wrought-Iron  =  i). 


THICK- 
NESS. 
W.  G. 

0000 

ooo 

oo 

0 

I 

2 

3 

4 

5 

6 

7 

INCH. 

•454 

29/64 

.425 

*7/64/ 

.380 

•340 

11/32 

.300 

.284 

•259 

.238 

<5/64/ 

.220 

7/32  / 

.203 
13/64 

.180 

3/i  6  b. 

INT. 

DIAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

inches. 

y& 

3M4 

2.84 

2-33 

1.92 

i-53 

1.41 

1.  21 

1.05 

•934 

.809 

.667 

1A 

3-49 

2.91 

2.44 

1.99 

1.84 

1.  60 

1.27 

1.  12 

.941 

y% 

4-54 

4-13 

3-49 

2-95 

2-45 

2.27 

2.00 

1.77 

i.  60 

i-43 

1.  21 

l/2 

5-23 

4-78 

4.06 

3-47 

2.91 

2.71 

2-39 

2.13 

i-93 

1-73 

1.49 

H 

5-93 

5-42 

4.64 

3-99 

3-37 

3-H 

2.78 

2.50 

2.26 

2.04 

I.76 

y* 

6.63     6.07 

5.22 

4-5° 

3.83 

3-57 

3.r7 

2.86 

2.60 

2-35 

2.03 

y& 

7.32     6.71 

5-79 

5.02 

4.29 

4.00 

3-57 

3.22 

2-93 

2.66 

2.3I 

1 

8.02     7.36 

6-37 

5-53 

4-74 

4-43 

3.96 

3-57 

3.26 

2.97 

2.58 

i# 

8.71 

8.00 

6-95 

6.05 

5-20 

:  4.86 

4-35 

3-94 

3.60 

3.28 

2.85 

i^ 

9.40 

8.65 

7-52 

6-57 

5-65 

\   5-29 

4-75 

4-3° 

3-93 

3.58 

3-13 

\y^ 

IO.  I 

9-3° 

8.10 

7.08 

6.  ii 

;  5-72 

5-14 

4.66 

4.26 

3.89 

3-40 

l& 

10.8 

9-94 

8.68 

7.60 

6.57 

\  6.16 

5-53 

5.02 

4.60 

4.20 

3-68 

1% 

"•5 

10.6 

9.26 

8.12 

7.02 

!  6.59 

5-93 

5-39 

4-93 

4.51 

3-95 

ify( 

12.  1 

II.  2 

9-83 

8.63 

7.48 

7.02 

6.32 

5-75 

5-27 

4.82 

4.22 

iH 

12.8 

11.9 

10.4 

9-15 

7-93 

7-45 

6.71 

5.60 

5.12 

4-5° 

2 

13-5 

12.5 

II.  0 

9.66 

8-39 

7-88 

7.11 

6.47 

5-93 

5-43 

4-77 

2% 

14.2 

13.2 

ii.  6 

10.2 

8.84 

8.31 

7-5° 

6.83 

6.27 

5-74 

5-04 

2X 

14.9 

13.8 

12.  1 

10.7 

9-30 

8.74 

7.89 

7.19 

6.60 

6.05 

5-32 

15.6 

14.5 

12.7 

II.  2 

9-75 

9-17 

8.29 

7.56 

6.94 

6.36 

5-59 

2>f 

I6.3 

13-3 

II.7 

10.2 

\  9.60 

8.68 

7.92 

7.27 

6.67 

5.86 

2% 

17.0 

15.8 

13-9 

12.2 

10.7 

10.0 

9.07 

8.28 

7.60 

6.97 

6.14 

17.7 

16.4 

14-5 

12.8 

II.  I 

10.5 

9-47 

8.64 

7-94 

7.28 

6.41 

2,7/% 

18.4 

17.1 

15.0 

*3'3 

11.57 

[    10.9 

9.86 

9.00 

8.27 

7-59 

6.68 

3 

I9.I 

17.7 

I5.6 

13-8 

12.  0 

j    "-3 

10.3 

9-36 

8.61 

7.90 

6-95 

3X 

20.4 

19.0 

16.8 

14.8 

12.9 

12.2 

ii.  i 

10.  1 

9-27 

8.52 

7-50 

21.8 

20.3 

17.9 

15-9 

13-9 

I3-  * 

ii.  8 

10.8 

9-94 

9-13 

8.04 

3K 

23.2 

21.6 

19.1 

16.9 

I4.8 

13-9 

12.6 

11.5 

10.6 

9-75 

8-59 

4 

24.6 

22.9 

20.2 

17.9 

15-7 

14.8 

13-4 

12.3 

II-3 

10.4 

9-13 

4# 

25-9 

24.2 

21.4 

19.0 

16.6 

15.6 

14.2 

13.0 

12.  0 

II.  0 

9.67 

4/i 

27-3 

25-4 

22.5 

20.  o 

17-5 

16.5 

15.0 

13-7 

12.7 

ii.  6 

IO.2 

4^ 

28.7 

26.7 

23-7 

21.0 

18.4 

17.4 

15.8 

14.4 

13-3 

12.2 

10.8 

5 

30.1 

28.0 

24-8 

22.1 

19-3 

18.2 

16.6 

i5-i 

14.0 

12.8 

"•3 

s# 

3J-5 

29-3 

26.0 

23.1 

20.  2 

19.1 

17-3 

15-9 

I4.6 

13-5 

11.9 

sK 

32*8 

30.6 

27.  I 

24.1 

21.  1 

20.0 

18.1 

16.6 

I4.I 

12.4 

5^ 

34-2 

31-9 

28.3 

25-2 

22.1 

20.8 

18.9 

17-3 

16.0 

14.7 

12.9 

6 

35-6 

33-2 

29-5 

26.2 

23.0 

21.7 

19.7 

18.0 

16.6 

15-3 

13-5 

WEIGHT   OF  COPPER   PIPES   AND  CYLINDERS. 

Table  No  94  (continued}. 

LENGTH,   i  FOOT.     Thickness  by  Holtzapffel's  Wire- Gauge  (Table  No.  13). 
Specific  Weight- 1. 16  (Specific  Weight  of  Wrought  Iron  =  i). 


263 


THICK- 
NESS. 

8 

9 

10 

ii 

12 

13 

14 

15 

16 

17 

18 

19 

20 

W.  G. 

INCH. 

ix1?5* 

.148 

•  134 

9/64-5 

.120 

.109 

7/64 

•095 

.083 

5/64/ 

.072 

.065 

.058 

.049 

3/64/ 

.042 

3/64  ^ 

•035 

INT. 

DlAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

inches. 

ft 

.581 

.491 

.422 

•357 

.310 

.254 

.210 

•  173 

.150 

.129 

.IO4 

.086 

.068 

1 

•832 

1.08 

.716 
.941 

.626 
.830 

•540 

.722 

.476 
.641 

•543 

£ 

.282 
•391 

•249 
.348 

.217 

•305 

.178 
•253 

.149 
.213 

.121 
•175 

i-33 

1.17 

1.03 

.904 

.807 

.687 

.588 

.500 

•447 

•393 

.327 

.277 

.228 

8 

1.58 

i-39 
1.62 

1.24 
1.44 

1.09 
1.27 

.972 
I.I4 

.831 
•975 

.714 
.840 

.610 

.719 

.644 

.481 
•  570 

.402 
.476 

•341 

.281 

•334 

y& 

2.09 

1.84 

1.65 

i-45 

1.30 

1.  12 

.966 

.828 

•743 

.658 

•550 

.468 

.387 

i 

2-34 

2.05 

1.85 

1.63 

1.47 

1.26 

1.09 

.938 

.842 

.746 

.625 

.532 

.440 

1/8 

2.59 

2.27 

2.05 

1.82 

1.63 

I.4I 

.22 

•05 

.940 

•834 

.699 

.596 

•493 

2.84 

2.49 

2.26 

2.00 

1.  80 

i-55 

•34 

.16 

.04 

.922 

•774 

•659 

•547 

j3^ 

3-°9 

2.72 

2.46 

2.18 

1.97 

1.70 

•47 

•27 

.14 

.01 

.848 

•  723 

.600 

i*5 

3-34 

2-94 

2.67 

2.36 

2.13 

1.84 

.60 

.38 

.24 

.10 

.922 

.787 

•653 

i# 

3-59 

3-17 

2.87 

2-55 

2.30 

1.99 

.72 

.48 

•34 

.19 

•997 

.851 

.706 

;| 

3-84 
4.09 

3-39 
3-62 

3-07 
3-28 

2-73 
2.91 

2.46 
2.63 

2.13 
2.27 

•85 
•97 

•59 
.70 

•43 
•53 

.27 

•36 

.07 
•15 

•915 

.978 

•759 
.812 

2 

4-34 

3-84 

3-48 

3-09 

2.79 

2.42 

2.  IO 

.81 

•63 

•45 

.22 

1.04 

.865 

2^ 

4-59 

4.07 

3.69 

3-27 

2.96 

2.56 

2.23 

1.92 

•73 

•54 

•29 

.11 

.919 

2^ 

4.84 

4.29 

3*89 

3-45 

3.12 

2.71 

2-35 

2.03 

•83 

•63 

•38 

•  17 

.972 

2^ 

5-09 

4-52 

4.09 

3-64 

3-29 

2.85 

2.48 

2.14 

•93 

•7i 

•45 

•23 

•03 

2>£ 

5-34 

4-74 

4-30 

3-82 

3-45 

3.00 

2.60 

2.25 

2.03 

.80 

•53 

•30 

.08 

2^ 

5-59 

4-97 

4-5° 

4.00 

3-62 

2-73 

2.36 

2.13 

.89 

.60 

.36 

.13 

2^ 

5-19 

4.71 

4.18 

3-79 

3'-28 

2.86 

2.47 

2.22 

•  98 

.68 

•43 

.18 

2% 

6.09 

5-42 

4.91 

4-37 

3-95 

3-43 

2.98 

2.58 

2.32 

2.07 

•75 

•49 

.24 

3 

6-34 

5.66 

5-" 

4-55 

4.12 

3-57 

3-n 

2.69 

2.42 

2.16 

.82 

•55 

•29 

3/< 

6.85 

6.ii 

5-52 

4.91 

4-45 

3-86 

3-36 

2.91 

2.62 

2-33 

1.96 

.68 

.40 

3X 

7-35 

6.56 

5-93 

5.28 

4-78 

4-15 

3-6i 

3.12 

2.82 

2.51 

2.  II 

.81 

•51 

3^ 

7.85 

7.01 

6-33 

5-64 

5-" 

4-44 

3-87 

3-34 

3.01 

2.68 

2.26 

•94 

.62 

4 

8.35 

7.46 

6.74 

6.01 

5-44 

4-73 

4.12 

3-56 

3-21 

2.86 

2.4I 

2.06 

•73 

4X 

8.85 

7.91 

7.14 

6-37 

5-77 

5.02 

4-37 

3-78 

3.41 

3-04 

2.56 

2.19 

•84 

4/^ 

9-35 

8.36 

7.55 

6.74 

6.  10 

5-30 

4.62 

4.00 

3.61 

3.21 

2.71 

2.32 

•94 

4* 

9-85 

8.81 

7.96 

7.10 

6-43 

5-59 

4.87 

4.22 

3.80 

3-39 

2.86 

2-45 

2.05 

5 

10.4 

9.26 

8.36 

7.46 

6.77 

5.88 

4-44 

4.00 

3-56 

3.01 

2-57 

2.16 

5X 

10.9 

9.71 

8-77 

7.83 

7.10 

6.17 

5-38 

4.66 

4.20 

3-74 

3-15 

2.70 

2.27 

50 

11.4 
11.9 

10.2 

10.6 

9.18 
9.58 

8.19 
8.56 

6.46 
6-75 

& 

4.88 
5-09 

4.40 

4-59 

3-92 
4.09 

3-30 
3-45 

2.83 
2.96 

2.38 
2.48 

6 

12.4 

II.  I 

9-99 

8.92 

I'.lg 

7.04 

6.14 

5-31 

4-79 

4.27 

3-6o 

3-09 

2.58 

264  WEIGHT   OF   METALS. 

Table  No.  94  (continued). 

LENGTH,  i  FOOT.     Thickness  by  Holtzapffel's  Wire-Gauge  (Table  No.  13). 
Specific  Weight^  1. 16  (Specific  Weight  of  Wrought- Iron  =  i). 


THICK- 
NESS. 
W.  G. 

oooo 

ooo 

00 

o 

i 

2 

3 

4 

5 

6 

7 

•454 

.425 

.380 

•340 

.300 

.284 

•259 

^238 

.220 

•  203 

.180 

INCH. 

26/64 

Hf- 

11/32 

T9/64/ 

k/ 

7/3*  / 

13/64 

3/i6  b. 

INT. 

DlAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

inches. 

&l/z 

38.4 

35-8 

31-8 

28.3 

21.3 

19-5 

18.0 

14.6 

7 

41.  1 

38.3 

34-i 

30.3 

26.6 

25-1 

22.8 

20.9 

19-3 

17.8 

15.7 

43-9 

40.9 

36-4 

32.4 

28.4 

22.4 

20.6 

16.8 

8  2 

46.6 

43-5 

38.7 

34-5 

30.3 

28.6 

26.0 

23-8 

22.O 

20.2 

17.9 

9 

52.1 

48.7 

43-3 

38.6 

33-9 

32.0 

29.1 

26.7 

24.6 

22.7 

20.1 

10 

57-7 

53-8 

47-9 

42.7 

37-5 

35-5 

32.3 

29.6 

27-3 

25-2 

22.2 

ii 

63-2 

59-o 

52.5 

46.8 

41.2 

38.9 

35-4 

32.5 

30.0 

27-7 

24.4 

12 

68.7 

64.2 

57-2 

51.0 

44-8 

42.4 

38.6 

35-4 

32.7 

30.1 

26.6 

13 

74-2 

69-3 

61.8 

55.1 

48.5 

45-8 

41.7 

38.3 

35-3 

32-6 

28.8 

14 

79-7 

74-5 

66.4 

59-2 

52.1 

49-3 

44-9 

41.2 

38.0 

35-  i 

31.0 

15 

85.2 

79-6 

71.0 

63-4 

55-8 

52-7 

48.0 

44.1 

40.7 

37-6 

33-2 

16 

90.7 

84.8 

75-6 

67-7 

59-4 

56.2 

51-2 

46.9 

43-4 

40.0 

35-4 

17 

96.3 

90.0 

80.2 

7!.8 

63.0 

59-6 

54-3 

49-8 

46.0 

42.5 

37-5 

18 

101.8 

95-  i 

84.9 

76.0 

66.7 

63.  i 

57-4 

52.7 

48.7 

45-o 

39-7 

19 

107.3 

100.3 

89.5 

80.  i 

70.3 

66.5 

60.6 

55-6 

51.4 

47-4 

41.9 

20 

II2.8 

105.5 

94.1 

84.2 

74-o 

70.0 

63-7 

58.5 

54-o 

49-9 

44.1 

21 
22 
23 

118.3 
123.8 
129.3 

110.7 
115.8 
120.9 

98.7 

103.3 
107.9 

88.3 

92.5 
96.6 

77-6 
81.3 
84-9 

73-4 
76.9 
80.3 

66.9 
70.0 

73-2 

61.4 

64-3 
67.2 

56.7 

a? 

52.4 
54-9 
57-3 

46.3 
48.5 
50.7 

24 

134.8 

126.1 

II2.6 

100.6 

88.6 

83-8 

76.3 

70.1 

64.7 

59-8 

52.9 

26 

146.0 

136.4 

121.  8 

1  08.  8 

95-9 

90.7 

82.6 

75-9 

70.1 

64.7 

57-2 

28 

157.2 

146.7 

131.0 

117.1 

103.1 

97-6 

89.0 

81.7 

75-4 

69.7 

61.6 

30 

168.4 

157.1 

140.2 

125.4 

110.4 

104.5 

95-3 

87.5 

80.8 

74-6 

66.0 

32 

179-6 

167.4 

149.5 

133-6 

117.7 

111.4 

101.6 

93-3 

86.2 

79-6 

70.4 

34 

190.7 

177.7 

141.9 

125.0 

118.3 

107.9 

99-i 

91-5 

84-5 

74-7 

36 

201.9 

1  88.0 

167.9 

150.1 

132-3 

125.2 

114.2 

104.9 

96.9 

89.5 

79.1 

WEIGHT   OF   COPPER   PIPES   AND   CYLINDERS. 
Table  No.  94  (continued}. 

LENGTH,  I  FOOT.     Thickness  by  Holtzapffel's  Wire-Gauge  (Table  No.  13). 
Specific  Weight  =  1. 16  (Specific  Weight  of  Wrought  Iron=  i). 


265 


THICK 

NESS. 

W.  G. 

8 

9 

10 

ii 

12 

13 

M 

15 

16 

17 

18 

19 

20 

INCH. 

^165 

9lf 

•134 

9/64  ^ 

.120 

.109 

7/64 

•095 

.083 

S/64/ 

•072 
5/64  ^ 

.065 

.058 

.049 

.042 

.035 

INT. 

DIAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

inches. 

6>^ 

13-4 

12.0 

10.8 

9.65 

8.75 

7.61 

6.64 

5-75 

5.19 

4.62 

3-90 

3-34 

2.8o 

7 

14.4 

12.9 

u.6 

10.4 

9.42 

8.19 

7.14 

6.19 

5.58 

4.97 

4-20 

3-6o 

3-oi 

7/^ 

15-4 

13-8 

12.47 

II.  I 

10.  1 

8.77 

7.65 

6.63 

5.98 

5-33 

4.49 

3.85 

3-23 

8 

16.4 

14.7 

13.2 

11.8 

10.74 

9-34 

8.15 

7.06 

6.37 

5.68 

4-79 

4.10 

3-43 

9 

18.4 

I6.5 

14.9 

13-3 

12.  1 

10.5 

9.16 

7-94 

7.16 

6.38 

5-39 

4.61 

3-86 

10 

20.4 

18.2 

16.5 

14.8 

13-4 

11.7 

10.2 

8.81 

7-95 

7.08 

5.12 

4.28 

ii 

22.4 

20.0 

18.1 

16.2 

14.7 

12.8 

II.  2 

9.69 

8.74 

7-79 

6  58 

5.63 

4.70 

12 

24.4 

21.8 

19.8 

17.7 

16.0 

14.0 

12.2 

10.6 

9.53 

8.49 

7.18 

6.14 

5.13 

I3 

26.4 

23-6 

21.4 

19.1 

17.4 

15.1 

13.2 

11.4 

10.3 

9.20 

7-77 

6.65 

5-55 

H 

28.4 

25.4 

23.0 

20.  6 

18.7 

16.3 

14.2 

12.3 

ii.  i 

9.90 

8-37 

7.16 

5.98 

15 

16 

30-4 
32.4 

27.2 
29.0 

24.6 
26.3 

22.1 
23-5 

20.0 
21-3 

17.4 

18.6 

15-2 

16.2 

13.2 
14.1 

11.9 
12.7 

10.6 

8.96 
9.56 

6.40 
6.82 

17 

34-4 

30.8 

27.9 

25.0 

22.7 

19.7 

17.2 

14.9 

13-5 

12.  1 

10.2 

8.69 

7-27 

18 

36.4 

32-6 

29-5 

26.4 

24.0 

20.9 

18.2 

14-3 

12.7 

10.7 

9.20 

7.69 

19 

38.4 

34-4 

31.2 

27.9 

25.3 

22.0 

19.2 

16.7 

I5-I 

13-4 

"•3 

9.71 

8.12 

20 

40.4 

36-2 

32.8 

29.3 

26.6 

23.2 

20.2 

17.6 

15-9 

I4.I 

11.9 

10.2 

8.54 

21 

42.4 

38.0 

34-4 

30.8 

27-9 

24-3 

21.3 

18.4 

16.6 

14.8 

12.5 

10.7 

8.96 

22 

44-4 

39-8 

36.0 

32.3 

29-3 

25-5 

22.3 

19-3 

17.4 

15-5 

II.  2 

9-39 

23 

46.4 

41.6 

37.7 

33.7 

30.6 

26.7 

23.3 

20.2 

18.2 

16.2 

13-7 

ii.  8 

9.81 

24 

48.5 

43-4 

39-3 

35-2 

31-9 

27.8 

24-3 

21.  1 

19.0 

16.9 

14.3 

12.3 

10.2 

26 

52.5 

47.0 

42.6 

38.1 

34-6 

30.1 

26.3 

22.8 

20.6 

18.4 

15.5 

13.3 

II.  I 

28 

56.5 

50.6 

45-8 

41.0 

37-2 

32.4 

28.3 

24.6 

22.2 

19.8 

16.7 

14-3 

II.9 

30 

60.5 

54-2 

49.1 

43-9 

39-9 

34-7 

30-3 

26.3 

23-7 

21.2 

17.9 

15-3 

12.8 

32 

64-5 

57-8 

52.3 

46.8 

42-5 

37-0 

32.3 

28.1 

25.3 

22.6 

19.1 

16.3 

13.6 

34 

68.5 

61.4 

55-6 

49.8 

45-i 

39-4 

34-4 

29.8 

26.9 

24.0 

20.3 

17.4 

14.5 

36 

72.5 

65.0 

58.8 

52.7 

47-8 

41.7 

36.4 

31.6 

28.5 

25.4 

21.5 

18.4 

15.3 

266 


WEIGHT   OF   METALS. 


Table  No.  95. — WEIGHT  OF  BRASS  TUBES, 

BY  EXTERNAL  DIAMETER. 

LENGTH,   I  FOOT.     Thickness  by  Holtzapffel's  Wire-Gauge  (Table  No.  13). 
Specific  Weight  =  1.1 1  (Specific  Weight  of  Wrought  Iron=i). 


THICK- 

NESS. 
W.  G. 

15 

16 

17 

18 

iQ 

20 

21 

22 

23 

24 

25 

INCH. 

.072 

5/64  ^ 

.065 
«/«*/ 

.058 
1/16  b. 

.049 

3/64/ 

.042 

3/64  b. 

•035 

'/32/ 

.032 

'/3» 

.028 
i/3a  b. 

.025 

i.6/64 

.022 

I-4/64 

.020 

1-3/64 

DlAM. 

Ibs. 

Ibs. 

lb. 

lb. 

lb. 

lb. 

lb. 

lb. 

lb. 

lb. 

lb. 

inches. 

# 

•037 

•°35 

.031 

.029 

.026 

.024 

3/i6 

.087 

.079 

.072 

.062 

•058 

.052 

.047 

.042 

•039 

K 

.130 

•"5 

.IO2 

.088 

.081 

.072 

.065 

.058 

•053 

5/i6 

.201 

.I87 

.172 

.150 

.132 

•"3 

.104 

.092 

.083 

.074 

.068 

H 

•255 

•234 

.214 

.186 

•163 

.138 

.128 

•H3 

.102 

.090 

.082 

7/16 

.306 

.281 

.256 

.221 

•193 

.164  • 

•151 

•133 

.120 

.106 

.097 

1A 

•358 

.329 

.298 

•257 

.224 

.189 

.174 

•154 

.138 

.122 

.III 

9/i6 

.411 

•376 

•340 

•293 

•254 

•215 

.197 

.174 

.156 

.138 

.126 

K 

.463 

.423 

.382 

.328 

.285 

.240 

.221 

.194 

.174 

•154 

.141 

W/i6 

tf 

•515 

.567 

.470 

•5!7 

.424 
.467 

.364 

•399 

•346 

.265 
.291 

.244 
.267 

•215 

•235 

.192 
.211 

.170 

.186 

•155 
.170 

J3/T6 

.620 

•  564 

.509 

•435 

•376 

-316 

.290 

•255 

•  229 

.202 

.184 

% 

.672 

.611 

•551 

.471 

.407 

•342 

.314 

.276 

.247 

.218 

.199 

J5/i6 

.724 

.658 

•593 

.506 

•437 

.367 

•337 

.296 

.265 

•234 

.213 

I 

•777 

.706 

•635 

•  542 

.468 

•393 

•  360 

.316 

•283 

.250 

.228 

Itf 

.881 

.801 

.719 

•  613 

•529 

•443 

.407 

•357 

•320 

.282 

•257 

I* 
Ifc 

.986 
1.09 

.896 
.991 

.804 
.888 

.684 
•755 

•59° 
.651 

•494 
•545 

•453 
.500 

.398 
•439 

•356 
•392 

•346 

.286 
•3*5 

iX 

1.20 

1.09 

.972 

.827 

.712 

•596 

.546 

•479 

•429 

•378 

•344 

W.  G. 

9 

IO 

ii 

12 

13 

14 

15 

16 

17 

18 

19 

INCH. 

.148 

9/64/ 

9/64  b. 

.120 
#*• 

.109 

7/64 

.095 

3/3»/ 

•083 

5/64/ 

.072 
S/64  b. 

.065 
'/*/ 

.058 

1/16  b. 

.049 

3/64/ 

.042 
3/64  b. 

DlAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

inches. 

I# 

1.90 

1.74 

I.58 

1-45 

1.28 

•13 

.986 

.896 

.804 

.684 

•590 

1/8 

2.  1  1 

1-93 

1.76 

1.  60 

1.41 

•25 

.991 

.99I 

.888 

•755 

.651 

I# 

2-33 

2.13 

1.94 

1.76 

i-55 

•37 

1.20 

1.09 

•  972 

i2l 

.712 

iH 

2-54 

2.32 

2.12 

1.92 

1.69 

•49 

1.30 

1.18 

i.  06 

.898 

•773 

i# 

2.76 

2.52 

2.30 

2.08 

1.83 

.61 

1.40 

.28 

.14 

.969 

•  834 

i# 

2.97 

2.71 

2.47 

2.24 

1.97 

•73 

I-5I 

•37 

•23 

.04 

•895 

2 

3-19 

2.91 

2.65 

2.39 

2.  IO 

•85 

1.61 

•47 

•3i 

.11 

•956 

2^ 

3-40 

3.10 

2.83 

2.55 

2.24 

•97 

1.72 

•56 

•39 

.18 

1.02  , 

2X 

3-62 

3-30 

3.01 

2.71 

2.38 

2.09 

1.82 

.66 

.48 

•25 

1.  08 

2^ 

3.83 

3-49 

3-19 

2.86 

2.52 

2.21 

i-93 

•75 

•56 

•33 

I.I4 

2^ 

4.04 

3-69 

3-37 

3.02 

2.66 

2-33 

2.03 

•85 

.65 

.40 

1.20 

WEIGHT   OF   BRASS   TUBES. 


267 


Table  No.  95  (continued). 

LENGTH,   I  FOOT.     Thickness  by  Holtzapffel's  Wire-Gauge  (Table  No.  13). 
Specific  Weight  — 1. 1 1  (Specific  Weight  of  Wrought  Iron=i). 


THICK- 
NESS. 
W.  G. 

3 

4 

5 

6 

7 

8 

9 

10 

ii 

12 

13 

INCH. 

Tf. 

•  238 

<5/64/ 

.220 

.203 

13/64 

.180 

3/i6  b. 

ii  ^b 

.148 

.134 

9/64  b. 

.120 

.109 

7/64 

.095 

3/32 

DIAM. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

inches. 

2 

5.24 

4.87 

4-55 

4.24 

3-80 

3-52 

3-19 

2.91 

2.65 

2-39 

2.10 

2% 

5-62 

5-22 

4.87 

4-54 

4.07 

3-76 

3-40 

3.10 

2.83 

2-55 

2.24 

2% 

5-99 

5-57 

5.19 

4-83 

4-33 

4.00 

3-62 

3-30 

3.01 

2.71 

2.38 

2-y& 

6-37 

5-5i 

5-13 

4-59 

4.24 

3.83 

3-49 

3-19 

2.86 

2.52 

2/2 

6-75 

6.26 

5-83 

5-42 

4.85 

4.48 

4.04 

3.69 

3-37 

3.02 

2.66 

2% 

7.12 

6.60 

6.14 

5-72 

5-12 

4.72 

4.26 

3-88 

3-55 

3.18 

2.79 

2¥ 

7-5° 

6-95 

6-47 

6.01 

5.38 

4.96 

4-47 

4.07 

3-73 

3-34 

2-93 

7.88 

7-30 

6.79 

6.31 

5-64 

5.20 

4.69 

4.27 

3-9i 

3-50 

3-07 

3 

8.25 

7.64 

7.11 

6.60 

5.90 

5-44 

4.90 

4-46 

4.09 

3-66 

3-21 

3X 

9.01 

8-33 

7-75 

7-19 

6.43 

5-92 

5-49 

4-85 

4-43 

3.98 

3.48 

3# 

9.76 
10.5 

9.02 
9.72 

8-39 
9-03 

7.78 
8-37 

6-95 

7-47 

688 

6.07 
6.65 

5-24 
5-63 

4.78 
5.12 

4-30 
4.61 

3.76 

4.04 

4 

ii.  3 

10.4 

9-67 

8.96 

8.00 

7^6 

7.24 

6.  02 

5-46 

4-93 

4.31 

4X 

12.0 

11.  i 

10.3 

9-55 

8.52 

7-83 

7.82 

6.41 

5.80 

5-25 

4-59 

4-^2 

12.8 

ii.  8 

10.9 

10.  1 

9.04 

8.31 

8.41 

6.80 

6.15 

4.87 

4^ 

13-5 

12.5 

ii.  6 

10.7 

9.56 

8-79 

8-99 

7.19 

6.49 

5!  88 

5 

14-3 

13.2 

12.2 

"•3 

10.  1 

9.27 

9-57 

7-58 

6.83 

6.  20 

5-42 

5X 

15.0 

13-9 

I2.9 

11.9 

10.6 

9-75 

IO.2 

7-97 

7.17 

6.51 

5-69 

5</ 

15.8 

14.6 

13-5 

12.5 

n.  i 

10.2 

10.7 

8.36 

7-52 

6.83 

5-97 

16.5 

15-3 

14.  1 

11.7 

10.7 

II.3 

8-75 

7.86 

7.15 

6.25 

6" 

17-3 

15-9 

14.8 

13-7 

12.2 

II.  2 

11.9 

9.14 

8.20 

7.46 

6.52 

268 


WEIGHT   OF   METALS. 


Table  No.  96. — WEIGHT  OF  ONE  SQUARE  FOOT  OF  SHEET  BRASS. 
Thickness  by  Holtzapffell's  Wire-Gauge  (Table  No.  13). 


Thickness. 

Weight  of 
i  Square 
Foot. 

Thickness. 

Weight  of 
i  Square 
Foot. 

Thickness. 

Weight  of 
i  Square 
Foot. 

No.W.G. 

inch. 

pounds. 

No.W.G. 

inch. 

pounds. 

No.W.G. 

inch. 

pounds. 

3 

•259 

10.9 

II 

.120 

5-05 

19 

.042 

1.77 

4 

•238 

IO.O 

12 

.109 

4-59 

20 

•035 

1.47 

5 

.220 

9.26 

13 

•095 

4.00 

21 

.032 

i-35 

6 

.203 

8-55 

14 

.083 

3-49 

22 

.028 

1.18 

7 

.l8o 

7.58 

J5 

.072 

3-03 

23 

.025 

1.05 

8 

.165 

6-95 

16 

.065 

2.74 

24 

.022 

.926 

9 

.148 

6.23 

i7 

.058 

2.44 

25 

.020 

.842 

10 

•134 

5-64 

18 

.049 

2.06 

Table  97. — SIZE  AND  WEIGHT  OF  TIN  PLATES. 


Mark. 

Size  of  Sheets. 

Number 
of  Sheets 

Weight 

in  a  Box. 

per  Box. 

inches,    inches. 

sheets. 

pounds. 

I  C 

14  x    10 

225 

112 

IX 

J?               JJ 

140 

IXX 

J)               JJ 

„ 

161 

IXXX 

57                   J) 

5) 

182 

IXXXX 

JJ                   ?> 

JJ 

203 

SD  C 

15    x    ii 

200 

168 

SDX 

))          )•> 

„ 

189 

SDXX 

))         » 

210 

SDXXX 

„         ,, 

J; 

23I 

S  D  XXXX 

» 

jj 

252 

DC 

I7    X     I2# 

IOO 

98 

DX 

5>                   5J 

J? 

126 

DXX 

•>•>                   » 

M 

147 

DXXX 

•)•>                   •)•) 

J) 

168 

DXXXX 

J>                   J) 

JJ 

169 

WEIGHT  OF   TIN   AND   LEAD   PIPES. 


269 


Table  No.  98. — WEIGHT  OF  TIN  PIPES. 
As  manufactured. 

I    FOOT   IN    LENGTH. 


Diameter 

THICKNESS. 

Diameter 

THICKNESS. 

Externally 

3/32"  inch. 

J^  inch. 

X  inch. 

inches. 

Ibs. 

Ibs. 

inches. 

Ibs. 

* 

.148 



*% 

5-°4 

% 

.384 

.472 

*% 

5.67 

ti 

.620 

.787 

m 

6.30 

I 

.856 

T.I03 

3 

6-93 

& 

1.095 

I.4I7 

ilA 

7.56 

5# 

1.328 

1.732 

3^ 

8.19 

& 

1.564 

2.047 

2 

1.  802 

2.362 

Table  No.  99. — WEIGHT  OF  LEAD  PIPES. 
As  manufactured. 


WEIGHT  AND  THICKNESS  OF  PIPE. 

Bore. 

Length. 

Calcu- 

Calcu- 

Calcu- 

Calcu- 

Weight. 

lated 
Thick- 

Weight. 

lated 
Thick- 

Weight. 

lated 
Thick- 

Weight. 

lated 
Thick- 

ness. 

ness. 

ness. 

ness. 

inches. 

feet. 

Ibs. 

inch. 

Ibs. 

inch. 

Ibs. 

inch. 

Ibs. 

inch. 

H 

15 

14 

.097 

16 

.112 

18 

.124 

22 

.146 

y% 

5) 



17 

.101 

21 

.121 

30 

.140 

N 

„ 

24 

.112 

28 

.147 

32 

.l8l 

36 

•215 

i 

5) 

36 

.136 

42 

.156 

56 

.2OO 

64 

.225 

ijf 

12 

36 

•139 

42 

.l6o 

48 

.l8o 

S2 

•193 

T-Yz 

„ 

48 

.I56 

56 

.179 

72 

.224 

84 

.257 

ify 

„ 



72 

.199 

84 

.228 

96 

.256 

2 

jj 

72 

.I78 

84 

.204 

96 

.231 

112 

.266 

2^ 

IO 



84 

.200 

96 

.227 

112 

.26l 

3 

„ 



— 

112 

.218 

140 

•275 

3/2 

„ 



— 

I30 

•225 

160 

•273 

4 

,, 



— 

170 

•257 

22O 

•327 

4/2 

» 



— 

170 

.232 

220 

•295 

4?/| 

s/l6  inch  thick.     Weight  per  lineal  foot  22.04  Ibs. 

4/ 

»                                 „                   23.25    „ 

A  3/i 

,,                                                           })                                  2/1.  /1C      .. 

5 

2*  ( 

36   „ 

*  j'^ 

2/0  WEIGHT   OF   METALS. 

Table  No.  100. — ENGLISH  ZINC  GAUGE.     (London  Zinc  Mills.} 


Ap- 

7  Ft.  X  2  Ft.  8  Ins. 

7  Feet  X  3  Feet. 

8  Feet  X  3  Feet. 

8£ 

Gauge 
No 

proxi- 
mate 
Weight 
per  Sq. 
Foot. 

i,oooths 
of 
an  Inch. 

Approximate 
,,,  •  ,        Number 

•asLjSss 

Appro 

Weight 
Der  Sheet. 

ximate 
Number 
of  Sheets 
in  loCwts. 

Appro 

Weight 
per  Sheet. 

ximate 
Number 
of  Sheets 
in  icCwts. 

Gauge 
No 

Nearest 
Birmingha 
Wire  Gau§ 

ozs. 

Ibs.   ozs. 

Ibs.  ozs. 

Ibs.  ozs. 

I 

*x 

.004 

2       IO 

427 

— 

— 

— 

— 

I 

41 

2 

3% 

.006 

3    J3 

294 

— 

— 

— 

— 

2 

38 

3 

3% 

.007 

4     IS 

227 

— 

— 

3 

37 

4 

43A 

.008 

— 

— 

6       4 

180 

— 

— 

4 

34 

5 

s% 

.010 

—              — 

7       9 

148 

— 

— 

5 

31 

6 

6% 

.Oil 

7     14         142 

8     14 

126 

10         2 

Ill 

6 

30 

7 

73/4 

.013 

9       i         124 

10       3 

no 

II       IO 

96 

7 

29 

8 

9 

.015 

10       8         107 

ii     13 

95 

13       8 

83 

8 

28 

9 

10 

.017 

ii     ii           96 

13          2 

85 

15         0 

75 

9 

27 

10 

ntf 

.019 

13       7 

83 

!5         2 

74 

17       4 

65 

IO 

25 

ii 

13 

.021 

15       3 

74 

I7          I 

66 

19       8 

57 

ii 

24 

12 

15 

.025 

17       8 

64 

19       II 

57 

22         8 

5° 

12 

23 

13 

17 

.028 

— 

22         5 

50 

25       8 

44 

13 

22 

J4 

*9 

.031 

— 

— 

24      15 

45 

28       8 

39 

J4 

21 

15 

22 

.036 

— 

— 

28       14 

39 

33      o 

34 

15 

20 

16 

25 

.041 

— 

— 

32       13 

34 

37       8 

30 

16 

19 

17 

28 

.046 

— 

— 

36      12 

30 

42      o 

27 

17 

18 

18 

31 

.051 

— 

— 

40      II 

28 

46       8 

24 

18 

— 

J9 

35 

•059 

— 

— 

45    15 

24 

52       8 

21 

19 

17 

20 

39 

.065 

— 

— 

5i       3 

22 

58       8 

r9 

20 

16 

21 

43 

.072 

— 

— 

56       7 

20 

64       8 

*7 

21 

15 

Sheets  thicker  than  above  are  rolled  to  Birmingham  Wire  Gauge 

Table  No.  IOOA. — "VM"  ZINC  GAUGE.     (Vieille-Montagne). 


Approximate 
Thickness. 

Approximate 
Weight  per 
Sqi  are  Foot. 

36  Ins.  X  72  Ins. 

36  Ins.  X  84  Ins. 

36  Ins.  X  96  Ins. 

Gauge. 

Approximate 
Weight  of  Sheets. 

1>  M"hV 
Q.  J-.  c 

Approximate 
Weight  of  Sheets. 

&?•* 
s.5! 

Approximate 
Weight  of 
Sheets. 

§>?•§ 

§=••5, 

D.I.  c 

Thou- 

,    3 

'3  5  °*  aj 

sandths 

W       yj-4) 

<"  °  W 

<«  °W 

«  °w 

of  an 

.H*~*3.5 

flj  O     " 

JJlrf 

V     Q         • 

Inch. 

HP 

Lbs. 

Ozs. 

Drms. 

Lbs. 

Ozs. 

Drms. 

Is1 

Lbs. 

Ozs. 

Drms. 

r3j3 

Lbs. 

Ozs. 

Drs. 

!§- 

about 

about 

about 

i 

0.004 

0.  100 

_ 

2 

5 

>•  Nos.  i  and  2  are  only  rolled  to  order  and  special  dimensions. 

2 

.006 

.141 

— 

3 

4 

1 

3 

.007 

.171 

— 

3 

15 

4 

6 

14 

249 

5 

ii 

213 

5 

14 

8 

187 

4 

.008 

.209 

— 

4 

13 

5 

6 

IO 

204 

6 

I 

175 

3 

8 

153 

5 

.010 

.247 

— 

5 

ii 

6 

6 

6 

172 

7 

7 

J47 

8 

8 

8 

129 

6 

.Oil 

.291 

— 

6 

ii 

7 

8 

6 

146 

8 

i 

7 

126 

10 

— 

8 

no 

7 

.013 

•337 

— 

7 

12 

8 

ii 

8 

126 

10 

12 

108 

ii 

IO 

— 

95 

8 

.015 

.386 

— 

8 

14 

9 

15 

12 

no 

ii 

i 

6 

95 

13 

5 

— 

83 

9 

.018 

•450 

— 

10 

5 

ii 

9 

10 

95 

13 

8 

9 

81 

15 

7 

8 

71 

10 

.020 

.500 

— 

ii 

7 

12 

14 

86 

15 

— 

3 

73 

17 

2 

8 

64 

ii 

.023 

-580 

— 

13 

5 

X4 

15 

10 

74 

T-7 

7 

9 

63 

19 

15 

8 

55 

12 

.026 

.660 

— 

15 

2 

17 

o 

4 

65 

19 

I3 

10 

56 

22 

II 

— 

49 

13 

.029 

.740 

I 

— 

15 

19 

0 

14 

57 

22 

3 

ii 

50 

25 

6 

8 

43 

14 

.032 

.820 

I 

2 

12 

21 

I 

8 

S2 

24 

9 

12 

45 

28 

2 

— 

39 

15 

.038 

•95° 

I 

5 

12 

24 

7 

8 

45 

28 

8 

12 

39 

32 

10 

— 

34 

16 

•043 

i.  080 

I 

8 

12 

27 

13 

8 

39 

32 

7 

12 

34 

37 

2 

— 

30 

17 

.048 

I.2IO 

I 

ii 

II 

31 

2 

6 

35 

36 

5 

7 

30 

8 

8 

27 

18 

•053 

1.340 

i 

14 

II 

34 

8 

6 

40 

4 

7 

27 

46 

— 

8 

24 

19 

.058 

1.470 

2 

i 

II 

37 

I4 

6 

29 

44 

3 

7 

25 

So 

8 

8 

22 

20 

•063 

I.  600 

2 

4 

IO 

41 

3 

4 

27 

48 

i 

2 

23 

54 

15 

— 

20 

21 

.O7O 

1.780 

2 

8 

12 

45 

13 

8 

24 

53 

7 

12 

21 

61 

2 

— 

18 

22 

.077 

1.960 

2 

12 

14 

50 

7 

2 

22 

58 

6 

19 

67 

5 

— 

16 

23 

.084 

2.I4O 

3 

I 

I 

55 

3 

2 

2O 

64 

6 

5 

73 

9 

8 

15 

24 

.091 

2.320 

3 

5 

3 

59 

13 

16 

18 

69 

12 

11 

79 

12 

8 

14 

25 

.098 

2.500 

3 

9 

5 

64 

7 

IO 

17 

75 

3 

9 

15 

85 

15 

8 

13 

26 

.105 

2.680 

3 

7 

69 

i 

M 

16 

80 

10 

3 

92 

8 

12 

FUNDAMENTAL  MECHANICAL  PRINCIPLES. 


FORCES   IN   EQUILIBRIUM. 
SOLID  BODIES. 

Parallelogram  of  Forces. — When  a  body  remains  at  rest  whilst  being  acted 
on  by  two  or  more  forces,  it  is  said  to  be  in  a  state  of  equilibrium,  and  so  also 
are  the  forces.     Thus,  if  the  forces  P/,  Q  ^,  R  r,  Fig.  86,  acting  on  the  body 
p  q  r,  keep  it  at  rest,  they  are  in 
equilibrium,  and  any  two  of  them 
balance   the   third.     The   lines  of 
force,  if  produced,  meet  at  one  point 
o  within  the  body,  and  if  a  parallel-      R 
ogram  be  constructed  having  two 
adjacent  sides  proportional  to  and 

parallel  tO  tWO  Of  the  forces  respec-  Fig.  86.— Equilibrium  of  Forces. 

tively,  to  represent  them  in  magni- 
tude and  direction,  the  diagonal  of  the  parallelogram  will  represent  the  third 
force  in  magnitude  and  direction.     Let  the  lines  OP,  OQ,  Fig.  87,  represent 
the  forces  P/,  Q^  in  magni- 
tude and  direction,  and  com-  _p 
plete    the    parallelogram    by 

drawing  the  parallels  p  R,  Q  R,  iv _ ^ 

and  draw  o  R.     Then  o  R  re-  ^*x^ 

presents    in    magnitude   and 
direction  the  resultant  of  the 

tWO  forces;    and  RO  taken  in  Fig.  87.-Parallelogram  of  Forces. 

the  opposite  direction  repre- 
sents the  third  force  Rr,  Fig.  86.     If  it  be  applied  in  this  direction  to 
the  point  o,  as  indicated  by  a  dot  line  o  R',  it  would  balance  the  other  two. 
This  construction  is  called  the  parallelogram  of  forces,  and  is  applicable  to 
any  three  forces  in  equilibrium. 

Three  forces  in  equilibrium  may  also  be  represented  by  a  triangle,  or 
half  a  parallelogram.  For  example,  the  triangle  OPR  represents  by  its 
three  sides  the  forces  o  P,  o  Q,  o  R,  the  side  P  R  being  substituted  for  o  Q. 

Three  forces  in  equilibrium  must  be  in  the  same  plane. 

When  the  directions  of  three  forces  holding  a  body  at  rest,  and  also  the 
magnitude  of  one  of  them,  are  known,  the  magnitudes  of  the  other  two  can 
be  determined  by  constructing  a  parallelogram  in  the  manner  above  exem- 
plified, and  measuring  the  lengths  of  the  sides  and  the  diagonal. 


2/2 


FUNDAMENTAL   MECHANICAL   PRINCIPLES. 


Polygon  of  Forces. — Equilibrium  may  subsist  between  more  than  three 
forces,  which  need  not  necessarily  be  in  the  same  plane;  and  they  can,  like 
those  already  illustrated,  be  developed  in  direction  and  magnitude  by  diagram. 
Thus,  let  the  point  o,  Fig.  88,  representing  a  solid  body,  be  kept  at  rest  by  a 
number  of  forces,  OP,  o  Q,  o  R,  o  s,  o  T.  Find  the  equivalent  diagonal  o/  for 
the  first  two  forces ;  then  construct  the  parallelogram  and  diagonal  o  r  for  the 
resultant  of  o/  and  the  third  force  OR;  and  again  the  parallelogram  and 
diagonal  o  s  for  the  resultant  of  o  r  and  the  fourth  force  o  s.  The  last 
resultant  o  s  represents  in  one  the  four  distributed  forces  OP,  OQ,  OR,  os,  and 
it  balances  the  fifth  force  o  T  equal  and  opposite  to  it.  As  o  s  and  o  T  are 
in  the  same  straight  line,  their  resultant  is  of  course  nothing. 

The  several  forces  thus  dealt  with  may  be  combined  into  a  polygon  of 
forces.     Draw  o  P,  Fig.  89,  parallel  and  equal  to  o  P,  Fig.  88,  p  Q  parallel  and 

equal  to  o  Q,  Q  R  parallel  to  o  R,  R  s 
parallel  to  os;  then,  finally,  so, 
completing  the  polygon,  will  be 
parallel  and  equal  to  OT,  Fig.  88, 
the  last  of  the  series.  Professor 
Mosely  illustrates  the  polygon  of 
forces  by  the  united  action  of  a 
number  of  bell-ringers,  pulling  by  a 
number  of  ropes  joined  to  a  single 
rope.  The  polygon  constructed  as 
in  Fig.  90,  shows  successively  by 
corresponding  letters,  the  individual 
contributions  of  the  bell-ringers, 
combined  into  one  vertical  force. 

Again,  equilibrium  may  be  estab- 
lished between  a  number  of  forces 
acting  in  the  same  plane,  applied  to  different  points  in  a  body,  or  system  of 
bodies.  For  example,  let  the  forces  P  o,  Q  o,  R  o,  s  o,  and  T  o,  be  applied 

to  several  points,  o,  o,  o,  o,  o,  on  a  flat  board 
ABC,  Fig.  91,  by  means  of  cords  and  weights; 
it  will  settle  into  a  position  of  equilibrium, 
when  the  opposing  forces  arrive  at  a  balance 
between  themselves.  An  axis  or  pivot  may 
be  established  at  any  point,  M,  on  the  surface 
of  the  board,  without  disturbing  the  equilib- 
rium, and  it  may  be  viewed  as  a  centre  of 
motion  round  which  the  forces  tend  to  turn 
the  board,  some  in  one  direction,  the  others 
the  opposite  way,  balancing  each  other.  The 
effect  of  each  force  to  turn  the  body  about 

the  centre  is  represented  by  the  product  of  its  magnitude  by  the  leverage, 
or  perpendicular  distance  of  its  direction  from  the  centre;  draw  these 
perpendiculars,  and  multiply  each  force  by  its  perpendicular  or  leverage, 
then  the  resulting  products  will  be  divisible  into  two  sets,  tending  to  turn 
the  board  in  opposite  directions.  The  sum  of  the  first  set  of  products  is 
equal  to  the  sum  of  the  second  set,  as  is  proved  by  the  fact  of  equilibrium. 
Moments  of  Forces. — The  product  of  a  force  by  the  perpendicular  dis- 
tance of  its  direction  from  any  given  point,  is  called  the  moment  of  the 


Fig.  88. — Equilibrium  of  more  than  Three  Forces. 


Fig.  89. — Polygon  of  Forces. 


FORCES   IN    EQUILIBRIUM. 


273 


Fig.  90. — Bell-ringers]  Polygon  of  Forces. 


force  about  that  point;  and  the  equilibrium  above  discussed,  in  connection 
with  Fig.  91,  is  the  result  of  the  equality  of  moments. 

The  law  of  the  equality  of  moments  may  be  thus  set  forth: — If  any 
number  of  forces  acting  anywhere  in  the 
same  plane,  on  the  same  body  or  connected 
system  of  bodies,  be  in  equilibrium,  then 
the  sum  of  the  forces  tending  to  turn  the 
system  in  one  direction  about  any  point  in 
that  plane,  is  equal  to  the  sum  of  the  mo- 
ments of  those  forces  tending  to  turn  the 
system  in  the  other  direction. 

Such  balanced  forces  may  be  transferred 
to  a  single  point,  and  placed  about  it,  as  in 
Fig.  88,  parallel  to  their  directions  as  they 
stand;  and  they  will  continue  in  equilibrium, 
holding  the  point  at  rest.  A  polygon  of 
the  forces  p  q  r  s  t  within  Fig.  90,  may  be 
constructed  similarly  to  Fig.  89. 

Though  the  principle  of  the  polygon  of 
forces  be  sufficient  to  test  the  equilibrium 
of  a  system  of  forces  acting  at  one  point, 
yet  the  principle  of  the  equality  of  moments, 
in  addition,  is  necessary  to  establish  the 
equilibrium  of  a  system  applied  to  different  points.  The  two  principles 
conjointly  are  necessary,  and  they  are  sufficient,  as  conditions  of  equilibrium. 

The  Catenary. — When 
a  chain,  or  a  rope,  or  a 
flexible  series  of  rods,  is 
suspended  by  its  extremi- 
ties, supporting  weights 
distributed  along  its 
length,  in  a  state  of  rest, 
it  forms  a  polygon  of 
forces  in  equilibrium,  as 
in  Fig.  92.  If  all  the 
forces  except  those  which 
act  on  the  extremities  of 
the  chain,  be  combined 
into  a  resultant,  then  the 
two  extreme  sides  being 
produced,  will  meet  the 
direction  of  the  resultant 
at  one  point.  Thus,  in 
the  polygon,  Fig.  92, 
loaded  with  weights,  w,  w, 
&c.,  the  vertical  resultant 
of  these  weights  w'  w", 
passing  through  their  common  centre  of  gravity,  intersects  at  w"  the  two 
extreme  sections  PA,  P'B,  when  produced  downwards. 

Similarly,  in  the  catenary,  Fig.  93,  which  is  the  curve  assumed  by  a  rope 
or  other  flexible  medium  uniformly  loaded  and  suspended  by  the  two 

18 


Fig.  91. — Equality  of  Moments. 


FUNDAMENTAL   MECHANICAL   PRINCIPLES. 


extremities,  if  tangents  be  drawn  to  the  extremities  A,  B,  of  the  curve,  meeting 
at  w",  they  represent  the  directions  of  the  forces  sustaining  the  curve  at 


Fig.  92.— The  Catenary. 


Fig.  93.— The  Catenary. 


those  points,  and  they  intersect  at  the  same  point  w",  the  vertical  line  G  w" 
passing  through  the  centre  of  gravity  of  the  curve.     Let  the  weight  of  the 


Fig.  94. — Centrifugal  Forces  in  Equilibrium. 


Fig.  95.— Parallel  Forces  in  Equilibrium. 


curve  be  represented  by  G  w",  and  complete  the  parallelogram  M  N,  then 
w"  M  and  w"  N  represent  in  force  and  direction  the  tension  at  the  points 
B  and  A. 

Centrifugal  Forces  in  Equilibrium. — When  a  cylindrical  vessel  is  exposed 
to  a  uniform  internal  pressure,  as  the  pressure  of  steam  within  a  boiler,  for 
example,  the  pressure  is  balanced  by  the  resistance  of  cohesion  of  the 
material  of  the  boiler.  Let  A  B  c  D,  Fig.  94,  be  the  section  of  a  cylindrical 
boiler,  the  radial  pressure  of  the  steam  may  be  represented  by  the  arrows, 
which  are  equal  and  opposite  in  direction.  The  tension  on  the  metal  in 
resisting  the  internal  pressure  at  any  particular  section  B  D,  is  equal  to  the 
sum  of  the  pressures  resolved  into  the  direction  at  right  angles  to  B  D,  or 
parallel  to  AC,  according  to  the  triangles,  or  half-parallelograms  of  force 
attached  to  each  oblique  arrow.  The  total  vertical  pressure  thus  obtained 
by  the  resolution  of  forces  is  equal  to  the  total  vertical  pressure  which 


FORCES   IN   EQUILIBRIUM. 


275 


would  be  exerted  on  the  sectional  line  B  D  if  it  be  supposed  to  be  a  rigid 
diaphragm  across  the  boiler,  which  is  easily  calculated.  If  the  radial 
pressure  be,  for  example,  100  Ibs.  on  each  square  inch  of  surface,  then  the 
total  pressure,  or  the  tension  on  the  two  sides  at  B  and  D,  would  be 
TOO  x  BD  on  each  inch  of  length  of  the  two  sides;  that  is  to  say,  if  the 
diameter  B  D  be  equal  to  60  inches,  the  tension  on  the  two  sides  would  be 
60  x  100  =  6000  pounds  for  each  inch  of  length. 

A  similar  argument  applies  to  the  tension  on  the  rim  of  a  revolving  fly- 
wheel. 

Parallel  Forces. — Systems  of  parallel  forces  are  particular  cases  of  the 
foregoing. — Let  A,  B,  c,  D,  E,  F,  Fig.  95,  be  a  system  of  parallel  forces  in  equili- 
brium; and  MN  a  line  perpendicular  to  them  in  the  same  plane,  and  cut 
by  them  at  the  points  a,  b,  c,  d,  e,  f.  They  may  act  at  any  points  in  their 
lines  of  direction  without  disturbing  the  equilibrium,  and  they  may  be  sup- 
posed to  be  applied  at  those  points  in  the  line  M  N.  Then,  the  sum  of  the 
moments  of  the  three  forces  A,  B,  c,  acting  in  one  direction,  with  respect  to 
any  point  M  as  a  centre,  is  equal  to  the  sum  of 
the  moments  of  the  forces  D,  E,  F,  opposed  to 
them.  Further,  the  sum  of  the  simple  forces 
A,  B,  c,  irrespective  of  their  moments,  is  equal  to 
that  of  the  forces  D,  E,  F,  so  that  the  fact  of  their 
being  in  equilibrium  resolves  itself  into  a  case  of 
action  and  reaction,  for  the  two  equivalent  forces 
representing  the  two  opposing  sums,  act  in  the 
same  straight  line  in  opposite  directions. 

When  three  parallel  forces  balance  each  other, 
acting  on  a  straight  line,  two  of  them  must  be 
opposed  to  the  third;  and  the  third  must  act 
between  the  other  two,  being  equal  and  opposite 
to  their  resultant.  Let  A,  B,  c,  Fig.  96,  be  three 
such  forces  applied  to  the  line  E  G  F,  at  the  points  Fis-  96.— Three  Parallel  Forces  in 
E,G,F  respectively;  then,  with  respect  to  the 

point  G,  the  moment  of  the  force  B  is  nothing,  because  it  passes  through 
that  point  and  has  no  leverage  on  it.  There  remain  the  moments  of  the 
extreme  forces,  A  and  c,  which  are  equal  to  each  other,  that  is  to  say 

AXEG  =  CXFG. 
From  this  it  follows,  by  proportion,  that 

A  :  c  :  :  F  G  :  E  G, 

and  that  the  extreme  forces  are  to  each  other  inversely  as  their  distances 
from  the  middle  force. 

In  general,  of  three  parallel  forces  acting  in  equilibrium  on  an  inflexible 
line,  the  first  in  order  is  to  the  third  as  the  distance  of  the  third  from  the 
second,  is  to  that  of  the  first  from  the  second. 

The  sum  of  the  first  and  third  is  equal  to  the  second;  and  when  the 
distances  or  leverages  are  equal,  the  first  and  third  forces  are  equal  to  each 
other. 

If  the  position  of  the  line  E  F  be  inclined  to  the  direction  of  the  three 
forces,  and  changed  to  E'  F',  Fig.  96, the  forces  A,  B,  c,  continue  in  equilibrium; 


276 


FUNDAMENTAL   MECHANICAL   PRINCIPLES. 


Fig.  97. — Parallelepiped  of  Forces. 


for  the  perpendicular  lines  G  E  and  G  F  continue,  as  before,  to  be  the  lever- 
ages of  the  extreme  forces  A  and  c,  on  the  central  point  G. 

When  three  forces  not  in  the  same  plane  act  on  one  point,  there  cannot 
be  equilibrium  between  them.  Two  of  these  may  be  reduced  to  their 

resultant  by  parallelogram,  and 
this  resultant  reduced  with  the 
third  force  to  a  final  resultant 
For  example,  let  the  lines  o  P, 
OQ,  OR,  Fig.  97,  represent  in 
magnitude  and  direction  three 
forces  not  in  one  plane  acting 
on  the  point  o.  By  parallel- 
ogram, o  s  is  the  resultant  of  the 
two  forces  o  P,  o  Q,  and  o  T  is  the 
final  resultant  of  o  s  and  the  third 
force  o  R.  That  is  to  say,  o  T  is  the  resultant  of  the  three  given  forces. 

If  parallelograms  be  formed  from  each  two  of  the  three  forces,  they  form, 
when  duplicated,  a  parallelepiped  of  forces,  of  which  the  diagonal  is  found 
by  the  final  resultant  o  T,  and  the  principle  of  the  parallelepiped  of  forces 
may  be  thus  defined: — If  three  forces  be  represented  in  magnitude  and 
direction  by  three  adjacent  edges  of  a  parallelepiped,  their  resultant  is 
represented  in  magnitude  and  direction  by  the  adjacent  diagonal  of  the 
solid. 

There  must  be  at  least  four  forces  to  produce  equilibrium  about  a  point, 
when  the  forces  are  not  in  the  same  plane. 

The  triangle  OST,  Fig.  97,  comprises  in  its  three  sides  the  resultant  of 
the  first  and  second  forces,  the  third  force,  and  the  resultant  of  the  three. 
If  the  first  resultant  o  s  be  replaced  by  the  two  lines  o  Q  and  Q  s,  which 
represent  the  first  and  second  forces,  they  form  the  four-sided  figure  o  Q  s  T, 
the  polygon  of  the  four  equilibrating  forces. 

A  greater  number  of  forces  than  four  acting  on  a  point  may  be  reduced 
in  like  manner. 

FLUID  BODIES. 

The  characteristic  property  of  fluids  is  the  capability  of  transmitting  the 
pressure  which  is  exerted  upon  a  part  of  the  surface  of  the  fluid,  in  all 
directions,  and  of  the  same  intensity: — the  same  pressure  per  square  inch  or 
per  square  foot. 

The  pressure  of  water  in  a  vessel,  caused  by  its  own  gravity,  increases  in 
proportion  to  the  depth  below  the  surface;  and  the  pressure  on  a  horizontal 
surface,  say,  the  bottom,  is  equivalent  to  the  weight  of  the  superincumbent 
column  of  water,  and  the  intensity  of  the  pressure  is  independent  of  the 
form  of  the  vessel.  The  same  rule  applies  when  the  pressure  is  from  below 
upwards. 

The  same  rule  also  applies  when  the  surface  is  either  vertical  or  inclined, 
and  the  mean  height  of  the  superincumbent  column  of  water  is  measured 
by  the  depth  of  the  centre  of  gravity  of  the  given  surface  below  the  surface 
of  the  water. 

The  water  in  open  tubes  communicating  with  each  other,  when  in  a  state 
of  equilibrium,  stands  at  the  same  level  in  the  tubes,  whatever  may  be  the 
relative  diameters  of  the  tubes. 


MOTION. — GRAVITY.  277 

The  height  of  the  superincumbent  column  of  water  is  called  the  head  of 
water. 

The  buoyancy,  or  the  upward  force  with  which  water  presses  a  body 
immersed  in  it,  from  below  upwards,  is  equal  to  the  weight  of  water  dis- 
placed by  the  body,  or  of  a  quantity  of  water  equal  in  volume  to  the  sub- 
merged body,  or  submerged  portion  of  a  body.  The  resultant  of  the 
upward  pressure  passes  through  the  centre  of  gravity  of  the  water  displaced; 
and  also,  when  the  floating  body  is  at  rest,  through  the  centre  of  gravity  of 
the  body. 

This  resultant  line  is  called  the  axis  of  floatation,  and  the  horizontal 
section  of  the  body  at  the  surface  of  the  water  is  the  plane  of  floatation. 

Bodies  float  either  in  an  upright  position  or  in  an  oblique  position.  A 
body  floats  with  stability,  when  it  strives  to  maintain  the  position  of  equili- 
brium, and  when  it  can  only  be  moved  out  of  this  position  by  force,  and 
will  return  to  it  when  the  force  is  withdrawn.  The  metacentre  is  the  point 
at  which  the  axis  of  floatation  intersects  the  axis  of  a  symmetrical  body,  as  a 
ship,  when  inclined.  If  the  metacentre  lies  above  the  centre  of  gravity  of 
the  ship,  the  ship  floats  with  stability;  if  below,  the  ship  is  unstable;  if  the 
centres  coincide,  which  they  must  do  in  a  cylinder  or  a  sphere,  for  example, 
the  body  floats  indifferently  in  any  position. 

For  the  weight,  volume,  and  pressure  of  water  and  air,  see  Water  and 
Air  as  standards  of  measure,  page  124. 

MOTION. 

The  motion  of  a  body  is  uniform,  when  the  body  passes  through  equal 
spaces  in  equal  intervals  of  time. 

Velocity  is  the  measure  of  motion,  and  is  expressed  by  the  number  of 
feet  or  other  unit  of  length  moved  through  per  second  or  other  unit  of  time. 

Motion  is  accelerated,  when  the  body  moves  through  continually  increased 
spaces  in  equal  intervals  of  time,  like  a  railway  train  starting  from  a  station. 
Motion  is  retarded,  when  the  body  moves  through  continually  decreased 
spaces  in  equal  intervals  of  time,  like  a  railway  train  arriving  at  a  station. 
The  acceleration  and  retardation  are  uniform,  when  the  spaces  moved  through 
increase  or  decrease  by  equal  successive  amounts,  like  a  body  falling  by 
the  action  of  gravity,  or,  on  the  contrary,  projected  upwards  in  opposition 
to  gravity. 

GRAVITY. 

When  bodies  fall  freely  near  the  surface  of  the  earth,  the  motion,  as 
already  said,  is  uniformly  accelerated;  equal  additions  of  velocity  being  made 
to  the  motion  of  the  body  in  equal  intervals  of  time. 

During  the  ist  second  of  time,  the  body,  starting  from  a  state  of  rest, 
falls  through  16.095  feet>  or>  say  16.1  feet;  during  the  ad  second,  it 
falls  through  three  times  16.1  feet;  during  the  3d  second,  it  falls  through 
five  times  16.1  feet,  and  so  on.  The  body  having,  in  the  ist  second,  fallen 
through  1 6.  i  feet,  from  a  state  of  rest,  with  a  motion  uniformly  accelerated, 
it  would  move  through  32.2  feet  in  the  next  second,  with  the  velocity 
thus  acquired,  without  any  additional  stimulus  from  gravity;  that  is  to 
say,  the  velocity  acquired  at  the  end  of  the  ist  second  is  32.2  feet  per  second. 
During  the  2d  second,  it  in  fact  acquires  an  additional  velocity  of  32.2  feet 
per  second,  making  up,  at  the  end  of  this  second,  a  final  velocity  of  64.4 


2?8  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

feet  per  second.  In  like  manner  the  body  acquires  an  additional  velocity 
of  32.2  feet  per  second  during  the  3d  second,  making  a  final  velocity  of 
three  times  32.2  feet,  or  96.6  feet  per  second.  And  so  on. 

Each  of  these  additional  velocities  is  acquired  in  falling  through  16.1  feet 
additional  to  the  space  fallen  through  in  virtue  of  the  velocity  acquired  at 
the  beginning  of  each  second. 

The  relations  of  height  fallen,  velocity  acquired,  and  time  of  fall,  are 
simply  exhibited  in  the  following  manner : — 

During  the  successive  seconds  the  heights  fallen  through  are  consecutively 
as  follow: — 

time,  i,  i,  i,  i  second, 

height  of  fall,          16.1,  16.1  x  3,      16.1  x  5,      16.1  x  7  feet. 

And  reckoning  the  totals  from  the  commencement  of  the  fall, 

total  times,  i,  2,  3,  4  seconds, 

total  height  of  fall,       16.1,  16.1  x  4,      16.1  x  9,      16.1  x  16  feet. 

or  16.1,  1 6.  i  x  22,     16.1  x  32,     16.1  x  42  feet._ 

or  16.1,  64.4,          144-9;          257-6  feet. 

Showing  that  the  total  height  fallen  is  as  the  square  of  the  total  time. 

Again,  during  the  successive  seconds,  the  successive  additional  velocities 
acquired  are: — 

time,  i,  i,  i,  i  second. 

velocities  acquired,       32.2,        32.2,        32.2,        32.2  feet  per  second. 

And  the  total  or  final  velocities  acquired,  reckoning  from  the  commence- 
ment of  the  fall,  are : — 

total  times,  i,          2,  3,  4  seconds. 

final  velocities,     32.2,     32.2  x  2,     32.2  x  3,     32.2  x  4  feet  per  second, 
or  32.2,     64.4,  96.6,         128.8  feet  per  second. 

Showing  that  the  velocity  acquired  is  in  direct  proportion  to  the  time  of  the 
fall. 

The  above  relations  of  time,  height,  and  velocity  are  brought  together 
for  comparison,  thus : — 

time  as,                             i,  2,  3,  4,   &c. 

velocity  acquired  as,        i,  2,  3,  4,   &c. 

height  of  fall  as,               i,  4,  9,  16,   &c. 

or  as  i,  22,  32,  42,  &c. 

Showing  that,  whilst  the  velocity  increases  simply  with  the  time,  the  height 
of  fall  increases  as  the  square  of  the  time,  and  as  the  square  of  the  velocity. 
The  force  of  gravity  is  expressed  by  the  velocity  communicated  by  gravity 
to  a  body  falling  freely  in  a  second,  namely,  32.2  feet  per  second,  and  is 
symbolized  by  g. 

The  foregoing  relations  of  time,  velocity,  and  height  of  fall,  are  comprised 
in  the  six  following  propositions  with  their  answers — applicable  to  the 
condition  of  a  body  falling  freely.  They  are  much  used  in  mechanical 
calculations. 

i  and  2.     Given  the  time,  to  find  the  velocity  and  the  height. 

3  and  4.     Given  the  velocity,  to  find  the  time  and  the  height. 

5  and  6.     Given  the  height,  to  find  the  time  and  the  velocity. 


GRAVITY.  2/9 

RULES  FOR  THE  ACTION  OF  GRAVITY. 

Putting  /  =  the  time  of  falling  in  seconds,  z>  =  the  velocity  in  feet  per 
second,  ^  =  the  height  of  fall  in  feet,  and  g  =  gravity  or  32.2;  then, 

RULE  i.  Given  the  time  of  fall,  to  find  the  velocity  acquired  by  a  falling 
body.  Multiply  the  time  in  seconds  by  32.2,  the  product  is  the  final 
velocity  in  feet  per  second.  Or 

#  =  32.2  /  ......................................  (  i  ) 

RULE  2.  Given  the  time  of  fall,  to  find  the  height  of  the  fall.  Multiply 
the  square  of  the  time  in  seconds  by  16.1.  The  product  is  the  height  of 
fall  in  feet.  Or 

*=i6.i  /2  .....................................   (2) 

RULE  3.  Given  the  velocity,  to  find  the  time  of  falling.  Divide  the 
velocity  in  feet  per  second  by  32.2.  The  quotient  is  the  time  in  seconds. 
Or 


RULE  4.  Given  the  velocity,  to  find  the  height  of  fall  "due"  to  the 
velocity.  Square  the  velocity  in  feet  per  second,  and  divide  by  64.4.  The 
quotient  is  the  height  of  fall  in  feet.  Or 


RULE  5.  Given  the  height  of  fall,  to  find  the  time  of  falling.  Divide  the 
height  in  feet  by  16.1,  and  find  the  square  root  of  the  quotient.  The  root 
is  the  time  in  seconds.  Or 


J. 


16.1 


(5) 


RULE  6.  Given  the  height  of  fall,  to  find  the  velocity  due  to  the  height. 
Multiply  the  height  in  feet  by  64.4,  and  find  the  square  root  of  the  product 
The  root  is  the  velocity  in  feet  per  second. 

Or,  multiply  the  square  root  of  the  height  in  feet  by  8.025;  tne  product 
is  the  velocity  in  feet  per  second. 

Note. — It  is  usual  to  take  the  integer  8  only  for  the  multiplier. 

Symbolically,  these  operations  are  expressed  as  follows : — 


32.2 


or  in  a  round  number  v  =  8  \J   h    (6) 

The  above  rules  are  applicable,  inversely,  to  the  motion  of  bodies  pro- 
jected upwards  and  uniformly  retarded  by  gravity.  The  height  to  which  a 
body  projected  vertically  upwards  by  an  initial  impulse,  will  ascend,  is  equal 
to  the  height  through  which  the  body  must  fall  in  order  to  acquire  the 
initial  velocity,  and  the  same  rule  (4)  applies  in  these  two  cases. 


280 


FUNDAMENTAL   MECHANICAL    PRINCIPLES. 


The  following  table,  No.  101,  gives  the  velocity  acquired  by  a  falling  body 
in  falling  freely  through  a  given  height.  Table  No.  102  gives,  conversely, 
the  height  of  fall  due  to  a  given  velocity.  Table  No.  103  gives  the  fall  and 
the  final  velocity  due  to  a  given  time  of  falling  freely. 


Table  No.  101. — VELOCITY  ACQUIRED  BY  FALLING  BODIES,  DUE  TO  GIVEN 

HEIGHTS  OF  FALL. 


\/  h. 


Height  of 
Fall. 

Velocity 
in  Feet 
per  Second. 

Height  of 
Fall. 

Velocity 
in  Feet 
per  Second. 

Height  of 
Fall. 

Velocity 
in  Feet 
per  Second. 

Height  of 
Fall. 

Velocity 
in  Feet 
per  Second. 

feet. 

ft.  per  sec. 

feet. 

ft.  per  sec. 

feet. 

ft.  per  sec. 

feet. 

ft.  per  sec. 

.01 

.803 

3-o 

13.90 

23 

38.49 

50 

56.74 

.02 

I.I4 

3-5 

15.01 

24 

39.31 

IOO 

80.25 

•03 

i-39 

4.0 

16.05 

25 

40.12 

150 

98.28 

.04 

1.61 

4-5 

17.03 

26 

40.92 

200 

">5 

•05 

i.  80 

5-o 

17.99 

27 

41.70 

300 

139.0 

.06 

1.97 

5-5 

18.82 

28 

42,47 

400 

160.5 

.07 

2.12 

6.0 

19.66 

29 

43-22 

500 

179.9 

.08 

2.27 

6-5 

20.46 

3° 

43-95 

600 

196.6 

.09 

2.41 

7.0 

21.23 

3i 

44.68 

700 

212.3 

.1 

2-54 

7-5 

21.97 

32 

45-39 

800 

226.9 

.2 

3.20 

8.0 

22.69 

33 

46.10 

90O 

240.7 

•3 

4.40 

8-5 

23.40 

34 

46.79 

IOOO 

253-8 

•4 

5-07 

9.0 

24.07 

35 

47-47 

1500 

310.8 

•5 

5.68 

9-5 

24-73 

36 

48.15 

2000 

358.9 

.6 

6.22 

10 

25.38 

37 

48.81 

2500 

401.2 

•7 

6.7I 

ii 

26.62 

38 

49-47 

3000 

439-5 

.8 

7.l8 

12 

27.80 

39 

50.11 

3500 

474-7 

•9 

7.6l 

I3 

28.93 

40 

50.75 

4000 

5°7-5 

I.O 

8.03 

14 

30.03 

4i 

51-38 

4500 

538.3 

1.2 

8.79 

15 

31.08 

42 

52.01 

500O 

567.4 

1.4 

9-5° 

16 

32.10 

43 

52.62 

6000 

621.6 

1.6 

10.15 

i7 

33-09 

44 

53-23 

7OOO 

671.4 

1.8 

10.77 

18 

34.05 

45 

53.83 

8000 

717.8 

2.0 

"•35 

19 

34.98 

46 

54-43 

90OO 

761.3 

2.25 

12.04 

20 

35.89 

47 

55-02 

IOOOO 

802.5 

2.50 

12.69 

21 

36.77 

48 

55-6o 

2-75 

I3-31 

22 

37.64 

49 

56-17 

GRAVITY. 


28l 


Table  No.  102. — HEIGHT  OF  FALL  DUE  TO  GIVEN  VELOCITIES. 


64.4' 


Velocity 
in  Feet 
per  Second. 

Height  of 
Fall. 

Velocity 
in  Feet 
per  Second. 

Height  of 
Fall. 

Velocity 
in  Feet 
per  Second. 

Height  of 
Fall. 

Velocity 
in  Feet 
per  Second. 

Height  of 
Fall. 

ft.  per  sec. 

feet. 

ft.  per  sec. 

feet. 

ft.  per  sec. 

feet. 

ft.  per  sec. 

feet. 

•25 

.OOIO 

19 

5-6l 

46 

32.9 

73 

82.7 

•50 

.0039 

20 

6.21 

47 

34-3 

74 

85.0 

•75 

.0087 

21 

6.85 

48 

35-8 

75 

87.4 

I.OO 

.Ol6 

22 

7.52 

49 

37-3 

80 

99.4 

1-25 

.024 

23 

8.21 

50 

38.8 

85 

112.  2 

1.50 

•035 

24 

8.94 

51 

40.4 

90 

125.8 

J-75 

.048 

25 

9.71 

52 

42.0 

95 

I4O.I 

2 

.062 

26 

10.5 

53 

43-6 

100 

155-3 

2-5 

.097 

27 

u-3 

54 

45-3 

i°5 

I7I.2 

3 

.I4O 

2'8 

II.  2 

55 

47.0 

no 

187.9 

3-5 

.190 

29 

I3-I 

56 

48.7 

115 

205.4 

4 

.248 

3° 

14.0 

57 

50-4 

I2O 

223.6 

4-5 

.314 

3i 

14.9 

58 

52.2 

I30 

262.4 

5 

.388 

32 

15-9 

59 

54-1 

140 

304.3 

6 

•559 

33 

16.9 

60 

55-9 

150 

349-4 

7 

.761 

34 

17.9 

61 

57-8 

J75 

475-5 

8 

•994 

35 

I9.O 

62 

59-7 

200 

621 

9 

1.26 

36 

2O.  I 

63 

61.6 

300 

1397 

10 

i-55 

37 

21.3 

64 

63.6 

400 

2484 

ii 

1.88 

38 

22.4 

65 

65.6 

500 

3882 

12 

2.24 

39 

23.6 

66 

67.6 

600 

559° 

13 

2.62 

40 

24.9 

67 

69.7 

700 

7609 

14 

3-°4 

4i 

26.1 

68 

71.8 

800 

9938 

J5 

3-49 

42 

27.4 

69 

73-9 

900 

12578 

16 

3-98 

43 

28.7 

70 

76.! 

IOOO 

15528 

i7 

4.49 

44 

30.1 

7i 

78.3 

18 

5-°3 

45 

31-4 

72 

80.5 

282 


FUNDAMENTAL   MECHANICAL   PRINCIPLES. 


Table  No.   103. — HEIGHT  OF  FALL  AND  VELOCITY  ACQUIRED,  FOR 
GIVEN  TIME  OF  FALL. 


Time  of 
Fall. 

Height  of 
Fall. 

Velocity 
acquired  in 
Feet  per 
Second. 

Time  of 
Fall. 

Height  of 
Fall. 

Velocity 
acquired  in 
Feet  per 
Second. 

Time  of 
Fall. 

Height  of 
Fall. 

Velocity 
acquired  in 
Feet  per 
Second. 

seconds. 

feet. 

ft.  per  sec. 

seconds. 

feet. 

ft.  per  sec. 

seconds. 

feet. 

ft.  per  sec. 

I 

16.1 

32.2 

12 

23l8 

386.4 

23 

8517 

740.6 

2 

64.4 

64.4 

13 

2721 

418.6 

24 

9273 

772.8 

3 

144.9 

96.6 

14 

3I56 

450.8 

25 

IO062 

805.0 

4 

257.6 

128.8 

I5 

3623 

483.0 

26 

10884 

837.2 

5 

402.5 

161.0 

16 

4122 

5I5-2 

27 

H737 

869.4 

6 

579-6 

193.2 

17 

4653 

547-4 

28 

12622 

901.6 

7 

788.9 

225.4 

18 

5217 

579-6 

29 

13540 

933-8 

8 

1030 

257.6 

I9 

5812 

611.8 

30 

14490 

966.0 

9 

i3°4 

289.8 

20 

6440 

644.0 

31 

15473 

998.2 

10 

1610 

322.0 

21 

7IOO 

676.2 

32 

16487 

1030 

ii 

1948 

354-2 

22 

7792 

708.4 

ACCELERATED   AND   RETARDED    MOTION    IN    GENERAL. 

The  same  rules  and  formulas  that  have  been  applied  to  the  action  of 
gravity  are  applicable  to  the  action  of  any  other  uniformly  accelerating 
force  on  a  body,  the  numerical  constants  being  adapted  to  the  force.  If  an 
accelerating  or  retarding  force  be  greater  or  less  than  gravity;  that  is  to  say, 
than  the  weight  of  the  body,  the  constants  16.1,  32.2,  and  64.4  are  to  be 
varied  in  the  same  proportion. 

To  do  this,  multiply  the  constant  by  the  accelerating  force,  and  divide 
the  product  by  the  weight  of  the  body.  Let  f  be  the  accelerating  force, 
and  w  the  weight  of  the  body,  then  the  constant  becomes 


16.1  /          32.2  /         64.4  / 
LJL    or  * £_,  or     **J  • 

WWW 


..  (a) 


and  substituting  this  in  the  formulas  (i)  to  (6)  for  gravity,  the  following 
general  rules  and  formulas  are  arrived  at  for  the  action  of  uniformly  accel- 
erating or  retarding  forces.  The  rules  are  written  for  accelerating  forces, 
but  they  apply  by  simple  inversion  to  retarding  forces  also. 

GENERAL  RULES  FOR  ACCELERATING  FORCES. 

The  accelerating  force  and  the  weight  of  the  body  are  expressed  in  the 
same  unit  of  weight;  and  the  space  in  feet,  the  time  in  seconds,  and  the 
velocity  in  feet  per  second. 

In  the  following  rules  the  time  during  which  a  body  is  acted  on  by  an 
accelerating  force  is  called  the  time;  the  velocity  acquired  at  the  end  of  the 


ACCELERATED   AND   RETARDED   MOTION.  283 

time  is  called  the  final  velocity;  the  space  traversed  by  the  body  during  the 
time  is  called  the  space;  the  accelerating  force  is  called  the  force. 

t   =  the  time. 
v  —  the  final  velocity. 
s  -the  space. 
/  =  the  force. 
w  -  the  weight. 

RULE  7.  Given  the  weight,  the  force,  and  the  time;  to  find  the  final 
velocity.  Multiply  the  force  by  the  time  and  by  32.2,  and  divide  by  the 
weight.  The  quotient  is  the  final  velocity.  Or 


.._ 


RULE  8.  Given  the  weight,  the  force,  and  the  time;  to  find  the  sface. 
Multiply  the  force  by  the  square  of  the  time  and  by  16.1,  and  divide  by  the 
weight.  Or 

.  =  1*1/1*  ...................................   (8) 

w 

RULE  9.  Given  the  weight,  the  final  velocity,  and  the  force;  to  find  the 
time.  Multiply  the  final  velocity  by  the  weight,  and  divide  by  the  force, 
and  by  32.2.  The  quotient  is  the  time.  Or 

"5=7  ......................................  <» 

RULE  i  o.  Given  the  weight,  the  final  velocity,  and  the  force;  to  find  the 
space.  Multiply  the  weight  by  the  square  of  the  velocity,  and  divide  by 
the  force,  and  by  64.4.  The  quotient  is  the  space.  Or 

w  v1  /    \ 

s-       ....... 


RULE  ii.  Given  the  weight,  the  force,  and  the  space;  to  find  the  time. 
Multiply  the  weight  by  the  space,  and  divide  by  the  force;  find  the  square 
root  of  the  quotient,  and  divide  it  by  4.  The  last  quotient  is  the  time  in 
seconds.  Or 


f=  T,     ,    w  s 


RULE  12.  Given  the  weight,  \he  force,  and  the  space;  to  find  the  final 
velocity.  Multiply  the  space  by  the  force,  and  divide  by  the  weight;  find 
the  square  root  of  the  quotient,  and  multiply  by  8.  The  product  is  the  final 
velocity.  Or 

fs  t**\ 


w 


RULE  13.  Given  the  weight,  the  space,  and  \htfinal  velocity;  to  find  the 
force.  Multiply  the  weight  by  the  square  of  the  final  velocity,  and  divide 
by  the  space,  and  by  64.4.  The  quotient  is  the  force.  Or 


(I3) 


284  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

RULE  14.  Given  the  weight,  time,  and  final  velocity  ;  to  find  the  force. 
Multiply  the  weight  by  the  velocity,  and  divide  by  the  time,  and  by  32.2. 
Or 


IV    V 


Note  i.  When  the  accelerating  or  retarding  force  bears  a  simple  ratio  to 
the  weight  of  the  body,  the  ratio  may,  for  greater  readiness  in  calculation, 
be  substituted  in  the  quantities  (a)  representing  the  modified  constants,  for 
the  force  and  the  weight.  Suppose  the  accelerating  force  is  a  tenth  part  of 
the  weight,  then  the  ratio  is  i  to  10,  and 

16.1 
-  =  i.6i, 

10 


and  these  quotients  may  be  substituted  for  16.1,  32.2,  and  64.4  respectively 
in  the  formulas  for  the  action  of  gravity  (i)  to  (6),  to  fit  them  for  direct 
use  in  dealing  with  an  accelerating  force  one-tenth  of  gravity,  the  height  h 
in  those  formulas,  of  course,  being  taken  to  mean  space  traversed. 

Note  2.  The  tables,  Nos.  101-103,  pages  280-282,  for  the  relations  of 
the  velocity  and  height  of  falling  bodies,  may  be  employed  in  solving 
questions  of  accelerating  force  generally. 

Example.  A  ball  weighing  10  Ibs.  is  projected  with  an  initial  velocity  of 
60  feet  per  second  on  a  level  bowling-green,  and  the  frictional  resistance 
to  its  motion  over  the  green  is  i  Ib.  What  distance  will  it  traverse  before 
it  comes  to  a  state  of  rest?  By  rule  10, 


the  distance  traversed. 

Again,  the  same  result  may  be  arrived  at,  according  to  Note  i,  by 
multiplying  the  constant  64.4,  in  rule  4,  for  gravity,  by  the  ratio  of  the 
force  and  the  weight,  which  in  this  case  is  ^,  and  64.4x^  =  6.44. 
Substituting  6.44  for  64.4  in  that  rule  and  formula,  the  formula  becomes 

*  =  /-=  1^  =  559  feet, 

6.44         6.44 

the  distance  traversed,  as  already  found. 

But  the  question  may  be  answered  more  directly  by  the  aid  of  the  table 
for  falling  bodies  (No.  102,  page  281).  The  height  due  to  a  velocity  of 
60  feet  per  second,  is  55.9  feet;  and  it  is  to  be  multiplied  by  the  inverse 
ratio  of  the  accelerating  force  and  the  weight  of  the  body,  or  i2,  or  10; 
that  is, 

55.9  x  10  =  559  feet, 
the  distance  traversed. 

If  the  question  be  put  otherwise  —  What  space  will  a  ball  move  over 
before  it  comes  to  a  state  of  rest,  with  an  initial  velocity  of  60  feet  per 


GRAVITATION   ON   INCLINED   PLANES.  285 

second,  allowing  the  friction  to  be  i-ioth  the  weight  of  the  ball?  The 
answer  may  be  given,  that  the  friction,  which  is  the  retarding  force,  being 
i-ioth  of  the  weight,  that  is  of  gravity,  the  space  described  will  be  10  times 
as  much  as  is  necessary  for  gravity,  supposing  the  ball  to  be  projected 
vertically  upwards  to  bring  the  ball  to  a  state  of  rest.  The  height  due  to 
the  velocity  is  55.9  feet;  then 

55.9  x  10 -559  feet, 
the  space  described  by  the  ball. 

ACTION  OF  GRAVITY  ON  INCLINED  PLANES. 

If  a  body  freely  descend  an  inclined  plane  by  the  force  of  gravity  alone, 
the  velocity  acquired  by  the  body  when  it  arrives  at  the  foot  of  the  plane, 
is  that  which  it  would  acquire  by  falling  freely  through  the  vertical  height. 
Or,  the  velocity  is  that  "  due  "  to  the  height  of  the  plane. 

The  time  occupied  in  making  the  descent  is  greater  than  that  due  to  the 
height,  in  the  ratio  of  the  length  of  the  plane,  or  distance  travelled,  to 
the  height.  The  time  is  therefore  directly  in  proportion  to  the  length  of 
the  plane,  when  the  height  is  the  same. 

The  impelling  or  accelerating  force  by  gravitation  acting  in  a  direction 
parallel  to  the  plane,  is  less  than  the  weight  of  the  body,  in  the  ratio  of  the 
height  of  the  plane  to  its  length.  It  is,  therefore,  inversely  in  proportion 
to  the  length  of  the  plane,  when  the  height  is  the  same. 

The  time  of  descent,  under  these  conditions,  is  inversely  in  proportion 
to  the  accelerating  force. 

If,  for  instance,  the  length  of  the  plane  be  five  times  the  height,  the  time 
of  making  freely  the  descent  on  the  plane  by  gravitation  is  five  times 
that  in  which  a  body  would  freely  fall  vertically  through  the  height;  and 
the  impelling  force  down  the  plane  is  I/5  th  of  the  weight  of  the  body. 

Problems  on  the  descent  of  bodies  on  inclined  planes  are  soluble  by  the 
aid  of  the  rules  7  to  14,  for  the  relations  of  accelerating  forces.  But, 
as  a  preliminary  step,  the  accelerating  force  is  to  be  determined,  by 
multiplying  the  weight  of  the  descending  body  by  the  height  of  the  plane, 
and  dividing  the  product  by  the  length  of  the  plane.  For  example,  let  a 
body  of  15  pounds  weight  gravitate  freely  down  an  inclined  plane,  the 
length  of  which  is  five  times  the  height,  the  accelerating  force  is  1 5  -=-  5  =  3 
pounds.  If  the  length  of  the  plane  be  100  feet,  the  height  is  20  feet,  and 
the  velocity  acquired  in  falling  freely  from  the  top  to  the  bottom  of  the 
plane  would  be,  by  rule  12, 

20   =35.776  feet  per  second. 
The  time  occupied  in  making  the  descent  is,  by  rule  n, 

?  =  X</  soo   =5.59  seconds. 
3 

Whereas,  for  a  free  vertical  fall  through  the  height,  20  feet,  the  time 
would  be,  by  rule  5, 


?=Xv     20   =  i.i  18  seconds, 
which  is  r/s  th  of  the  time  of  making  the  descent  on  the  inclined  plane. 


286  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

Special  Rules  for  the  Descent  on  Inclined  Planes. 

The  height  and  the  length  of  an  inclined  plane  may  be  substituted 
for  the  accelerating  force  and  the  weight  respectively  in  the  rule  (n), 
to  find  the  time.  Putting  7z  =  the  height  of  the  plane,  and  /=the  length 
of  the  plane,  the  formula  (n) 


becomes  ,=  .  = 


RULE  15.  —  Given  the  length  and  the  height  of  the  inclined  plane,  to  find 
the  time  in  which  a  body  would  freely  descend  by  gravitation.  Divide  the 
length  by  four  times  the  square  root  of  the  height;  the  quotient  is  the  time 
in  seconds. 

For  example,  the  length  of  the  plane  is  100  feet,  and  the  height  is  20 
feet,  and  the  time  is 

100 

t=  —7—    -  5.59  seconds, 
4v  20 

as  v/as  found  before. 

Again,  by  inversion  of  the  formula  (15), 


(16) 


l-^t\/    h  ,  and  then 

/2 


i6/2 


RULE  1 6. — Given  the  length  of  the  inclined  plane,  and  the  time  of 
free  descent  by  gravitation,  to  find  the  height  through  which  the  body 
descends.  Divide  the  square  of  the  length  by  the  square  of  the  time 
in  seconds  and  by  1 6 ;  the  quotient  is  the  length  of  the  inclined  plane. 

For  example,  the  length  of  the  plane  is  100  feet,  and  the  time  of  descent 
is  5.59  seconds;  then  the  vertical  height  of  the  descent  is 

o 

h=  -  =  20  feet,  the  height. 

5.592xi6 

/ 

AVERAGE  VELOCITY  OF  A  MOVING  BODY  .  UNIFORMLY  ACCELERATED 

OR  RETARDED. 

The  average  velocity  of  a  moving  body  uniformly  accelerated  or  retarded, 
during  a  given  time  or  in  a  given  space,  is  equal  to  half  the  sum  of  the 
initial  and  final  velocities;  and  if  the  body  begin  from  a  state  of  rest  or 
arrive  at  a  state  of  rest,  the  average  speed  is  half  the  final  or  initial  velocity, 
as  the  case  may  be.  Thus,  in  the  example  of  a  ball  rolling,  the  initial 
speed  or  velocity  is,  in  either  case,  60  feet  per  second,  and  the  terminal 
speed  is  nothing;  the  average  speed  is  therefore  the  same,  namely,  one-half 
of  that,  or  30  feet  per  second. 


MASS.—  CENTRE  OF  GRAVITY.  287 

MASS. 

Weight  is  not  an  essential  property  of  a  body;  it  is  only  the  attraction  of  the 
earth  exerted  upon  the  body.  Suppose  the  attractive  force  to  be  suspended, 
then  the  body  would  cease  to  have  weight.  What  would  remain?  Mass, 
or  substance,  simply.  But,  though  weight  is  not  mass,  it  is  a  direct  measure 
of  mass,  in  the  same  locality,  or  wherever  the  force  of  gravitation  is  the 
same,  for  double  the  mass  has  twice  the  weight.  Weight  alone,  however, 
is  not  sufficient  as  a  universal  measure  of  mass,  since  the  weight  of  the 
same  mass  would  vary  according  to  the  force  of  gravitation  for  different 
situations.  The  mass,  therefore,  varies  inversely  as  the  force  of  gravitation, 
when  the  weight  remains  the  same.  That  force  is  measurable  by  the  height 
through  which  a  body  falls  in  a  given  time,  or  by  the  velocity  acquired  at 
the  end  of  that  time,  say,  a  second,  expressed  by  g.  In  its  most  general 
form,  then,  the  expression  for  the  mass  of  a  body  comprises^  the  weight 
directly  and  the  force  of  gravitation  inversely;  thus 


in  which  m  is  the  mass,  w  the  weight,  g  the  force  of  gravitation;  that  is  to 
say,  the  mass  of  a  body  is  equal  to  the  weight  of  the  body  divided  by  the 
force  of  gravity.  Since  the  weight  and  the  force  of  gravity  vary  in  the 

same  ratio,  the  mass  —  of  a  body  is  the  same  at  all  places.     As  the  quan- 

<5 

tity  of  matter  of  the  same  body  is  also  constant  whatever  place  it  occupies, 


the  mass  —  gives  an  exact  idea  of  the  quantity  of  matter,  and  is  a  measure 
of  it. 

MECHANICAL   CENTRES. 

There  are  four  mechanical  centres  of  force  in  bodies,  namely,  the  centre 
of  gravity,  the  centre  of  gyration,  the  centre  of  oscillation,  and  the  centre 
of  percussion. 

CENTRE  OF  GRAVITY. 

The  centre  of  gravity  is  the  physical  centre  of  a  body,  or  of  a  system  of 
bodies  in  rigid  connection  with  each  other,  about  which  the  gravitation  of 
the  several  particles  of  the  body  is  self-balanced,  and  at  which  it  can  be 
freely  suspended  or  supported  in  any  position  in  a  state  of  rest. 

In  various  calculations,  the  whole  weight  or  mass  of  a  body  is  considered 
as  placed  at  its  centre  of  gravity. 

To  find  the  centre  of  gravity  of  any  plane  figure  mechanically  :—  Suspend 
the  figure  by  any  point  near  its  edge,  and  mark  on  it  the  direction  of  a 
plumb-line  hung  from  that  point;  then  suspend  it  from  some  other  point, 
and  again  mark  the  direction  of  the  plumb-line  in  like  manner.  Then  the 
centre  of  gravity  of  the  surface  will  be  at  the  point  of  intersection  of  the 
two  marks  of  the  plumb-line. 

The  centre  of  gravity  of  parallel-sided  objects  may  readily  be  found  in 
this  way.  For  instance,  to  find  the  centre  of  gravity  of  the  arch  of  a  bridge; 
draw  the  elevation  upon  paper  to  a  scale,  cut  out  the  figure,  and  proceed 
with  it  as  above  directed,  in  order  to  find  the  position  of  the  centre  of 


288  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

gravity  in  elevation  for  the  model.  In  the  actual  arch,  the  centre  of  gravity 
will  have  the  same  relative  position  as  in  the  paper  model. 

In  regular  figures  or  solids  the  centre  of  gravity  is  the  same  as  their 
geometrical  centres.  Thus,  the  centre  of  gravity  of  a  straight  line,  a 
parallelogram,  a  prism,  a  cylinder,  a  circle,  the  circumference  of  a  circle,  a 
ring,  a  sphere,  and  a  regular  polygon,  is  the  geometrical  centre  of  these 
figures  and  solids  respectively. 

To  find  the  centre  of  gravity  of  a  triangle;  draw  a  straight  line  from  one 
of  its  angles  to  the  middle  of  the  opposite  side;  the  centre  of  gravity  will  be 
in  this  line  at  a  distance  of  two-thirds  of  its  length  from  the  angle.  Or, 
draw  a  straight  line  from  two  of  the  angles  to  the  middle  of  the  opposite 
sides  respectively;  the  point  of  intersection  of  the  two  lines  will  be  the 
centre  of  gravity. 

For  a  trapezium,  or  irregular  four-sided  figure,  draw  the  two  diagonals, 
and  find  the  centres  of  gravity  of  each  of  the  four  triangles  thus  formed; 
then  join  each  opposite  pair  of  these  centres  of  gravity.  The  joining  lines 
will  cut  each  other  in  the  centre  of  gravity  of  the  figure. 

For  a  cone  and  a  pyramid,  the  centre  of  gravity  is  in  the  axis  or  centre 
line,  at  a  distance  of  three-fourths  of  the  length  of  the  axis  from  the  vertex, 
or  one-fourth  from  the  base. 

For  an  arc  of  a  circle,  the  centre  of  gravity  lies  in  the  radius  bisecting  the 
arc,  and  the  distance  of  it  from  the  centre  of  the  arc  is  found  by  multiplying 
the  radius  by  the  chord  of  the  arc,  and  dividing  by  the  length  of  the  arc ;  the 
quotient  is  the  distance  of  the  centre  of  gravity  from  the  centre  of  the  circle. 

For  a  sector  of  a  circle,  the  centre  of  gravity  is  two-thirds  of  the  distance 
of  that  of  an  arc,  from  the  centre  of  the  circle.  It  is  found  independently 
by  multiplying  the  radius  by  twice  the  chord  of  the  arc,  and  dividing  by 
three  times  the  length  of  the  arc;  the  quotient  is  the  distance  of  the  centre 
of  gravity  from  the  centre  of  the  circle. 

For  a  parabolic  space,  the  centre  of  gravity  is  in  the  axis,  or  centre  line, 
and  its  distance  from  the  vertex  is  three-fifths  of  the  centre  line  or  axis. 

For  a  paraboloid,  the  centre  of  gravity  is  in  the  axis,  at  a  distance  from 
the  vertex  of  two-thirds  of  the  axis. 

For  two  bodies,  fixed  or  suspended  one  at  each  end  of  a  straight  bar,  the 
common  centre  of  gravity  is  in  the  bar,  at  that  point  which  divides  the 
distance  between  their  individual  centres  of  gravity,  in  the  inverse  ratio 
of  the  weights  respectively.  For  example,  if  two  weights  of  20  Ibs.  and 
10  Ibs.  be  suspended  on  a  bar  at  a  distance  of  9  feet  apart  between  their 
centres  of  gravity,  the  common  centre  of  gravity  will  divide  the  distance  in 
the  ratio  of  i  to  2,  being  3  feet  from  the  heavier  weight,  and  6  feet  from 
the  lighter.  In  this  example,  the  weight  of  the  bar  is  neglected;  but  it  may 
be  allowed  for  according  to  the  following  direction. 

For  more  than  two  bodies  connected  in  one  system,  the  common  centre 
of  gravity  may  be  found  by  finding,  in  the  first  place,  the  common  centre 
of  gravity  of  two  of  them,  and  then  finding  that  of  these  two  jointly  with 
a  third,  and  so  on  to  the  last  body  in  the  group. 

CENTRE  OF  GYRATION. — RADIUS  OF  GYRATION. — MOMENT  OF  INERTIA. 

The  centre  of  gyration,  or  revolution,  is  that  point  in  a  revolving  body,  or 
system  of  bodies,  at  a  certain  distance  from  the  axis  of  motion,  in  which 
the  whole  of  the  matter  in  revolution  may,  as  an  equivalent  condition,  be 


CENTRE  OF  GYRATION.  289 

conceived  to  be  concentrated,  just  as  if  a  pound  of  platinum  were  substituted 
for  a  pound  of  revolving  feathers,  whilst  the  moment  of  inertia  remains  the 
same.  The  work  accumulated  in  the  body,  of  which  the  moment  of 
inertia  is  a  measure,  remains  in  such  a  case  the  same,  at  the  same  angular 
velocity;  and,  as  a  necessary  consequence,  if  the  same  accelerating  force 
be  applied  to  the  body  at  the  centre  of  gyration,  as  would  actually  be 
expended  over  the  distributed  matter  of  the  body  to  communicate  to  it  its 
angular  velocity,  the  same  angular  velocity  would  be  generated. 

The  distance  of  the  centre  of  gyration  from  the  axis  of  motion  is  called 
the  radius  of  gyration  ;  and  the  moment  of  inertia  is  equal  to  the  product 
of  the  square  of  the  radius  of  gyration  by  the  mass  or  weight  of  the  body. 

The  moment  of  inertia  of  a  revolving  body  is  found  exactly  by  ascertain- 
ing the  moments  of  inertia  of  every  particle  separately,  and  adding  them 
together;  or,  approximately,  by  adding  together  the  moments  of  the  small 
parts  arrived  at  by  the  subdivision  of  the  body. 

RULE  i.  To  find  the  moment  of  inertia  of  a  revolving  body.  Divide 
the  body  into  small  parts  of  regular  figure.  Multiply  the  mass,  or  the 
weight,  of  each  part  by  the  square  of  the  distance  of  its  centre  of  gravity 
from  the  axis  of  revolution.  The  sum  of  the  products  is  the  moment  of 
inertia  of  the  body. 

Note.  —  The  value  of  the  moment  of  inertia  obtained  by  this  process 
will  be  more  nearly  exact,  the  smaller  and  more  numerous  the  parts  into 
which  the  body  is  divided. 

RULE  2.  To  find  the  length  of  the  radius  of  gyration  of  a  body  about  a 
given  axis  of  revolution.  Divide  the  moment  of  inertia  of  the  body  by  its 
mass,  or  its  weight,  and  find  the  square  root  of  the  quotient.  The  square 
root  is  the  length  of  the  radius  of  gyration;  or 


in  which  m  is  the  moment  of  inertia,  and  w  is  the  weight  of  the  body. 
Note.—  When  the  parts  into  which  the  body  is  divided  are  equal,  the 

radius  of  gyration  may  be  determined  by  taking  the  mean  of  all  the  squares 

of  the  distances  of  the  parts  from  the  axis  of  revolution,  and  finding  the 

square  root  of  the  mean  square. 

The  following  are  useful  examples  of  the  radius  of  gyration  of  bodies  :  — 
In  a  straight  bar,  or  a  thin  rectangular  plate,  revolving  about  one  of  its 

ends,  the  radius  of  gyration  is  equal  to  the  length  of  the  rod,  multiplied  by 

0-5773- 

In  a  straight  bar,  or  a  thin  rectangular  plate,  revolving  about  its  centre, 
the  radius  of  gyration  is  equal  to  half  the  length,  multiplied  by 

/Y/   y$,  or  0.5773. 

The  general  expression  for  the  radius  of  gyration  in  a  straight  bar  revolving 
on  any  point  of  its  length,  is 


//  as  + 
V  V3  (* 


in  which  a  and  b  are  the  lengths  of  the  two  parts  of  the  bar;  that  is  to  say, 

19 


290  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

divide  the  sum  of  the  cubes  of  the  two  parts  by  three  times  the  length  of 
the  bar,  and  extract  the  square  root  of  the  quotient.  The  root  thus  found 
is  equal  to  the  radius  of  gyration. 

In  a  circular  plate,  a  solid  wheel  of  uniform  thickness,  or  a  solid  cylinder 
of  any  length,  revolving  on  its  axis,  the  radius  of  gyration  is  equal  to  the 
radius  of  the  object,  multiplied  by 

y^  or  0.7071. 

In  a  plane  ring,  like  the  rim  of  a  fly-wheel,  revolving  on  its  axis,  the  radius 
of  gyration  is  equal  to  the  square  root  of  half  the  sum  of  the  squares  of  the 
inside  and  outside  radius  of  the  rim. 

In  a  thin  circular  plate,  put  in  motion  round  one  of  its  diameters, 
the  radius  of  gyration  is  equal  to  half  the  radius  of  the  circle. 

For  the  circumference  of  a  circle,  revolving  about  a  diameter,  the  radius 
of  gyration  is  equal  to  the  radius  multiplied  by  0.7071. 

In  the  circumference  of  a  circle  revolving  about  its  own  axis,  the  radius 
of  gyration  is  equal  to  the  radius  of  the  circle. 

In  a  solid  sphere  revolving  about  a  diameter,  the  radius  of  gyration  is 
equal  to  the  radius  multiplied  by 


2/s,  or  °-6325- 

In  the  surface  of  a  sphere,  or  an  insensibly  thin  spherical  shell,  the 
radius  of  gyration  is  equal  to  the  radius  multiplied  by 

A/    ^3,  or  0.8165. 

In  a  cone  revolving  about  its  axis,  the  radius  of  gyration  is  equal  to  the 
radius  multiplied  by  V.^or  .5477. 

CENTRE  OF  OSCILLATION. 

The  centre  of  oscillation  of  a  body  vibrating  about  a  fixed  axis  or  centre 
of  suspension,  by  the  action  of  gravity,  is  that  point  in  which,  if,  as  an 
equivalent  condition,  the  whole  matter  of  the  vibrating  body  were  concen- 
trated, the  body  would  continue  to  vibrate  in  the  same  time.  It  is  the 
resultant  point  of  the  whole  vibrating  energy,  or  of  the  action  of  gravity  in 
causing  oscillation.  As  the  particles  of  the  body  further  from  the  centre  of 
suspension  have  greater  velocity  of  vibration  than  those  nearer  to  it,  it  is 
apparent  that  the  centre  of  oscillation  is  more  distant  than  the  centre  of 
gravity  is  from  the  axis  of  suspension,  but  it  is  situated  in  the  centre  line 
which  passes  from  the  axis  through  the  centre  of  gravity.  It  differs  also 
from  the  centre  of  gyration  in  this,  that  whilst  the  motion  of  oscillation  is 
produced  by  the  gravity  of  the  body,  that  of  gyration  is  caused  by  some 
other  force  acting  at  one  place  only. 

The  radius  of  oscillation,  or  the  distance  of  the  centre  of  oscillation  from 
the  axis  of  suspension,  is  a  third  proportional  to  the  distance  of  the  centre 
of  gravity  from  the  axis  of  suspension  and  the  radius  of  gyration.  Hence 
the  following  rule  for  finding  the  radius  of  oscillation : — 


CENTRE   OF  OSCILLATION. — THE  PENDULUM.  291 

RULE  3.  To  find  the  radius  of  oscillation  in  a  body  vibrating  on  an 
axis.  Square  the  radius  of  gyration  of  the  body,  and  divide  by  the  distance 
of  the  centre  of  gravity  from  the  axis  of  suspension.  The  quotient  is  the 
radius  of  oscillation.  Or, 

-r,    ,.        f       .,,    .  radius2  of  gyration.  /     >. 

Radius  of  oscillation  = si : — ? r- (  3  ) 

distance  of  centre  of  gravity  from  axis. 

If  the  axis  of  suspension  be  in  the  vertex  or  uppermost  point  of  a  plane 
figure,  and  the  motion  be  edgewise,  then, 

In  a  right  line,  or  straight  rod,  the  radius  of  oscillation  is  two-thirds 
of  the  length  of  the  rod. 

In  a  circle  suspended  at  the  circumference,  the  radius  of  oscillation  is 
three-fourths  of  the  diameter. 

In  a  rectangle  suspended  by  one  of  its  angles,  it  is  two-thirds  of  the 
diagonal. 

In  a  parabola  suspended  by  the  vertex,  it  is  five-sevenths  of  the  axis 
plus  one-third  of  the  parameter. 

In  a  parabola  suspended  by  the  middle  of  its  base,  it  is  four-sevenths 
of  the  axis  plus  half  the  parameter. 

But,  if  the  oscillation  of  the  plane  figure  be  sidewise,  then, 

In  an  isosceles,  or  equal-sided  triangle,  it  is  three-fourths  of  the  height 
of  the  triangle. 

In  a  circle  it  is  five-eighths  of  the  diameter. 

In  a  parabola  it  is  five-sevenths  of  the  axis. 

In  a  sector  of  a  circle  suspended  by  the  centre,  it  is  three-fourths  of  the 
radius  multiplied  by  the  length  of  the  arc,  and  divided  by  the  length  of  the 
chord. 

In  a  cone  it  is  four-fifths  of  the  axis,  plus  the  quotient  obtained  by 
dividing  the  square  of  the  radius  of  the  base  by  five  times  the  axis. 

In  a  sphere  it  is  two-fifths  of  the  square  of  the  radius  divided  by  the  sum 
of  the  radius  and  the  length  of  the  cord  by  which  the  sphere  is  suspended, 
plus  the  radius  and  the  length  of  the  cord.  For  example,  in  a  sphere 
1 6  inches  in  diameter,  suspended  by  a  cord  25  inches  long,  the  radius  of 
oscillation  is 

2  x  82 

8  +  25  =  0.78  +  33  =  33.78  inches, 


5(8  +  25) 

or  0.78  inch  below  the  centre  of  the  sphere. 

It  may  be  noted  that  the  depression  of  the  centre  of  oscillation  below 
the  centre  of  the  sphere,  namely,  0.78  inch,  is  signified  in  the  first  quantity 
in  this  equation. 

The  Pendulum. 

A  "simple  pendulum"  is  the  most  elementary  form  of  oscillating  body, — 
consisting  theoretically  of  a  heavy  particle  attached  to  one  end  of  a  cord, 
or  an  inflexible  rod,  without  weight,  and  caused  to  vibrate  on  an  axis  at 
the  other  end,  or  the  centre  of  suspension. 

If  an  ordinary  pendulum  be  inverted,  so  that  the  centre  of  oscillation  shall 
become  the  centre  of  suspension,  then  the  first  centre  of  suspension  will 
become  the  new  centre  of  oscillation,  and  the  pendulum  will  vibrate  in  the 


FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

same  time  as  before.  This  reciprocal  action  of  the  pendulum  is  a  property 
of  all  pendulous  bodies,  and  it  is  known  as  the  reciprocity  of  the  pendulum. 
The  time  of  vibration  of  an  ordinary  pendulum  depends  on  the  angle  or 
the  arc  of  vibration,  and  is  greater  when  the  arc  of  vibration  is  greater,  but 
in  a  very  much  smaller  proportion;  and  if  this  arc  do  not  exceed  4°  or  5°, 
that  is  to  say,  from  2°  to  2^°  on  each  side  of  the  vertical  line,  the  time  of 
vibration  is  sensibly  the  same,  however  the  length  of  the  arc  may  vary 
within  that  limit.  This  property  of  a  pendulum,  of  equal  times  of  vibration, 
is  known  as  isochronism. 

To  construct  a  pendulum  such  that  the  time  of  vibration  shall  be  the 
same  whatever  the  magnitude  of  the  angle  of  vibration  may  be,  it  is  neces- 
sary to  cause  the  pendulum  to  vibrate,  not  in  a  circular  arc,  but  in  a 
cycloidal  curve.  For  this  object  the  pendulum  is  suspended  by  a  flexible 
thread  or  rod,  which  oscillates  between  two  cycloidal  surfaces,  on  which  it 
alternately  laps  and  unlaps  itself;  these  are  generated  by  a  circle  of  which 
the  diameter  is  equal  to  half  the  length  of  the  pendulum.  By  means  of 
the  circle  o  B,  Fig.  98,  for  example,  of  which  the  diameter  is  half  the  length 

of  the  pendulum,  describe  the  right  and 
left  cycloidal  curves  OCA,  OC'A',  on  the 
horizontal  line  A  A';  and  draw  the  tangent 
c  B  c',  touching  the  cycloids  at  the  middle 
of  their  lengths.  The  half-lengths  o  c,  o  c', 
are  equal  to  twice  the  diameter  of  the 
generating  circle  OB,  and  consequently 
equal  to  the  length  of  the  pendulum,  which 
Fig.  98.— Cycloidal  Pendulum.  will  vibrate  in  equal  times,  on  the  centre 

of  suspension  o,  between  the  entire  half- 
lengths  o  c,  o  c',  or  in  any  shorter  path.  The  curve  c  P  c'  thus  described 
by  the  pendulum,  is  itself  a  cycloidal  curve,  and  is  a  duplicate  of  the  other 
cycloids.  Though  a  cycloidal  motion  of  the  pendulum  is  necessary  to  render 
it  isochronous  for  all  angles  of  vibration,  yet  taking  very  small  arcs  of  the 
cycloidal  path  on  either  side  of  the  vertical  line,  they  do  not  sensibly  differ 
from  the  circular  arcs  which  would  be  described  by  an  ordinary  pendulum 
of  the  same  length  (o  P)  swinging  freely.  Hence  the  reason  that  the  ordinary 
pendulum  vibrates  in  equal  times  when  its  vibrations  do  not  exceed  4°  or 
5°  in  extent. 

The  length  of  the  pendulum  vibrating  seconds  at  the  level  of  the  sea 
in  the  latitude  of  London  is  39.1393  inches,  nearly  a  metre;  at  Paris 
it  is  39.1279;  at  Edinburgh  it  is  39.1555  inches;  at  New  York,  39.10153 
inches;  at  the  equator  it  is  39.027  inches,  and  at  the  pole  it  is  39.197 
inches.  Generally,  if  the  force  of  gravity,  or  the  length  of  the  seconds 
pendulum  at  the  equator  be  represented  by  i,  the  gravity,  or  the  length  of 
pendulum  at  other  latitudes  will  be  as  follows : — 

Length  of  Seconds  Pendulum. 

At  the  equator i.ooooo 

„    30°  latitude 00141 

„    45         »      00283 

,,    52         „       00357 

»    60         „      00423 

„    90        „      (the  pole) 00567 


THE  PENDULUM.  293 

According  to  these  ratios,  the  force  of  gravity,  and  the  length  of  the 
seconds  pendulum,  at  the  pole,  are  Vijetii  greater  than  at  the  equator;  there 
being  a  difference  of  length  of  between  a  fourth  and  a  fifth  of  an  inch. 

The  following  are  the  relations  of  the  lengths  of  pendulums  and  the  times 
of  their  vibrations,  that  is  to  say,  of  such  as  vibrate  through  equal  angles, 
or  of  which  the  total  angle  of  vibration  does  not  exceed  4°  or  5°:  — 

The  times  of  vibration  of  pendulums  are  proportional  to  the  square  root 
of  the  lengths  of  the  pendulums. 

Conversely,  the  lengths  of  pendulums  are  to  each  other  as  the  squares  of 
the  times  of  one  vibration,  or  inversely  as  the  squares  of  the  numbers  of 
vibrations  in  a  given  time. 

The  length  of  the  seconds  pendulum  at  London,  39.1393  inches,  may 
be  taken  as  a  datum  for  calculation  applicable  to  pendulums  of  different 
lengths,  and  to  different  times  of  vibration. 

RULE  4.  To  find  the  time  of  vibration  of  a  pendulum  of  a  given 
length.  Divide  the  square  root  of  the  given  length  in  inches  by  the 
square  root  of  39.1393,  or  6.2561.  The  quotient  is  the  time  of  a  vibration 
in  seconds.  Or 

' 


in  which  /  is  the  given  length  of  pendulum  in  inches,  and  /  the  time  of 
vibration  in  seconds. 

RULE  5.  To  find  the  number  of  vibrations  per  second  of  a  pendulum  of 
given  length.  Divide  6.2561  by  the  square  root  of  the  length  in  inches. 
The  quotient  is  the  number  of  vibrations  per  second. 

For  the  number  of  vibrations  per  minute.  Divide  375.366  by  the 
square  root  of  the  length  in  inches.  The  quotient  is  the  number  of 
vibrations  per  minute.  Or 

«-^  (per  second);  .......................  (5) 

minute).  ......................  (  5  j 


v// 
in  which  n  is  the  number  of  vibrations. 

RULE  6.  To  find  the  length  of  a  pendulum  when  the  time  of  a 
vibration  is  given.  Multiply  the  square  of  the  time  of  one  vibration  in 
seconds  by  39.1393.  The  product  is  the  length  of  the  pendulum  in 
inches.  Or 

/=/2x  39.1393 (6) 

RULE  7.  To  find  the  length  of  a  pendulum  when  the  number  of 
vibrations  per  second  is  given.  Divide  39.1393  by  the  square  of  the  num- 
ber of  vibrations  in  a  second.  The  quotient  is  the  length  of  the  pendulum 
in  inches. 

When  the  number  of  vibrations  per  minute  is  given.  Divide  140,900 
by  the  square  of  the  number  of  vibrations  in  a  minute.  The  quotient  is 
the  length  of  the  pendulum  in  inches.  Or 

/_       39.1393  .   (7) 

n2  (per  second) 3" 

7-        140,900  .-  (  7  ) 

n2  (per  minute) 


294  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

A  pendulum  may  be  shortened  and  yet  vibrate  in  the  same  time  as 
before,  by  the  action  of  a  second  weight  fixed  on  the  pendulum  rod  above 
the  centre  of  suspension.  Here  the  upper  weight  counteracts  the  lower, 
and  there  is  only  the  balance  of  gravitating  force  due  to  the  preponderance 
of  the  lower  weight  available  for  vibrating  both  masses.  The  mass  being 
thus  increased  while  the  gravitating  force  is  diminished,  a  longer  time  is 
required  for  each  vibration  when  the  length  of  pendulum  remains  unaltered, 
or  the  pendulum  may  be  shortened  so  that  the  time  of  the  vibrations  con- 
tinues the  same.  By  varying  the  height  of  the  upper  weight  above  the 
centre  of  suspension,  and  thus  varying  the  level  of  the  common  centre  of 
gravity,  the  period  of  vibration  is  varied  in  proportion. 

RULE  8.  To  find  the  weight  of  the  upper  bob  of  a  compound  pendulum 
necessary  to  vibrate  seconds,  when  the  weight  of  the  lower  bob  is  given, 
and  the  respective  distances  of  the  bobs  from  the  centre  of  suspension. 
Multiply  the  distance  in  inches  of  the  lower  bob  from  the  centre  of  suspen- 
sion by  39.1393,  and  from  the  product  subtract  the  square  of  that  distance 
(i);  again,  multiply  the  distance  in  inches  of  the  upper  bob  from  the 
centre  of  suspension  by  39.1393,  and  add  the  square  of  that  distance 
(2);  multiply  the  lower  weight  by  the  remainder  (i),  and  divide  by  the 
sum  (2).  The  quotient  is  the  weight  of  the  upper  bob.  Or 


in  which  D  and  d  are  the  respective  distances  of  the  lower  and  upper  bobs 
from  the  centre  of  suspension,  and  W,  w,  their  respective  weights. 

Thus,  by  means  of  a  second  bob,  pendulums  of  small  dimensions 
may  be  made  to  vibrate  as  slowly  as  may  be  desired.  The  metronome, 
an  instrument  for  marking  the  time  of  music,  is  constructed  on  this 
principle,  the  upper  weight  being  slid  and  adjusted  on  a  graduated  rod 
to  measure  fast  or  slow  movements. 

THE  CENTRE  OF  PERCUSSION. 

If  a  blow  is  struck  by  an  oscillating  or  revolving  body  moving  about  a 
fixed  centre,  the  percussive  action  is  the  same  as  if  the  whole  mass  of 
the  body  were  concentrated  at  the  centre  of  oscillation.  That  is  to  say, 
the  centre  of  percussion  is  identical  with  the  centre  of  oscillation,  and 
its  position  is  found  by  the  same  rules  as  for  the  centre  of  oscillation. 
If  an  external  body  is  so  struck  that  the  mean  line  of  resistance  passes 
through  the  centre  of  percussion,  then  the  whole  force  of  percussion  is 
transmitted  directly  to  the  external  body;  on  the  contrary,  if  the  revolving 
body  be  struck  at  the  centre  of  percussion,  the  motion  of  the  revolving 
body  will  be  absolutely  destroyed,  so  that  the  body  shall  not  incline  either 
way,  just  as  if  every  particle  separately  had  been  struck. 

CENTRAL   FORCES. 

When  a  body  revolves  on  an  axis,  every  particle  moves  in  a  circle  of 
revolution,  but  would,  if  freed,  move  off  in  a  straight  line,  forming  a  tangent 
to  the  circle.  The  force  required  to  prevent  the  body  or  particle  flying 
from  the  centre  is  called  centripetal  force,  and  the  tendency  to  fly  from 
the  centre  is  centrifugal  force.  These  forces  are  equal  and  opposite  — 
examples  of  action  and  reaction  —  and  are  classed  as  central  forces. 


CENTRAL   FORCES.  2Q5 

Centrifugal  force  varies  as  the  square  of  the  speed  of  revolution. 
It  varies  as  the  radius  of  the  circle  of  revolution. 
It  varies  as  the  mass  or  the  weight  of  the  revolving  body. 
Let  c  be  the  centrifugal  force  in  pounds,  w  the  weight  of  the  revolving 
body  in  pounds,  r  the  radius  of  revolution  or  gyration  in  inches,  m  the  mass 

of  the  body  =  —  ,  in  which  ^  =  32.2  or  gravity;  and  v  the  linear  or  circum- 

e> 

ferential  velocity  in  feet  per  second;  then 

_  m  v2  _w  v1 

r        32.2  r 

That  is  to  say,  the  centrifugal  force  of  a  revolving  body  is  equal  to  the 
weight  of  the  body  multiplied  by  the  square  of  the  linear  velocity,  divided 
by  32.2  times  the  radius  of  gyration. 

If  the  height  due  to  the  velocity  be  substituted  for  the  velocity  in  the 

above  equation,  the  height  h  being  equal  to  -  ,  then 

64.4 

_2WV2'_2.  W  k 

64.4  r        r 
and 

c  :  w  :  :  2  h  :  r. 

That  is  to  say,  the  centrifugal  force  is  to  the  weight  of  the  body  as  twice  the 
height  due  to  the  velocity  is  to  the  radius  of  gyration. 

From  the  first  equation  the  following  rules  for  revolving  bodies  are 
deduced,  for  finding  one  of  the  four  elements  when  the  other  three  are 
given:—  namely,  the  centrifugal  force,  the  radius  of  gyration,  the  linear 
velocity,  and  the  weight. 

RULE  i.  For  the  centrifugal  force.  Multiply  the  weight  by  the  square 
of  the  speed,  and  divide  by  32.2  times  the  radius  of  gyration.  The  quotient 
is  the  centrifugal  force.  Or 


(!) 


32.2  r 

RULE  2.  For  the  linear  velocity.  Multiply  the  centrifugal  force  by  the 
radius  of  gyration,  and  by  32.2,  and  divide  by  the  weight;  and  find  the 
square  root  of  the  quotient.  The  root  is  the  velocity.  Or 


RULE  3.  For  the  weight.  Multiply  the  centrifugal  force  by  the  radius  of 
gyration,  and  by  32.2,  and  divide  by  the  square  of  the  velocity.  The 
quotient  is  the  weight.  Or 

(3) 


RULE  4.  For  the  radius  of  gyration.  Multiply  the  weight  by  the  square 
of  the  velocity,  and  divide  by  the  centrifugal  force,  and  by  32.2.  The 
quotient  is  the  radius  of  gyration.  Or 

wv*  ,   ; 

r- (4) 

32.2  c 


296  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

Note.  —  When  the  velocity  is  expressed  as  angular  velocity,  in  revolutions 
per  unit  of  time,  it  is  to  be  reduced  to  linear  or  circumferential  velocity  by 
multiplying  it  by  the  radius  of  gyration  and  by  6.2832;  or 

•v  —  6.2832  v'  r, 

in  which  v'  is  the  angular  velocity. 

By  substitution  and  reduction  in  equation  (i),  the  following  equation  in 
terms  of  the  angular  velocity  is  arrived  at:  — 


0.8156  c^wrv"2,  .......................................  (5) 

from  which  is  found 

..  .(6) 


. 

0.8156 

That  is  to  say,  the  centrifugal  force  is  equal  to  the  weight  multiplied  by  the 
radius  of  gyration  and  by  the  square  of  the  angular  velocity,  and  by  1.226. 

MECHANICAL   ELEMENTS. 

The  function  of  mechanism  is  to  receive,  concentrate,  diffuse,  and  apply 
power  to  overcome  resistance.  The  combinations  of  mechanism  are  num- 
berless; but  the  primary  elements  are  only  two,  namely,  the  lever  and  the 
inclined  plane.  By  the  lever,  power  is  transmitted  by  circular  or  angular 
action;  that  is  to  say,  by  action  about  an  axis;  by  the  inclined  plane,  it  is 
transmitted  by  rectilineal  action.  The  principle  of  the  lever  is  the  basis  of 
the  pulley  and  the  wheel  and  axle;  that  of  the  inclined  plane  is  the  basis  of 
the  wedge  and  the  screw. 

For  the  present,  frictional  resistance  and  the  weight  of  the  mechanism 
are  not  considered;  the  terminal  resistance  is  called  the  weight;  and  the 
elemental  mechanisms  are  to  be  treated  as  in  a  state  of  equilibrium,  in 
which  the  power  exactly  balances  the  weight  without  actual  movement. 
The  action,  or  work  done,  will  be  subsequently  treated. 

THE  LEVER. 

The  elementary  lever  is  an  inflexible  straight  bar,  turning  on  an  axis  or 
fixed  point,  called  the  fulcrum;  the  force  being  transmitted  by  angular 

motion  about   the   fulcrum,  from   the 
point  where  the  power  is  applied  to  the 

i     point  where  the  weight   is  raised,  or 
other  resistance  overcome.     There  are 
_     three  varieties  of  the  lever,  according 
a     as    the    fulcrum,   the   weight,   or    the 
power  is  placed  between  the  other  two, 
but   the  action  is,  in  every  case,  re- 
Fig.  99.—  Lever.  ducible  to  that  of  three  parallel  forces 

in  equilibrium  (page  275). 

First.  The  power  is  applied  at  one  end  a,  of  the  lever  a  b  c,  Fig.  99,  and 
transmitted  through  the  fulcrum,  b,  to  the  weight  at  the  other  end  c.  The 
moments  of  the  power  and  the  weight  about  the  fulcrum  are  equal,  or 

power  x  a  b  =•  weight  x  b  c. 
That  is,  the  product  of  the  power  by  its  distance  from  the  fulcrum  is  equal 


THE   LEVER. 


297 


to  the  product  of  the  weight  by  its  distance  from  the  fulcrum.     Conse- 
quently 

power  :  weight  :  :  b  c  :  a  b, 

that  is,  the  power  and  the  weight  are  to  each  other  inversely  as  their 
respective  distances  from  the  fulcrum. 

The  ratio  of  the  length  of  the  power  end  of  the  lever  to  the  length  of  the 
weight  end  is  called  the  leverage  of  the  power.  The  respective  lengths, 
Fig.  99,  being  7  feet  and  i  foot,  the  leverage  is  7  to  i,  or  7. 

The  three  varieties  of  the  lever  are 
grouped  together  in  Figs.  100,  101,  and 
1 02.  In  each  case,  the  lever  is  supposed 
to  be  8  feet  long  and  divided  into  feet. 
The  leverage,  in  the  first,  is  7  to  i,  or  7; 
in  the  second,  8  to  i,  or  8;  in  the  third, 
Y%  to  i,  or  y% :  showing  that,  in  the  first 
case,  the  power  balances  seven  times  its 
own  amount;  in  the  second  case,  eigh 
times  its  amount;  in  the  third  case,  only 


Fig.  100. — Lever,  ist  kind. 


Fig.  101. — Lever,  ad  kind. 


Fig.  102. — Lever,  3d  kind. 


one-eighth  of  itself,  because  it  is  nearer  to  the  fulcrum  than  the  weight. 

In  each  case  the  moments  of  the  power  and  the  weight  about  the  fulcrum 
are  equal,  for,  in  each  case, 


The  pressures  exerted  at  the  extremities  of  the  lever  act  in  the  same 
direction,  and  the  sum  of  them  is  equal  and  opposite  to  the  intermediate 
pressure,  whether  it  be  that  of  the  fulcrum,  the  weight,  or  the  power  ( — ). 
From  this  the  pressure  on  the  fulcrum  may  be  found.  If  it  be  in  the 
middle,  the  pressure  is  equal  to  the  sum  of  the  power  and  the  weight,  that 
is,  60  +  420  =  480  Ibs.  in  the  example  above;  if  at  one  end,  it  is  equal  to 
the  difference  of  them,  that  is,  it  is  480  -  60  =  420  Ibs.  when  the  weight  is 
in  the  middle,  and  it  is  60-7^  =  52^  Ibs.  when  the  power  is  in  the 
middle. 

From  the  equation  for  the  equality  of  moments, 


orPxL    =Wx/, 

in  which  L  and  /  are  the  respective  distances  of  the  power  and  the  weight 
from  the  fulcrum,  rules  may  be  formed  for  finding  any  one  of  the  four 
quantities,  when  the  other  three  are  given. 

RULE  i.  To  find  the  power.  Multiply  the  weight  by  its  distance  from 
the  fulcrum,  and  divide  by  the  distance  of  the  power  from  the  fulcrum. 
The  quotient  is  the  power. 


298  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

Or,  divide  the  weight  by  the  leverage.     The  quotient  is  the  power.     Or 


P  = 


d) 


RULE  2.  To  find  the  weight.  Multiply  the  power  by  its  distance  from 
the  fulcrum,  and  divide  by  the  distance  of  the  weight  from  the  fulcrum. 
The  quotient  is  the  weight. 

Or,  multiply  the  power  by  the  leverage.     The  product  is  the  weight.     Or 

TTT  -t        JLj  /  x 


RULE  3.  To  find  the  distance  of  the  power  from  the  fulcrum.  Multiply 
the  weight  by  its  distance  from  the  fulcrum,  and  divide  by  the  power.  The 
quotient  is  the  distance  of  the  power  from  the  fulcrum.  Or 


i-/    


W/ 


(3) 


RULE  4.  To  find  the  distance  of  the  weight  from  the  fulcrum.  Multiply 
the  power  by  its  distance  from  the  fulcrum,  and  divide  by  the  weight.  The 
quotient  is  the  distance  of  the  weight  from  the  fulcrum.  Or 

/    PL 

/=-w (4) 

If  the  weight  of  the  lever  be  included  in  such  calculations,  its  influence 
is  the  same  as  if  its  whole  weight  or  its  mass  were  collected  at  its  centre  of 
gravity.  Thus,  if  the  lever  of  the  first  kind,  Fig.  100,  weighs  30  Ibs.,  and  its 
centre  of  gravity  be  at  the  middle  of  its  length,  the  weight  of  the  lever 
co-operates  with  the  power,  at  a  mean  distance  of  3  feet  from  the  fulcrum. 
By  equality  of  moments 

(P  x  7)  x  (30  x  3)  =  W  x  i  =  420  Ibs.  x  i, 
and  P  x  7  =  420  -  90  =  330  Ibs.; 

therefore  P,  the  power  at  the  end  of  the  lever  required  to  balance  the 


Fig.  103. — Inclined  Lever. 


Fig.  104. — Inclined  Lever. 


weight,  is  only  330-^-7  =  47.1  Ibs.  in  co-operation  with  the  weight  of  the 

lever,  as  compared  with  60  Ibs.,  without  reckoning  the  aid  from  this  source. 

When  the  lever  is  inclined  to  the  direction  of  the  forces,  as  in  Fig.  103, 


THE   LEVER. 


299 


equilibrium,  or  the  equality  of  moments,  may  nevertheless  be  maintained. 
Drawing  the  horizontal  line  a'  b  cf  through  the  fulcrum,  to  meet  the  ver- 
ticals through  the  power  and  the  weight  at  a'  and  d,  the  moments  of  the 
power  and  the  weight  are  to  be  estimated  on  the  horizontal  lengths  a'  b,  b  <?\ 
and 

the  moment  P  x  a'  b  =  the  moment  W  x  b  S. 

The  equality  of  moments  may  be  proved  in  another  way.  Let  the 
power  and  the  weight  be  resolved,  in  order  to  find  the  pressures  on  the 
ends  of  the  lever,  at  right  angles  to  it,  and  thus  to  arrive  at  the  moments 
as  estimated  on  the  actual  length  of  the  lever.  Let  the  verticals  through 
the  ends  of  the  lever,  a  m  and  c  n,  'Fig.  104,  represent  the  power  and  the 
weight  respectively,  and  draw  a  P'  and  c  W  perpendicular  to  the  lever,  and 
m  P'  and  n  W  parallel  to  it,  completing  the  triangles  a  m  P',  c  n  W.  Then 
a  P'  and  c  W  are  the  components  of  the  power  and  the  weight  respectively 
tending  to  turn  the  lever;  and,  it  may  be  added,  they  bear  the  same  ratio 
to  each  other  as  the  power  and  the  weight.  Consequently,  if  these  com- 
ponents be  multiplied  by  the  respective  lengths  of  the  lever,  the  products 
mil  be  the  moments  of  the  components,  and  the  moments  will  be  equal;  or 

the  moment  a  P'  x  a  b  —  the  moment  c  W  x  b  c. 

These  two  methods  of  analyzing  and  finding  the  moments  of  the  forces 
acting  on  an  inclined  lever — one,  combining  a  reduced  length  of  lever  with 
the  whole  power  and  weight;  the  other,  combining  the  whole  length  of 
lever  with  a  reduced  power  and  weight — lead  to  one  conclusion,  that  a 
lever,  if  balanced  in  one  position,  is  balanced  in  every  other  position, 
when  the  forces  continue  to  act  in  parallel  lines. 


JV- 


(8 


o 


O 


Fig.  105. — Bent  Lever. 


Fig.  1 06.— Bent  Lever. 


The  conditions  of  equilibrium  in  a  bent  lever  may  be  defined  similarly. 
Let  the  lever  a  b  c,  Fig.  105,  be  bent  at  the  fulcrum  b;  draw  the  horizontal 
line  a'  b  c',  then  the  moments  of  the  power  and  the  weight  are  reckoned 
on  the  lines  a'  b,  b  c',  and  they  are  equal  to  each  other;  or 


3oo 


FUNDAMENTAL  MECHANICAL   PRINCIPLES. 


Again,  let  the  forces  acting  on  a  lever,  whether  straight  or  bent,  be 
otherwise  than  vertical  or  parallel.  When  the  arms  of  the  lever  are  at 
right  angles,  and  the  power  and  the  weight  applied  at  right  angles  to  the 
arms,  as  in  Fig.  106,  the  moments  are  reckoned  directly  on  the  arms,  a  b,  b  c, 
as  in  a  straight  lever;  and 

the  moment  P  x  a  b  =  the  moment  W  x  b  c. 

The  thrust,  or  pressure  on  the  fulcrum,  is  in  this  case  less  than  the  sum  of 
the  power  and  the  weight;  and  it  may  be  determined  by  constructing  a 

parallelogram  upon  the  two  arms  of  the 
lever,  the  arms  representing  inversely  the 
respective  forces.  That  is,  a  b  represents 
the  magnitude  and  direction  of  the  weight 
W,  and  b  c  those  of  the  power  P.  The 
diagonal  b  b',  of  the  parallelogram  repre- 
sents the  magnitude  and  direction  of  the 
third  force  acting  at  the  fulcrum  to  oppose 
the  other  two  and  maintain  equilibrium. 

When  the  same  lever  is  tilted  into  an 
oblique  position,  the  power  continuing  to 
act  horizontally  on  the  lever,  Fig.  107, 
draw  the  vertical  b'  c'  through  the  end  c  of 

the  lever,  and  produce  the  power  line  a  p  to  meet  it  at  b'.  Complete  the 
parallelogram  of  b '  d  b;  then  the  sides  a'  b  and  b  c'  are  the  perpendiculars 
to  the  directions  to  the  power  and  weight,  on  which  the  moments  are 
reckoned,  so  that 

the  moment  P  x  of  b  =  the  moment  W  x  b  <?. 
The  diagonal  b  b'  is  the  resultant  force  at  the  fulcrum. 


Fig.  107. — Bent  Lever. 


Fig.  108. — Bent  Lever. 


Fig.  109. — Serpentine  Lever. 


If  the  power  do  not  act  horizontally,  but  in  some  other  direction,  a  pt 
Fig.   1 08,  produce  the  power-line  pa  and  draw  baf  perpendicular  to  it. 


THE   LEVER.  30  1 

Draw  b  S  as  before;  then  the  moments  are  reckoned  on  the  perpendiculars 
b  a',  b  c,  and,  as  before, 


To  find  the  resultant  force  at  the  fulcrum.  On  the  fulcrum  b  as  a  centre 
describe  arcs  of  circles  with  the  radii  b  a'  and  b  S,  and  draw  b  a",  b  c" 
respectively  parallel  to  the  directions  of  the  weight  and  the  power,  to  cut 
the  arcs  at  a?  and  c".  Complete  the  parallelogram,  and  the  diagonal  b  V 
represents  in  magnitude  and  direction  the  resultant  force  at  the  fulcrum. 

In  this  solution  the  power  and  the  weight  are  assumed  to  act  exactly, 
or  sensibly,  in  the  same  plane. 

Again,  in  the  serpentine  lever  a  b  c,  Fig.  109,  supposed  to  be  a  pump- 
handle,  the  power  P  is  applied  obliquely  in  the  direction  a  P.  Produce 
P  a  and  W  c,  and  draw  the  perpendiculars  b  a',  b  c'  from  the  fulcrum  for 
the  lengths  of  the  moments,  then 


Construct  the  parallelogram,  as  in  the  foregoing  figure,  and  the  diagonal 
b  b"  represents  the  resultant  force  at  the 
fulcrum. 

By  similar  treatment  the  action  of  the 
forces  in  levers  of  the  second  and  third 
kinds  may  be  analyzed.  The  lever  of  the 
second  kind,  #  c  b,  Fig.  no,  in  an  oblique 
position,  is  acted  on  horizontally  by  the 
power  and  the  weight  at  a  and  c;  draw 
the  vertical  b  d  a',  then  b  c'  and  b  a'  are 
the  distances  at  which  the  forces  act  from  W 
the  fulcrum,  or  the  lengths  of  the  mo-  Fig.  no.-Leverofthe2dkind. 

ments,  and 


and  the  horizontal  resultant  force  at  the  fulcrum  is  equal  to  the  difference 
of  the  weight  and  the  power. 

If  more  than  two  forces  be  applied  to  a  lever  in  a  state  of  equilibrium, 
the  sum  of  the  moments  tending  to  turn  the  lever  in  one  direction  is  equal 
to  the  sum  of  those  tending  in  the  opposite  direction. 

If  two  or  more  levers  are  connected  consecutively  one  to  the  other,  so 
that  they  act  as  one  system,  with  the  power  and  the  weight  at  the  extremi- 
ties, then,  in  equilibrium,  the  ratio  of  the  power  to  the  weight  is  the  product 
of  the  separate  inverse  ratios  of  all  the  levers.  For  example,  in  a  connected 
series  of  three  levers,  having  each  their  arms  in  the  ratio  of  2  to  i,  the 
combined  inverse  ratio  is  found  by  multiplying  2  by  2  and  by  2  ;  thus 

first  lever  .....................................  2  to  i  ratio, 

second  lever  ....................  ............  2  to  i  ratio, 

third  lever  .............................  2  to  i  ratio, 


compound  ratio 8  to  i. 

That  is,  the  power  is  to  the  weight  as  i  to  8. 


302 


FUNDAMENTAL   MECHANICAL   PRINCIPLES. 


THE  PULLEY. 

The  pulley  is  a  wheel  over  which  a  cord,  or  chain,  or  band  is  passed,  in 
order  to  transmit  the  force  applied  to  the  cord  in  another  direction.  It  is 
equivalent  to  a  continuous  series  of  levers,  with  equal  arms  on  one  fulcrum 
or  axis,  and  affords  a  continuous  circular  motion  instead  of  the  intermittent 
circular  motion  of  one  lever.  The  weight  W,  Fig.  in,  is  sustained  by  the 
power  P,  by  means  of  a  cord  passed  over  the  pulley  A,  in  fixed  supports, 
and  the  centre  line  a  b  c  represents  the  element  of  the  lever,  from  the  ends 
of  which  the  power  and  the  weight  may  be  conceived  to  depend,  turning 
on  the  fulcrum  b.  By  equality  of  moments,  Px#£  =  Wx#£y  and  the 
arms  or  radii  a  b,  b  c  being  equal,  the  power  is  equal  to  the  weight,  and  the 
counter-pressure  at  the  fulcrum  is  equal  to  twice  the  weight. 

When  the  power  and  weight  act  in  directions  at  an  angle  with  each  other, 
as  in  Fig.  112,  the  acting  radii  ab,  be,  representing  the  element  of  a  bent 


Fig.  113. — Pulley. 


lever,  are  lines  drawn  from  the  centre  perpendicular  to  the  directions  of  the 
power  and  weight.     The  power  is  equal  to  the  weight,  but  the  counter- 
pressure  on  the  fulcrum  is  less  than  twice  the  weight,  and  is  represented  by 
the  diagonal  b  b'  of  the  parallelogram  formed  by  the 
radii  b  a',  b  /,  drawn  from  the  fulcrum  parallel   to 
the  directions  of  the  (power  and  the  weight  respec- 
tively. 

Another  construction  for  the  parallelogram  of 
forces  in  the  action  of  the  pulley  is  obtained  by 
producing  the  directions  of  the  power  and  the  weight 
beyond  the  pulley,  Fig.  113,  intersecting  each  other 
at  I/,  then  forming  the  parallelogram,  and  drawing 
the  diagonal  W  as  the  resultant  pressure  on  the 
fulcrum. 

Thus  the  single  fixed  pulley  acts  like  a  lever  of  the 
first  kind,  and  simply  changes  the  direction  of  force, 
without  modifying  the  intensity  of  the  power. 

But  the  pulley  may  be  employed  as  a  lever  of  the 
second  kind  by  suspending  the  weight  to  the  axis 
fixing  one  end  of  the  cord  to  a  point  as  a  fulcrum 
114,  the  weight  W  is  suspended   from  the  axis  c, 


Fig.  114. — Movable  Pulley, 
as  a  lever  of  the  2d  kind. 


of  the  pulley,  and 
point.     Thus,  in  Fig. 


THE    PULLEY. 


303 


the  cord  is  fixed  to  the  point  b',  and  the  power  P  acts  through  the  diameter 
a  c  b,  in  which  b  is  the  fulcrum.     By  equality  of  moments, 


that  is,  the  product  of  the  power  by  the  diameter  of  the  pulley  is  equal  to 
the  product  of  the  weight  by  the  radius  of  the  pulley, 
and  the  leverage  being  as  2  to  i,  the  power  is  only 
half  the  weight. 

In  acting  as  a  lever  of  the  third  kind,  the  power  is 
applied  to  the  axis  a,  Fig.  115,  one  end  of  the  cord 
being  fixed  at  b\  and  the  weight  attached  at  the  other 
end,  c.  In  this  case,  by  equality  of  moments  the 
product  of  the  power  by  the  radius  of  the  pulley  is 
equal  to  that  of  the  weight  by  the  diameter,  and  the 
leverage  being  as  i  to  2,  the  power  is  twice  the  weight. 

These  demonstrations  with  respect  to  movable 
pulleys  only  apply  to  cases  of  parallel  cords;  that  is 
to  say,  when  the  direction  of  the  power  is  parallel  to 
that  of  the  weight.  If,  on  the  contrary,  they  be  inclined 
to  each  other,  as  in  Fig.  116,  in  which  the  weight  is  suspended  by  the 
axis,  the  power  becomes  greater  than  half  the  weight,  as  is  shown  by  the 
parallelogram  of  which  the  diagonal  c'  c"  represents  the  weight,  and  the 
sides  <?  b",  S  a",  the  pull  on  the  fulcrum,  and  the  power  exerted  to  sustain 
the  weight.  Each  of  these  sides  is  greater  than  half  the  diagonal. 


Fig.  1 15.— Movable  Pulley, 
as  a  lever  of  the  ad  kind. 


Fig.  1 16.— Movable  Pulley. 


Fig.  117.— Pulley-Blocks. 


Combinations  of  Pulleys. — Fast  and  Loose  Pulleys. — In  these  last  two 
applications  of  the  pulley,  it  becomes  movable  when  in  action,  and  by  com- 
bining two  or  more  movable  pulleys  on  the  same  or  different  axles  in  one 
block,  with  one  cord,  the  gain  of  power  may  be  increased  in  the  same  pro- 
portion. The  movable  block  A,  Fig.  117,  carrying  the  weight,  is  used 


304 


FUNDAMENTAL  MECHANICAL   PRINCIPLES. 


with  a  fixed  counterpart  B,  the  rope  is  attached  by  one  end  to  the  fixed 
block,  and  is  passed  over  the  movable  and  fixed  pulleys,  from  one  to  the 
other  in  succession,  the  power  being  applied  to  the  other  end,  as  at  P. 
This  system  is  known  as  fast  and  loose  pulley-blocks. 

The  fixed  end  of  the  rope  is  sometimes  attached  to  the  movable  block. 

RULE  i.  To  find  the  power  necessary  to  balance  a  weight  or  resistance 
by  means  of  a  system  of  fast  and  loose  pulleys.  Divide  the  weight  by  the 
number  of  ropes  by  which  it  is  carried;  that  is,  the  number  of  ropes  which 
proceed  from  the  movable  block.  The  quotient  is  the  power  required  to 
balance  the  weight. 

When  the  fixed  end  of  the  rope  is  attached  to  the  fixed  block,  the  num- 
ber of  ropes  proceeding  from  the  loose  block  is  twice  the  number  of  mov- 
able pulleys,  and  the  power  may  be  found  by  dividing  the  weight  by  twice 
the  number  of  movable  pulleys. 

When  the  end  of  the  rope  is  attached  to  the  movable  block,  the  divisor 
may  be  taken  at  twice  the  number  of  movable  pulleys  plus  i. 

Or,  putting  n  for  the  number  of  movable  pulleys;  if  the  fixed  end  of  the 
rope  is  attached  to  the  fixed  block, 


and  if  the  fixed  end  of  the  rope  be  attached  to  the  movable  block, 

W 


2  n+  i 


(la) 


RULE  2.  To  find  the  weight  or  resistance  that  will  be  balanced  by  a 
given  power,  by  means  of  a  system  of  fast  and  loose  pulleys.  Multiply  the 
power  by  the  number  of  ropes  proceeding  from  the 
movable  block.  The  product  is  the  required  weight. 

Or,  when  the  rope  is  attached  to  the  fixed  block, 
multiply  the  power  by  twice  the  number  of  movable 
pulleys. 

Or,  when  the  rope  is  attached  to  the  movable 
block,  multiply  the  power  by  twice  the  number  of 
movable  pulleys  plus  i. 

Or,  in  the  first  case, 


in  the  second  case, 


(2) 


(20) 


Fig.  1 18.— Movable  Pulleys. 


Again,  a  combination  may  be  formed  of  a  num- 
ber of  movable  pulleys,  as  in  Fig.  1 18,  each  of  which, 
A,  B,  C,  is  suspended  by  a  cord,  with  one  end  fixed 
to  the  roof  and  the  other  end  fixed  to  the  axis  of 
the  next  pulley.  The  weight  W  is  hung  to  the  axis 

of  the  first  pulley  A,  which  delivers  half  the  weight  to  the  second  pulley  B, 
which  delivers  half  of  the  weight  hanging  to  it,  or  one-fourth  of  the  first 
weight  W,  to  the  third  pulley  C;  from  which  only  one-eighth  of  the  first 
weight  passes  over  the  guide  or  neutral  pulley  D  to  the  power  P.  In 


THE    PULLEY.  305 

general  the  divisor  for  the  power  is  2",  or  the  ;zth  power  of  2,  n  being  the 
number  of  movable  pulleys. 

RULE  3.  To  find  the  power  necessary  to  balance  a  weight  by  means  of  a 
system  of  separate  movable  pulleys,  with  separate  cords  consecutively  con- 
nected as  above  described.  Divide  the  weight  by  that  power  of  2  of  which 
the  index  is  the  number  of  movable  pulleys.  The  quotient  is  the  power  or 
force  required  to  balance  the  weight. 

Or,  divide  and  subdivide  the  weight  successively  by  2  as  many  times  as 
there  are  movable  pulleys  to  find  the  power  required.  Or 

w 


RULE  4.  To  find  the  weight  that  can  be  balanced  by  a  given  power,  by 
means  of  a  system  of  separate  movable  pulleys  as  above  described.  Mul- 
tiply the  power  by  that  power  of  2  of  which  the  index  is  the  number  cf 
movable  pulleys.  The  product  is  the  weight  required. 

Or,  multiply  the  power  successively  by  2  as  many  times  as  there  are 
pulleys.  Or 

W-Px  2" (4) 

Note. — It  is  necessary  that  the  cords  should  be  parallel  to  each  other, 
as  in  the  illustration,  in  order  that  the  above  rules,  3  and  4,  may  apply. 

WHEEL  AND  AXLE. 

The  wheel  and  axle  may  be  likened  to  a  couple  of  pulleys  of  different 
diameters  united  together  on  one  axis,  of  which  the  larger,  a,  Fig.  119,  is  the 
wheel,  and  the  smaller,  c,  the  axle,  with  a  common  ful- 
crum, b;  the  centre  line  abc  representing  the  elements  of 
a  lever.     The  weight  W  on  the  axle  at  c  is  balanced  by 
the  power  P,  on  the  wheel  at  a.    The  moments  are  equal, 
or 


and  the  power  is  to  the  weight  inversely  as  their  distances 
from  the  centre;  or 

P  :  W  :  :  b  c  :  a  b. 

T 

If  a  crank  handle  be  substituted  for  the  wheel,  making 
a  windlass,  the  radius  of  the  crank  is  substituted  for  that       l[s-  119-—  wheel 

...  ...  .  .  ,  .          _    ,  aiiu  Axle. 

of  the  wheel  in  estimating  the  ratio  of  the  forces. 

Of  the  four  elements,  namely,  the  radius  of  the  wheel  or  crank,  the  radius 
of  the  axle  or  roller,  the  power,  and  the  weight,  if  three  be  given,  the  fourth 
can  be  found  as  follows,  putting  R  and  r  for  the  respective  radii. 

RULE  i.  To  find  the  power.  Multiply  the  weight  by  the  radius  of  the 
axle,  and  divide  by  the  radius  of  the  wheel.  The  quotient  is  the  power. 
Or 

P-Wx       ......................................  (i) 


RULE  2.  To  find  the  weight.     Multiply  the  power  by  the  radius  of  the 

20 


306  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

wheel,  and  divide  by  the  radius  of  the  axle.     The  quotient  is  the  weight. 
Or 


(2) 


RULE  3.  To  find  the  radius  of  the  wheel.  Multiply  the  weight  by  the 
radius  of  the  axle,  and  divide  by  the  power.  The  quotient  is  the  radius  of 
the  wheel.  Or 

R-       ..........................................  (3) 


RULE  4.  To  find  the  radius  of  the  axle.  Multiply  the  power  by  the 
radius  of  the  wheel,  and  divide  by  the  weight.  The  quotient  is  the  radius 
of  the  axle.  Or 

PR 


Note.  —  The  diameters  of  the  wheel  and  the  axle  or  roller  may  be 
employed  in  the  calculations  instead  of  the  radii. 

INCLINED  PLANE. 

The  inclined  plane  is  a  sloper  or  a  flat  surface  inclined  to  the  horizon,  on 
which  weights  may  be  raised.  By  such  substitution  of  a  sloping  path  for 
a  direct  vertical  line  of  ascent,  a  given  weight  can  be  raised  by  a  power 
which  is  less  than  the  weight  itself. 

There  are  three  elements  of  calculation  in  the  inclined  plane  :  —  the  plane 
itself,  A  B,  Fig.  120;  the  base,  or  horizontal  length,  A  C;  and  the  height  or 

vertical  rise  B  C;  together  forming  a  right- 
angled  triangle.  The  weight  W  is  to  be 
raised  through  a  height  equal  to  C  B,  and 
for  that  object  is  drawn  up  the  slope  from 
A  to  B.  It  is  partly  supported  during  the 
ascent,  and  it  is  in  virtue  of  this  degree  of 
support  given  to  the  weight  that  such  a 
"  dead  pull  "  as  that  of  a  direct  vertical 
*A  lift  is  avoided,  and  that  it  can  be  raised 

Fig.  i20.-inciined  Plane.  by  a  power  much  less  than  its  own  weight. 

Let  the  weight  W  be  kept  at  rest  on  the 

incline  by  the  power  P,  acting  in  the  line  b  P',  parallel  to  the  plane.  Draw 
the  vertical  line  ba  to  represent  the  weight;  also  bb'  perpendicular  to  the 
plane,  and  complete  the  parallelogram  b'  c.  Then  the  vertical  weight  b  a 
is  equivalent  to  b  b',  which  is  the  measure  of  support  given  by  the  plane  to 
the  weight,  and  b  c,  which  is  the  force  of  gravity  tending  to  draw  the  weight 
down  the  plane.  The  power  required  to  maintain  the  weight  in  equilibrium 
is  represented  by  this  force  be.  Thus,  the  power  and  the  weight  are  in 
the  ratio  of  b  c  to  b  a. 

Since  the  triangle  of  forces  a  b  c  is  similar  to  the  triangle  of  the  incline 
ABC,  the  latter  may  be  substituted  for  the  former  in  determining  the 
relative  magnitude  of  the  forces,  and 

P  :  W  ::  be  :  ab  :  :  BC  :  A  B, 


THE   INCLINED   PLANE.  307 

that  is,  the  power,  acting  parallel  to  the  inclined  plane,  is  to  the  weight,  as 
the  height  of  the  plane  to  its  length.  Then,  by  equality  of  moments, 

PxAB  =  WxBC, 
or  P  x  length  of  inclined  plane  =  W  x  height  of  inclined  plane (  a  ) 

For  example,  take  the  length  of  the  inclined  plane,  24  feet;  the  height, 
2  feet;  and  the  weight  to  be  raised,  360  Ibs.  The  power  required  to 
balance  the  weight  is  equal  to  360  x  2  -=-  24  =  30  Ibs. 

Again,  the  base,  A  C,  of  the  inclined  plane,  represents  the  element  of 
the  pressure  of  the  weight  on  the  inclined  plane. 

It  is  thus  seen  that  the  sides  of  the  triangle  formed  by  an  inclined  plane, 
its  base,  and  its  height,  are  respectively  proportional  as  follows : — 

The  inclined  plane  to  the  weight  at  rest  on  the  plane. 

The  base  to  the  pressure  of  the  weight  on  the  plane. 

The  height  to  the  power  acting  parallel  to  the  plane. 

When  the  power  acts  in  a  direction  parallel  to  the  base,  as  in  Fig.  1 2 1 ,  in 
which  the  power  P  supports  the  weight 
W  in  the  direction  b  P',  parallel  to  the 
base;  draw  the  vertical  ba  to  represent 
the  weight,  and  the  line  bb'  perpen- 
dicular to  the  incline,  and  complete  the 
parallelogram  b'  c.  The  weight  b  a,  de- 
composed, is  equivalent  to  b  b',  the  per- 
pendicular to  the  incline,  representing 
the  pressure  of  the  weight  upon  the 
plane,  and  be,  the  force  of  traction,  or 
the  power  P.  Here  the  pressure  b  b'  on  Fis-  '"--inclined  Plane, 

the  plane  is  greater  than  the  weight,  and 

the  power  b  c  is  greater  than  when  the  line  of  traction  is  parallel  to  the 
incline. 

The  triangles  a  b  c,  A  B  C,  being  similar,  the  ratios  of  the  power  and  the 
weight  are  as  follows : — 

P  :  W  ::  be  :  ab  ::  BC   :  AC; (b) 

that  is,  they  are  to  each  other  as  the  height  of  the  plane  to  its  base;  and 
the  inclined  plane,  the  base,  and  the  height,  are  respectively  proportional 
as  follows : — 

The  inclined  plane  to  the  pressure  of  the  weight  on  the  plane. 
The  base  to  the  weight  at  rest  on  the  plane. 

The  height  to  the  power  acting  parallel  to  the  base. 

If  the  power  be  applied  in  any  direction  above  that  which  is  parallel  to 
the  incline,  though  the  pressure  of  the  weight  on  the  plane  will  be  less  than 
the  weight  itself,  yet,  as  in  the  previous  case,  the  power  is  greater  than  is 
necessary  when  it  acts  in  a  direction  parallel  to  the  plane.  Thus,  in 
Fig.  122,  in  which  the  power  P  acts  at  a  divergent  angle  in  the  direction 
b  P7,  draw  the  vertical  b  a,  the  perpendicular  b  b',  to  the  plane,  and  the 
extension  of  the  power  line  to  c,  and  complete  the  parallelogram.  Then, 
the  weight  is  represented  by  b  a,  the  pressure  on  the  incline  by  b  b',  and  the 
power  by  a  I/  or  b  c. 


308 


FUNDAMENTAL   MECHANICAL   PRINCIPLES. 


Fig.  122. — Inclined  Plane. 


For  comparison,  the  parallelogram  that  would  represent  the  relative 
forces  arising  from  a  power  acting  parallel  to  the  plane,  is  added  on  the 
figure  in  dotted  lines  extending  to  the  angles  b"  and  /.  It  shows  that  the 

pressure  on  the  plane  is  greater 
than  when  the  power  is  di- 
vergent, but  that  the  power 
is  less. 

It  follows  that  the  longer 
the  inclined  plane,  when  the 
height  is  the  same,  the  less  is 
the  power  required  to  balance 
the  weight ;  in  fact,  the  power 
simply  varies  in  the  inverse 
ratio  of  the  length  of  the 
plane. 

If  two  inclines,  A  B  and 
B  D,  of  unequal  lengths  and 
the  same  height,  be  united 
back  to  back  on  the  line  BC, 
then  two  weights,  W  and  Wx, 

on  the  respective  planes,  connected  by  a  cord  over  a  pulley  at  the  summit 
B,  will  balance  each  other,  when  they  are  in  the  ratio  of  the  lengths  of  the 
planes  on  which  they  rest.  That  is, 

W  :  W7  :  :  A  B  :  B  D. 

From  the  formula  ( a  ),  rules  may  be  formed  for  finding  one  of  the  following 
four  elements  when  the  other  three  are  given,  namely,  the  length  of  the 

inclined  plane,  the  height  of 
it,  the  weight,  and  the  power 
to  balance  the  weight  when 
acting  in  a  direction  parallel 
to  the  incline. 

RULE  i.  To  find  the  power. 
Multiply  the  weight   by   the 
height  of  the  plane,  and  divide 
Fig.  123.— Double  inclined  Plane.  by  the  length.     The  quotient 

is  the  power. 

RULE  2.  To  find  the  weight.  Multiply  the  power  by  the  length  of  the 
plane,  and  divide  by  the  height.  The  quotient  is  the  weight. 

RULE  3.  To  find  the  height  of  the  inclined  plane.  Multiply  the  power 
by  the  length,  and  divide  by  the  weight.  The  quotient  is  the  height. 

RULE  4.  To  find  the  length  of  the  inclined  plane.  Multiply  the  weight 
by  the  height  of  the  plane,  and  divide  by  the  power.  The  quotient  is  the 
length. 

Identity  of  the  Inclined  Plane  and  the  Lever. 

Though  the  inclined  plane  is  distinguished  from  the  lever  in  the  mode  of 
operation,  inasmuch  as  there  is  no  motion  about  a  mechanical  centre,  as  in 
the  lever,  yet  the  conditions  of  equilibrium  on  the  inclined  plane  may  be 
established  on  the  principle  of  the  lever.  Suppose  a  round  weight  W  kept 
at  rest  on  the  incline  A  B  by  a  power  P  parallel  to  the  incline,  passing 


LEVERAGE   ON    THE   INCLINED   PLANE. 


3C9 


through  the  centre  a.     Draw  a  b  perpendicular  to  the  incline;  the  point  b 

is  the  point  of  contact  of  the  weight  with  the  incline.     Draw  the  vertical 

line  a  d,  and  the  perpendicular  b  c  to  it.     Then  the  lines  a  b,  b  c  form  a  bent 

lever  a  be,  of  which  b  is  the  fulcrum,  and 

a  b,  be  the  arms.     The  weight  acts  at  the 

extremity  c  of  the  short  arm,  and  the  power 

at  the  extremity  a  of  the  long  arm ;  and  the 

power  and   the  weight   are   to   each   other 

inversely  as  the  relative  arms  of  the  lever, 

ab,  be.    Now,  as  a  b  c  and  A  B  C  are  similar 

triangles,  the  arms  ab,bc  are  to  each  other 

as  the  length  and  the  height  A  B,  B  C,  of 

the  incline,  and 


P   :  W  ::  be  :  ab  ::  BC   :  A  B; 


Fig.  124. — Leverage  on  an  Inclined  Plane. 


that  is,  the  power  is  to  the  weight  as  the  height  of 
length,  which  is  the  proportion  already  established 

The  ratio  of  the  length  of  an  inclined  plane  to 
the  leverage  of  the  plane,  and  the  products  of  the 
the  plane,  and  of  the  weight  into  the  height  of  the 
moments  of  the  power  and  the  weight. 

Suppose,  again,  that  the  power  is  applied  at  P, 
(7  P,  passed  round  and  over  the  weight  parallel 


the  inclined  plane  to  its 
( a  )  page  307). 
its  height  may  be  called 
power  into  the  length  of 
plane,  may  represent  the 

Fig.  125,  through  a  cord 
to  the  incline;  then  the 


C~  A 

F'g.  125. — Leverage  on  .in  Inclined  Plane. 


Fig.  126.— Wedge. 


diameter  of  the  weight  a  b  becomes  the  long  arm  of  the  lever  a  be,  through 
which  the  power  acts,  being  double  the  length  of  the  arm  ab.  Fig.  124, 
where  the  power  is  applied  at  the  centre  of  the  weight.  By  thus  doubling 
the  leverage,  the  power  is  halved,  and  the  ratio  of  the  power  to  the  weight 
is  as  half  the  height  of  the  plane  to  its  length. 

In  this  case  there  is  the  action  of  a  movable  pulley  combined  with  an 
inclined  plane;  the  rolling  weight  moved  by  a  cord  lapped  round  it,  repre- 
senting a  movable  pulley  with  the  weight  attached  to  the  axle.  Thus  the 
leverage  of  the  power  on  the  inclined  plane  can  be  doubled. 

THE  WEDGE. 

The  wedge  is  a  pair  of  inclined  planes  united  by  their  bases,  or  "back  to 
back,"  as  A  B  C  B',  Fig.  126.  Whereas  inclined  planes  are  fixed,  wedges  are 
moved,  and  in  the  direction  of  the  centre  line  C  A,  against  a  resistance 
equally  acted  on  by  both  planes  of  the  wedge.  The  function  of  the  wedge 


310  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

is  to  separate  two  bodies  by  force,  or  divide  into  two  a  single  body.  In 
some  cases  the  wedge  is  moved  by  blows,  as  in  splitting  timber;  in  others 
it  is  moved  by  pressure.  The  action  by  simple  pressure  is  now  to  be  con- 
sidered. 

The  pressure  P  is  applied  to  a  wedge  at  the  head  B  B'  at  right  angles  to 
the  surface,  and  the  resistance  or  "  weight "  to  be  overcome  is  opposed  to 
the  wedge  and  acts  at  right  angles  to  the  faces  A  B,  A  B',  at  the  middle 
points  of  which,  a,  a,  it  is  supposed  to  be  concentrated.  Whilst  the  wedge 
and  the  power  move  through  a  space  equal  to  the  length  of  the  wedge  A  C, 
the  weight  is  moved  or  overcome  through  a  space  equal  to  the  breadth  of 
the  wedge  B  B';  and,  as  the  power  is  to  the  weight  inversely  as  the  spaces 
described,  they  are  to  each  other  directly  as  the  breadth  to  the  length  of 
the  wedge.  That  is, 

P   :  W  :  :  B  B'  :  A  C, 

and  the  product  of  the  power  by  the  length  of  the  wedge  is  equal  to  the 
product  of  the  weight  by  the  breadth  of  the  wedge ;  or 

PxAC-WxBB', 
or  P  x  length  =  W  x  breadth  of  wedge (  c  ) 

By  the  aid  of  the  parallelogram  the  same  conclusions  are  arrived  at. 
Thus,  in  Fig.  126,  produce  the  directions  of  the  two  resistances,  W  a,  W  a,  to 
meet  in  the  middle  of  the  wedge  at  b,  complete  the  parallelogram,  and  draw 
the  diagonals  a  c  a  and  b  b' .  The  diagonal  b  bf  is  the  resultant  of  the  two 
forces  a  &,  a  b,  and  represents  the  pressure  on  the  head  of  the  wedge.  Again, 
in  the  triangle  a  b  c,  whilst  a  b  represents,  in  magnitude  and  direction,  the 
perpendicular  pressure  of  the  resistance  on  the  wedge,  a  c,  which  is  perpen- 
dicular to  the  centre  line  of  the  wedge,  represents,  in  magnitude  and 
direction,  the  force  applied  in  overcoming  the  resistance.  The  ratio  of  the 
power  to  the  weight  is  therefore  as  b  b'  to  a  c.  And,  as  the  triangle  abb'  is 
similar  to  the  triangle  A  B  B', 

P  :  W  ::  b  b'  :  etc  ::  B  B'  :  AC; 

that  is,  the  power  is  to  the  weight  as  the  breadth  of  the  wedge  to  its  length. 

From  the  formula  (  c  ),  the  following  rules  for  wedges  acting  under  pres- 
sure, as  distinct  from  impact,  are  deduced : — 

RULE  i.  To  find  the  weight  or  transverse  resistance.  Multiply  the 
power  by  the  length  of  the  wedge,  and  divide  by  the  breadth  of  the  head. 
The  quotient  is  the  weight. 

RULE  2.  To  find  the  power.  Multiply  the  weight  or  transverse  resistance 
by  the  breadth  of  the  head,  and  divide  by  the  length  of  the  wedge.  The 
quotient  is  the  power. 

RULE  3.  To  find  the  length  of  the  wedge.  Multiply  the  weight  by  the 
breadth  of  the  wedge,  and  divide  by  the  power.  The  quotient  is  the 
length  of  the  wedge. 

RULE  4.  To  find  the  breadth  of  the  wedge.  Multiply  the  power  by  the 
length  of  the  wedge,  and  divide  by  the  weight.  The  quotient  is  the  breadth 
of  the  wedge. 

Note. — i.  The  length  of  the  wedge  is  taken  as  the  distance  from  the 
head  to  the  point  of  intersection  of  the  sides. 


THE   SCREW.  311 

2.  The  power  may  be  applied  at  the  point  of  the  wedge  by  drawing, 
instead  of  at  the  head  by  pressing. 

3.  The  power  may  be  applied  in  a  direction  parallel  to  one  of  the  sides 
of  the  wedge,  and  the  relation  of  the  power  to  the  weight  may  be  found  by 
construction,  in  the  same  manner  as  for  the  inclined  plane,  when  the  power 
is  applied  in  a  direction  parallel  to  the  base.    See  proportion  ( b ),  page  307, 

THE  SCREW. 

The  screw  is  an  inclined  plane  lapped  round  a  cylinder.  Take,  for 
example,  an  inclined  plane  ABC,  Fig.  127,  and  bend  it  into  a  circular  form, 
resting  on  its  base,  Fig.  128,  so  that  the  ends  meet.  The  incline  may  be 


Fig.  127. 

continued  winding  upwards  round  the  same  axis,  and  thus  winding  or 
helical  inclined  planes  of  any  required  length  and  height  may  be  con- 
structed. The  helix  thus  arrived  at  being  placed  upon  a  solid  cylinder, 
and  the  dead  parts  of  the  helix  removed,  the  product  is  an  ordinary  screw. 
The  inclined  fillet  is  the  "  thread  "  of  the  screw,  and  the  screw  is  called 
"external."  But  the  thread  may  also  be  applied 
within  a  hollow  cylinder,  and  then  it  is  "  internal," 
such  as  an  ordinary  "  nut "  is. 

The  distance  of  two  consecutive  coils  apart, 
measured  from  centre  to  centre,  or  from  upper  side 
to  upper  side, — literally  the  height  of  the  inclined 
plane, — for  one  revolution,  is  the  "pitch"  of  the 
screw. 

The  effect  of  a  screw  is  estimated  in  terms  of  the 
pitch  and  the  radius  of  the  handle  employed  to  turn  either  it  or  the  nut, 
one  on  the  other;  and  the  leverage  of  the  power  is  the  ratio  of  the  circum- 
ference of  the  circle  described  by  the  power  end  of  the  handle  to  the  pitch. 
The  radius  is  to  be  measured  to  the  central  point  where  the  power  is 
applied. 

The  circumference  being  equal  to  the  radius  multiplied  by  twice  3.1416, 
or  6.28, 

P  :  W  ::/  :  rx6.28, 
in  which/  is  the  pitch  and  r  the  radius;  also 

6.28  Pr  =  Wxt-.  .  (<t) 


Fig.  128. 


that  is,  6.28  times  the  product  of  the  power  by  the  radius  of  the  handle  is 
equal  to  the  product  of  the  weight  by  the  pitch.  Whence  the  following 
rules  relative  to  the  power  of  a  screw,  for  finding  any  one  of  those  four 
quantities  when  the  other  three  are  given : — 

RULE  i.  To  find  the  power.     Multiply  the  weight  by  the  pitch,  and 


312  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

divide  by  the  radius  of  the   handle  and  by  6.28.     The  quotient  is  the 
power.     Or 


6.28  r 


RULE  2.  To  find  the  weight.     Multiply  the  power  by  the  radius  and  by 
6.28,  and  divide  by  the  pitch.     The  quotient  is  the  weight.     Or 


RULE  3.  To  find  the  pitch.  Multiply  the  power  by  the  radius  of  the 
handle  and  by  6.28,  and  divide  by  the  weight.  The  quotient  is  the  pitch. 
Oi 

6.28  P  r  .     x 

L-W~  .....................................   (3) 

RULE  4.  To  find  the  radial  length  of  the  handle.  Multiply  the  weight 
by  the  pitch,  and  divide  by  the  power  and  by  6.28.  The  quotient  is  the 
length  of  the  handle.  Or 


Note.  —  When  the  power  is  applied  through  a  wheel  fixed  to  the  screw, 
the  acting  diameter  of  the  wheel  may  be  substituted  for  the  radius  in  the 
above  rules  and  formulas,  and  the  constant  becomes  3.14. 

Similarly,  should  the  power-wheel  be  fixed  to  the  nut  so  as  to  turn 
it  upon  the  screw,  instead  of  the  screw  within  the  nut,  the  same  sub- 
stitutions may  be  made. 

WORK. 

Work  consists  of  the  sustained  exertion  of  pressure  through  space. 

The  English  unit  of  work  is  one  foot-pound;  that  is,  a  pressure  of  one 
pound  exerted  through  a  space  of  one  foot. 

The  French  unit  of  work  is  one  kilogrammetre;  that  is,  a  pressure  of  one 
kilogramme  exerted  through  a  space  of  one  metre. 

One  kilogrammetre  is  equal  to  7.233  foot-pounds. 

In  the  performance  of  work  by  means  of  mechanism,  the  work  done 
upon  the  weight  is  equal  to  the  work  done  by  the  power.  This  prin- 
ciple of  the  equality  of  work  is  deducible  from  the  principle  of  the 
equality  of  moments,  and  is  expressed  generally  by  the  equation 

PxH  =  Wx^,  .......................................   (a) 

in  which  H  is  the  height  or  space  moved  through  by  the  power,  and 
h  the  height  or  space  moved  through  by  the  weight  at  the  same  time. 
It  signifies  that  the  product  of  the  power  by  the  space  through  which  it  has 
acted  is  equal  to  the  product  of  the  weight  by  the  space  through  which 
it  has  acted. 
Again, 

P  :  W  :  :  h  :  H, 

signifying  that  the  power  is  to  the  weight  inversely  as  the  respective  heights 
or  spaces  moved  through  by  them  in  the  same  time. 


WORK. — WORK   WITH   THE   MECHANICAL   ELEMENTS.         313 
WORK    DONE   WITH    THE    LEVER. 

On  the  principle  of  the  equality  of  moments,  the  power  and  the  weight 
in  the  lever,  neglecting  frictional  resistance,  are  to  each  other  inversely  as 
the  lengths  of  the  arms  upon  which  they  act,  that  is,  of  their  radii  of 
motion;  and  inversely  as  the  arcs  or  spaces  passed  through  or  described  by 
the  ends  of  the  arms.  If  the  weighted  lever,  Fig.  99,  page  296,  be  moved 
by  the  power,  the  power  descends  through  an  arc  at  a,  and  the  weight  is 
raised  through  an  arc  at  c.  These  arcs  may  be  taken  as  the  heights  moved 
through,  and  are  proportional  to  the  lengths  of  the  respective  arms,  a  b,  b  c. 
In  this  example,  these  are  as  7  to  i,  and  if  the  power  descend  7  inches  the 
weight  is  raised  only  i  inch;  but  the  weight  raised  is  seven  times  the  power 
applied,  and  "what  is  gained  in  power  is  lost  in  speed,"  or,  more  correctly, 
in  space  moved  through.  The  equality  of  work  thus  developed  from  the 
equality  of  moments  is  thus  expressed 

power  x  arc  a  =  weight  x  arc  c (a) 

To  show  this  arithmetically,  let  the  weight  be  raised  through  i  foot;  then, 
with  a  leverage  of  7  to  i,  the  power  descends  7  feet,  and  taking  it,  as  before, 
at  60  Ibs.,  the  weight  it  raises  will  be  60  Ibs.  x  7  =  420  Ibs.,  and  the  equation 
of  work  is 

60  Ibs.  x  7  feet        =       420  Ibs.  x  i  foot, 
(or  420  foot-pounds)         (or  420  foot-pounds). 

Again, 

power  :  weight  :  :  arc  c  :  arc  a, 

expressing  the  principle  of  virtual  velocities,  the  relative  velocities  being 
indicated  by  the  arcs  a,  c. 

WORK    DONE   WITH    THE    PULLEY. 

In  using  the  single  fixed  pulley,  Fig.  in,  page  302,  the  power  is  equal  to 
the  weight,  and  the  spaces  through  which  they  move  in  the  same  time  are 
equal. 

With  the  movable  pulley,  Fig.  114,  the  weight  is  suspended  at  the  axle, 
and  in  raising  the  weight  i  foot,  the  power  at  the  circumference,  with  a 
leverage  of  2,  passes  through  2  feet  and  is  only  half  the  weight.  If 
P  and  W  be  20  Ibs.  and  40  Ibs.  respectively,  the  equality  of  work  is  thus 
expressed — 

(P)  20  Ibs.  x  2  feet  =  (W)  40  Ibs.  x  i  foot  =  40  foot-pounds; 

and  by  means  of  this  pulley  a  weight  double  the  power  is  raised  half  the 
height  through  which  the  power  is  applied. 

Conversely,  when  the  weight  is  suspended  at  the  circumference  of  the 
movable  pulley,  Fig.  115,  and  the  power  applied  at  the  axle,  the  leverage 
is  % \  the  power  is  therefore  double  the  weight,  and  moves  through  i  foot 
whilst  the  weight  moves  through  2  feet.  Thus 

(P)  40  Ibs.  x  i  foot  =  (W)  20  Ibs.  x  2  feet --40  foot-pounds. 

In  a  system  of  fast  and  loose  pulley  blocks,  Fig.  117,  page  303,  the 
power  being  equal  to  the  weight  divided  by  the  number  of  ropes,  then,  by 


3H  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

equality  of  work,  the  space  through  which  the  power  is  moved  is  equal  to 
the  height  through  which  the  weight  is  raised,  multiplied  by  the  number  of 
ropes.  Suppose  that  there  are  three  movable  pulleys  and  six  ropes ;  if  the 
weight,  120  Ibs.,  be  raised  i  foot,  each  rope  is  shortened  i  foot  and  the 
power  is  moved  6  feet.  And 

(P)  20  Ibs.  x  6  feet  =  (W)  120  Ibs.  x  i  foot=  120  foot-pounds. 

WORK    DONE    WITH    THE   WHEEL   AND    AXLE. 

While  the  wheel,  Fig.  119,  page  305,  makes  one  revolution,  the  axle  also 
makes  one.  The  power  descends  or  traverses  a  space  equal  to  the  cir- 
cumference of  the  wheel  =  2  (a  b)  x  3.1416,  whilst  the  weight  is  raised  through 
a  space  equal  to  the  circumference  of  the  axle  =  2  (b  c)  x  3.1416.  If  the 
radius  of  the  wheel  be  i  foot  6  inches,  and  that  of  the  axle  3  inches,  the 
circumferences  are  9.42  feet  and  1.57  feet,  being  as  6  to  i;  and  the  power 
and  the  weight,  conversely,  are  as  i  to  6.  If  the  power  be  20  Ibs.,  then 

(P)  20  Ibs.  x  9.42  feet  =  (W)  120  Ibs.  x  1.57  feet. 
(188.4  foot-pounds)          (188.4  foot-pounds). 

WORK    DONE    WITH    THE    INCLINED    PLANE. 

The  weight  is  raised  in  opposition  to  gravity,  and  the  work  done  on  it  is 
expressed  by  the  product  of  the  weight  into  the  vertical  height  of  the 
inclined  plane.  The  work  done  by  the  power  is  expressed  by  the  product 
of  the  power  into  the  length  of  the  plane.  These  two  products  express 
equal  quantities  of  work,  and 

Px/=Wx>fc, 

as  before  intimated  at  (a),  page  307,  to  express  equality  of  moments. 

For  example,  the  length  of  the  plane  is  24  feet  and  the  height  2  feet; 
the  weight  is  120  Ibs.,  the  power  10  Ibs.  Then,  the  work  done  in  raising 
the  weight  up  the  whole  of  the  incline  is  240  Ibs.,  thus 

(P)  10  Ibs.  x  24  feet  =  (W)  120  Ibs.  x  2  feet. 
(240  foot-pounds)          (240  foot-pounds). 

The  power  is  here  supposed  to  be  applied  in  a  direction  parallel  to  the  plane. 
If  applied  in  a  direction  at  an  angle  to  the  plane,  as  in  Fig.  122,  page  308, 
it  is  to  be  resolved  into  its  components,  parallel  and  perpendicular  to  the 
plane.  Draw  the  line  b  c  parallel  to  the  incline;  then  the  power  applied, 
b  c,  is  equivalent  to  the  force  actually  expended  b  c,  and  to  the  pressure 
without  motion  c  c',  The  consumption  of  power  is  expressed  by  the  pro- 
duct of  its  parallel  equivalent,  b  c,  into  the  length  of  the  plane.  Taking, 
for  example,  as  above,  the  weight,  120  Ibs.,  and  the  active  power,  10  Ibs., 
represented  by  the  parallel  force  b  c ;  then  the  amount  of  the  horizontal 
force,  or  the  power  applied,  b  c,  is  found  by  proportion,  thus 

A  C   :  A  B  :  :  b  c    :  b  c; 

that  is,  the  parallel  and  horizontal  forces  are  to  each  other  as  the  base  to 
the  length  of  the  incline. 


WORK   IN   MOVING   BODIES.  315 

WORK   DONE   WITH    THE    WEDGE. 

Supposing  the  wedge  driven  by  a  constant  pressure  through  a  distance 
equal  to  its  length,  the  work  done  by  the  power  is  expressed  by  the  power 
into  the  length,  and  the  work  done  on  the  weight  is  expressed  by  the  pro- 
duct of  the  weight  into  the  breadth  of  the  wedge.  By  equality  of  work, 


as  before  stated,  in  expressing  equality  of  moments. 

If  the  wedge  be  driven  for  only  a  part  of  its  length,  the  work  done  by 
the  power  is  in  the  proportion  of  the  part  of  the  length  driven;  and  the 
work  done  on  the  weight  is  similarly  in  the  proportion  of  the  part  of  the 
breadth  by  which  the  resisting  surfaces  are  separated. 

WORK   DONE   WITH    THE   SCREW. 

In  one  revolution  of  the  screw,  the  weight  is  raised  through  a  height 
equal  to  the  pitch  of  the  thread,  whilst  the  power  acts  through  the  circum- 
ference of  the  circle  described  by  the  point  at  which  it  is  applied  to  a  lever. 
The  products  of  the  power  and  the  weight  by  the  spaces  described  by 
them  are  equal,  or 

Px6.28  r  =  Wx/, 
as  before  stated  (page  311)  to  express  equality  of  moments. 

WORK  DONE  BY  GRAVITY. 

The  work  done  by  gravity  on  a  falling  body  is  equal  to  the  weight  of  the 
body  multiplied  by  the  height  through  which  it  falls. 

WORK   ACCUMULATED    IN    MOVING    BODIES. 

The  quantity  of  work  stored  in  a  body  in  motion  is  the  same  as  that 
which  would  be  accumulated  in  it  by  gravity  if  it  fell  from  such  a  height  as 
would  be  sufficient  to  give  it  the  same  velocity;  in  short,  from  the  height 
clue  to  the  velocity.  (See  GRAVITY,  page  277).  The  accumulated  work 
expressed  in  foot-pounds,  is  equal  to  the  height  so  found  in  feet,  multiplied 
by  the  weight  of  the  body  in  pounds.  The  height  due  to  the  velocity  is 
equal  to  the  square  of  the  velocity  divided  by  64.4,  and  the  work  and  the 
velocity  may  be  found  directly  from  each  other,  according  to  the  following 
rules  :  — 

RULE  i.  Given  the  weight  and  velocity  of  a  moving  body,  to  find  the 
work  accumulated  in  it.  Multiply  the  weight  in  pounds  by  the  square  of 
the  velocity  in  feet  per  second,  and  divide  by  64.4.  The  quotient  is  the 
accumulated  work  in  foot-pounds. 

Or,  putting  U  for  the  work,  v  for  the  velocity,  and  w  for  the  weight, 

U  =  !^...  ..  (i) 

64.4 

Or,  secondly  :  —  Multiply  the  weight  in  pounds  by  the  height  in  feet  due 
to  the  velocity.  The  product  is  the  accumulated  work  in  foot-pounds.  Or, 
putting  h  for  the  height, 

U  =  wxh  .......................................  (  i  a) 


316  FUNDAMENTAL   MECHANICAL   PRINCIPLES. 

WORK  DONE  BY  PERCUSSIVE  FORCE. 

If  a  wedge  be  driven  by  blows  or  strokes  of  a  hammer  or  other  heavy 
mass,  the  effect  of  the  percussive  force  is  measured  by  the  quantity  of  work 
accumulated  in  the  striking  body.  This  work  is  calculated  by  the  preceding 
rules,  from  the  weight  of  the  body  and  the  velocity  with  which  the  blow  is 
delivered,  or  directly  from  the  height  of  the  fall,  if  gravity  be  the  motive 
power. 

The  useful  work  done  through  the  wedge  is  equal  to  the  work  delivered 
upon  the  wedge,  supposing  that  there  is  no  elastic  or  vibrating  reaction 
from  the  blow,  just  as  if  the  work  had  been  delivered  by  a  constant  pres- 
sure equal  to  the  weight  of  the  striking  body,  exerted  through  a  space  equal 
to  the  height  of  the  fall,  or  the  height  due  to  its  final  velocity. 

Of  course,  in  order  to  give  effect  to  the  constant  pressure  on  the  wedge, 
now  imagined  to  be  brought  into  action,  the  pressure  would  require  to  be 
applied  to  the  resisting  medium  through  some  combination  of  the  mechanical 
elements. 

But  where  elastic  action  intervenes,  a  portion  of  the  work  delivered  is 
uselessly  absorbed  in  elastically  straining  the  resisting  body;  and  the  elastic 
action  may  be,  in  some  situations,  so  excessive  as  to  absorb  the  whole  of 
the  work  delivered.  In  this  case,  there  would  not  be  any  useful  work  done. 

These  remarks,  applied  to  the  action  of  a  blow  on  a  wedge,  are  applicable 
equally  to  the  action  of  a  blow  of  the  monkey  of  a  pile-driver  upon  a  pile. 
If  there  be  no  elastic  action,  the  work  delivered  being  the  product  of  the 
weight  of  the  monkey  by  the  height  of  its  fall,  is  equal  to  the  work  done  in 
sinking  the  pile:  that  is,  to  the  product  of  the  frictional  and  other  resistance 
to  its  descent  by  the  depth  through  which  it  descends  for  one  blow  of  the 
monkey. 

Supposing  that  the  pile  rests  upon  and  is  absolutely  resisted  by  a  hard 
unyielding  obstacle,  the  work  done  becomes  wholly  useless,  and  consists  of 
elastic  or  vibrating  action  j  or  it  may  be  that  the  head  of  the  pile  is  split 
open. 


HEAT. 


THERMOMETERS. 

The  action  of  Thermometers  is  based  on  the  change  of  volume  to  which 
bodies  are  subject  with  a  change  of  temperature,  and  they  serve,  as  their 
name  implies,  to  measure  temperature.  Thermometers  are  filled  with  air, 
water,  or  mercury.  Mercurial  thermometers  are  the  most  convenient,  because 
the  most  compact.  They  consist  of  a  stem  or  tube  of  glass,  formed  with  a 
bulbous  expansion  at  the  foot  to  contain  the  mercury,  which  expands  into 
the  tube.  The  stem  being  uniform  in  bore,  and  the  apparent  expansion  of 
mercury  in  the  tube  being  equal  for  equal  increments  of  temperature,  it 
follows  that  if  the  scale  be  graduated  with  equal  intervals,  these  will  indi- 
cate equal  increments  of  temperature.  A  sufficient  quantity  of  mercury 
having  been  introduced,  it  is  boiled  to  expel  air  and  moisture,  and  the  tube 
is  hermetically  sealed.  The  freezing  and  the  boiling  points  on  the  scale 
are  then  determined  respectively  by  immersing  the  thermometer  in  melting 
ice  and  afterwards  in  the  steam  of  water  boiling  under  the  mean  atmospheric 
pressure,  14.7  Ibs.  per  square  inch,  and  marking  the  two  heights  of  the 
column  of  mercury  in  the  tube.  The  interval  between  these  two  points  is 
divided  into  180  degrees  for  Fahrenheit's  scale,  or  100  degrees  for  the 
Centigrade  scale,  and  degrees  of  the  same  interval  are  continued  above  and 
below  the  standard  points  as  far  as  may  be  necessary.  It  is  to  be  noted 
that  any  inequalities  in  the  bore  of  the  glass  must  be  allowed  for  by  an 
adaptation  of  the  lengths  of  the  graduations.  The  rate  of  expansion  of 
mercury  is  not  strictly  constant,  but  increases  with  the  temperature,  though, 
as  already  referred  to,  this  irregularity  is  more  or  less  nearly  compensated 
by  the  varying  rates  of  expansion  of  glass. 

In  the  Fahrenheit  Thermometer,  used  in  Britain  and  America,  the  number 
o°  on  the  scale  corresponds  to  the  greatest  degree  of  cold  that  could  be 
artificially  produced  when  the  thermometer  was  originally  introduced.  32° 
("the  freezing-point")  corresponds  to  the  temperature  of  melting  ice,  and 
212°  to  the  temperature  of  pure  boiling  water — in  both  cases  under  the 
ordinary  atmospheric  pressure  of  14.7  Ibs.  per  square  inch.  Each  division 
of  the  thermometer  represents  i°  Fahrenheit,  and  between  32°  and  212° 
there  are  180°. 

In  the  Centigrade  Thermometer,  used  in  France  and  in  most  other 
countries  in  Europe,  o°  corresponds  to  melting  ice,  and  100°  to  boiling 
water.  From  the  freezing  to  the  boiling  point  there  are  100°. 

In  the  Reaumur  Thermometer,  used  in  Russia,  Sweden,  Turkey,  and 
Egypt,  o°  corresponds  to  melting  ice,  and  80°  to  boiling  water.  From  the 
freezing  to  the  boiling  point  there  are  80°. 


3l8  HEAT. 

Each  degree  Fahrenheit  is  f  of  a  degree  Centigrade,  and  y  of  a  degree 
Reaumur,  and  the  relations  between  the  temperatures  indicated  by  the 
different  thermometers  are  as  follows  :  — 


C.  being  the  temperature  in  degrees  Centigrade. 
R.  do.  do.  Reaumur. 

F.  do.  do.  Fahrenheit. 

That  is  to  say,  that  Centigrade  temperatures  are  converted  into  Fahrenheit 
temperatures  by  multiplying  the  former  by  9  and  dividing  by  5,  and  adding 
32°  to  the  quotient;  and  conversely,  Fahrenheit  temperatures  are  converted 
into  Centigrade  by  deducting  32°,  and  taking  -fths  of  the  remainder. 

Reaumur  degrees  are  multiplied  by  |-  to  convert  them  into  the  equivalent 
Centigrade  degrees;  conversely,  -fths  of  the  number  of  Centigrade  degrees 
give  their  equivalent  in  Re'aumur  degrees. 

Fahrenheit  is  converted  into  Reaumur  by  deducting  32°  and  taking  -|ths 
of  the  remainder,  and  Reaumur  into  Fahrenheit  by  multiplying  by  f  ,  and 
adding  32°  to  the  product. 

Tables  No.  104,  105  contain  equivalent  temperatures  in  degrees  Centigrade 
for  given  degrees  Fahrenheit,  from  o°  F.,  or  zero  on  the  Fahrenheit  scale,  to 
608°  F.  ;  and  conversely,  the  temperature  in  degrees  Fahrenheit  correspond- 
ing to  degrees  Centigrade,  from  o°  C.,  or  zero  on  the  Centigrade  scale,  to 
320°  C. 


EQUIVALENT   TEMPERATURES. 


319 


Table  No.   104. — EQUIVALENT  TEMPERATURES  BY  THE  FAHRENHEIT 
AND  CENTIGRADE  THERMOMETERS. 


Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

0 

-17.78 

+  38 

+  3-34 

+  76 

+  24.45 

+  114 

+  45.56 

+  I 

17.23 

39 

3-90 

77 

25.00 

i*5 

46.11 

2 

16.67 

40 

4-45 

78 

25.56 

116 

46.67 

3 

i6.n 

4i 

5.00 

79 

26.12 

117 

47-23 

4 

I5-56 

42 

5-56 

80 

26.67 

118 

47.78 

5 

15.00 

43 

6.ii 

81 

27.23 

119 

48.34 

6 

14-45 

44 

6.67 

82 

27.78 

I2O 

48.90 

7 

13.90 

45 

7-23 

83 

28.34 

121 

49-45 

8 

13-34 

46 

7-78 

84 

28.89 

122 

5O.OO 

9 

12.78 

47 

8-34 

85 

29-45 

I23 

50.56 

10 

12.23 

48 

8.89 

86 

30.00 

I24 

5I.II 

ii 

11.67 

49 

9-45 

87 

30-55 

I25 

5I-67 

12 

ii.  ii 

5° 

10.00 

88 

31.11 

126 

52.23 

J3 

10.56 

51 

10.56 

89 

31.67 

127 

52.78 

14 

10.00 

52 

II.  II 

90 

32.22 

128 

53-34 

15 

9-45 

53 

11.67 

9i 

32.78 

129 

53-90 

16 

8.89 

54 

12.23 

92 

33-33 

130 

54.45 

17 

8-34 

55 

12.78 

93 

33.80 

O  O        s 

131 

55-00 

18 

7.78 

56 

13-34 

94 

34-45 

132 

55.56 

J9 

7-23 

57 

13.90 

95 

35-°° 

i33 

56.11 

20 

6.67 

58 

14.45 

96 

35.56 

i34 

56.67 

21 

6.ii 

59 

15.00 

97 

36.11 

r35 

57-23 

22 

5-56 

60 

15.56 

98 

36.67 

136 

57.78 

23 

5.00 

61 

1  6.  1  1 

99 

37.23 

137 

58.34 

24 

4-45 

62 

16.67 

IOO 

37-78 

138 

58.90 

25 

3-90 

63 

17.23 

IOI 

38.34 

139 

59-45 

26 

3-34 

64 

17.78 

102 

38-9° 

140 

60.00 

27 

2.78 

65 

18.34 

103 

39-45 

141 

60.56 

28 

2.23 

66 

18.89 

104 

40.00 

142 

6i.n 

29 

1.67 

67 

*9-45 

105 

40.56 

i43 

61.67 

30 

i.  ii 

68 

20.00 

106 

41.11 

144 

62.23 

31 

0.56 

69 

20.56 

107 

41.67 

i45 

62.78 

32 

o.oo 

70 

21.  II 

108 

42.23 

146 

63-34 

33 

+  0.56 

7i 

21.67 

109 

42-78 

i47 

63.90 

34 

I.  II 

72 

22.23 

no 

43-34 

148 

64.45 

35 

1.67 

73 

22.78 

in 

43-9° 

149 

65.00 

36 

2.23 

74 

23.34 

112 

44-45 

J50 

65-56 

37 

2.78 

75 

23.90 

H3 

45.00 

*f* 

66.ii 

320 


HEAT. 


Table  No.   104  (continued). 
FAHRENHEIT  AND  CENTIGRADE. 


Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

+  I52 

+  66.67 

+  193 

+  89.45 

+  234 

+  112.23 

+  275 

+  135.00 

J53 

67.23 

I94 

9O.OO 

235 

112.78 

276     135.56 

i54 

67-78 

195 

90.56 

236 

113-34 

277 

136.11 

i55 

68.34 

196 

91.11 

237 

113.90 

278 

136.67 

156 

68.90 

I97 

91.67 

238 

114-45 

279     137.23 

i57 

69-45 

198 

92.23 

239 

115.00 

280 

137.78 

158 

70.00 

199 

92.78 

240 

II5-56 

28l 

138.34 

i59 

70.56 

200 

93-34 

241 

n6.ii 

282 

138.90 

160 

71.11 

201 

93-90 

242 

116.67 

283 

139-45 

161 

71.67 

2O2 

94-45 

243 

117.23 

284 

140.00 

162 

72.23 

203 

95.00 

244 

117.78 

285 

140.56 

163 

72.78 

204 

95.56 

245 

118.34 

286 

I4I.II 

164 

73-34 

205 

96.11 

246 

118.90 

287 

141.67 

165 

73-9° 

2O6 

96.27 

247 

119-45 

288 

142.23 

166 

74-45 

207 

97-23 

248 

120.00 

289 

142.78 

167 

75.00 

208 

97-78 

249 

120.56 

290 

143.34 

168 

75-56 

2O9 

98.34 

250 

121.  II 

29I 

I33-90 

169 

76.11 

210 

98.90 

25T 

121.67 

292 

144.45 

170 

76.67 

211 

99-45 

252 

122.23 

293 

145.00 

171 

77-23 

212 

100.00 

253 

122.78 

294 

145.56 

172 

77.78 

2I3 

100.56 

254 

123.34 

295 

146.11 

i73 

78.34 

214 

101.11 

255 

123.90 

296 

146.67 

i74 

78.90 

215 

101.67 

256 

124-45 

297 

I47.23 

*75 

79-45 

216 

102.23 

257 

I25.OO 

298 

147.78 

176 

80.00 

217 

102.78 

258 

125.56 

299 

148.34 

177 

80.56 

218 

103.34 

259 

126.11 

300 

148.90 

178 

8i.ii 

219 

103.90 

260 

126.67 

3OI 

149-45 

179 

81.67 

220 

104.45 

26l 

127.23 

302 

150.00 

180 

82.23 

221 

105.00 

262 

127.78 

303 

ISO'S6 

181 

82.78 

222 

105.56 

263 

128.34 

3°4 

I5I.II 

182 

83-34 

223 

106.11 

264 

128.90 

305 

151.67 

183 

83.90 

224 

106.67 

265 

129.45 

3°6 

152.23 

184 

84-45 

225 

107.23 

266 

130.00 

3°7 

152.78 

185 

85.00 

226 

107.78 

267 

130.56 

308 

153-34 

186 

85-56 

227 

108.83 

268 

I3I.II 

309 

I53.90 

187 

86.11 

228 

108.90 

269 

131.67 

3IO 

154.45 

188 

86.67 

229 

109.45 

270 

132.23 

311 

155-00 

189 

87.23 

230 

I  IO.OO 

271 

132.78 

312 

155.56 

190 

87.78 

23I 

110.56 

272 

133.34 

3i3 

156.11 

191 

88.34 

232 

III.  II 

273 

133.90 

$*4 

156.67 

192 

88.90 

233 

111.67 

274     134.45 

3J5 

I57-23 

EQUIVALENT   TEMPERATURES. 


321 


Table  No.  104  (continued}. 
FAHRENHEIT  AND  CENTIGRADE. 


Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

+  3l6 

+  I57-78 

+  357 

+  180.56 

+  398 

+  203.34 

+  439 

+  226.11 

3*7 

158.34 

358 

181.11 

399 

203.90 

440 

226.67 

3i8 

158.90 

359 

181.67 

400 

204.45 

441 

227.23 

3i9 

!59-45 

360 

182.23 

401 

205.OO 

442 

227.78 

320 

160.00 

361 

182.78 

402 

205.56 

443 

228.34 

321 

160.56 

362 

183.34 

403 

206.  II 

444 

228.90 

322 

i6i.n 

363 

183.90 

404 

206.67 

445 

229.45 

323 

161.67 

364 

184.45 

405 

207.23 

446 

23O.OO 

324 

162.23 

365 

185.00 

406 

207.78 

447 

230.56 

325 

162.78 

366 

185-56 

407 

208.34 

448 

231.11 

326 

163-34 

367 

i86.n 

408 

208.90 

449 

231.67 

327 

163.90 

368 

186.67 

409 

209.45 

45° 

232.23 

328 

164.45 

369 

187.23 

410 

210.00 

45  1 

232.78 

329 

165.00 

37o 

187.78 

411 

210.56 

452 

233-34 

33° 

165-56 

37i 

188.34 

412 

211.  II 

453 

233.90 

33i 

i66.n 

372 

188.90 

4i3 

211.67 

454 

234-45 

332 

166.67 

373 

189.45 

414 

212.23 

455 

235.00 

333 

167.23 

374 

190.00 

4i5 

212.78 

456 

235-56 

334 

167.78 

375 

190.56 

416 

213-34 

457 

236.11 

335 

168.34 

376 

191.11 

4i7 

213.90 

458 

236.67 

336 

168.90 

377 

191.67 

418 

214-45 

459 

237.23 

337 

169.45 

378 

192.23 

419 

2I5.OO 

460 

237.78 

338 

170.00 

379 

192.78 

420 

215.56 

461 

238.34 

339 

170.56 

380 

193-34 

421 

2l6.II 

462 

238.90 

340 

171.11 

38i 

193.90 

422 

216.67 

463 

239-45 

34i 

171.67 

382 

194-45 

423 

217.23 

464 

240.00 

342 

172.23 

383 

195.00 

424 

217.78 

465 

240.56 

343 

172.78 

384 

I95-56 

425 

218.34 

466 

241.11 

344 

173-34 

385 

196.1  1 

426 

218.90 

467 

241.67 

345 

173.90 

386 

196.67 

427 

219-45 

468 

242.23 

346 

174-45 

387 

197.23 

428 

220.OO 

469 

242.78 

347 

175.00 

388 

197.78 

429 

220.56 

470 

243-34 

348 

I75-56 

389 

198.34 

43° 

221.  II 

47i 

243.90 

349 

176.11 

39° 

198.90 

43i 

221.67 

472 

244.45 

350 

176.67 

39i 

I99.45 

432 

222.23 

473 

245.00 

35i 

177.23 

392 

200.00 

433 

222.78 

474 

245-56 

352 

177-78 

393 

200.56 

434 

223.34 

475 

246.11 

353 

178.34 

394 

201.  II 

435 

223.90 

476 

246.67 

354 

178.90 

395 

201.67 

436 

224.45 

477 

247.23 

355 

179-45 

396 

202.23 

437 

225.OO 

478 

247.78 

356 

180.00 

397 

202.78 

438 

225.56 

479 

248.34 

21 


322 


HEAT. 


Table  No.   104  (continued}. 
FAHRENHEIT  AND  CENTIGRADE. 


Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

Degrees 
Fahr. 

Degrees 
Centigrade. 

+  480 

+  248.90 

+  513 

+  267.23 

+  546 

+  285.56 

+  579 

+  303.90 

481 

249.45 

5.«4 

267.78  1 

547 

286.11 

580 

304.45 

482 

250.00 

515 

268.34 

548 

286.67 

581 

305.00 

483 

250.56 

516 

268.90 

549 

287.23 

582 

305.56 

484 

251.11 

517 

269.45 

550 

287.78 

583 

306.II 

485 

251.67 

518 

270.00 

551 

288.34 

584 

306.67 

486 

252.23 

5J9 

270.56 

SS2 

288.90 

585 

307.23 

487 

252.78 

520 

271.11 

553 

289.45 

586 

307.78 

488 

253-34 

521 

271.67 

554 

290.00 

587 

308.34 

489 

253-90 

522 

272.23 

555 

290.56 

588 

308.90 

49° 

254-45 

523 

272.78 

556 

291.11 

589 

309.45 

491 

255.00 

524 

273.34 

557 

291.67 

59° 

310.00 

492 

255-56 

525 

273.90 

558 

292.23 

59i 

310.56 

493 

256.11  ; 

526 

274.45 

559 

292.78 

592 

3II.II 

494 

256.67 

527 

275.00 

560 

293-34 

593 

311.67 

495 

257-23 

528 

275.56 

561 

293.90 

594 

312.23 

496 

257.78 

529 

276.11 

562 

294.45 

595 

312.78 

497 

258.34 

530 

276.67 

563 

295.00 

596 

3J3-34 

498 

258.90 

53i 

277.23 

564 

295.56 

597 

313-90 

499 

259-45 

S32 

277.78 

565 

296.11 

598 

314.45 

500 

260.00 

533 

278.34 

566 

296.67 

599 

3!5.oo 

So1 

260.56 

534 

278.90 

567 

297.23 

600 

3I5.56 

502 

261.11 

535 

279-45 

568 

297.78 

601 

316.11 

503 

261.67 

536 

28O.OO 

569 

298.34 

602 

316.67 

5°4 

262.23 

537 

280.56 

57o 

298.90 

603 

317-23 

505 

262.78 

538 

28l.II 

571 

299.45 

604 

317.78 

506 

263.34 

539 

281.67 

572 

300.OO 

605 

318.34 

507 

263.90 

540 

282.23 

573 

300.56 

606 

318.90 

508 

264.45 

54i 

282.78 

574 

301.11 

607 

319.45 

509 

265.00 

542 

283.34 

575 

301.67 

608 

320.00 

510 

265.56 

543 

283.90 

576 

302.23 

5" 

266.11 

544 

284.45 

577 

302.78 

512 

266.67 

545 

285.00  i 

578 

303.34 

EQUIVALENT  TEMPERATURES. 


323 


Table  No.  105. — EQUIVALENT  TEMPERATURES  BY  THE  CENTIGRADE  AND 
FAHRENHEIT  THERMOMETERS. 


Degrees 
Cent. 

Degrees 
Fahr. 

Degrees 
Cent. 

Degrees 
Fahr. 

Degrees 
Cent. 

Degrees 
Fahr. 

Degrees 
Cent. 

Degrees 
Fahr. 

-20 

-    4.0 

4-21 

+  69.8 

+  62 

+  143.6 

+  103 

+  217.4 

19 

2.2 

22 

71.6 

63 

1454 

104 

219.2 

18 

0.4 

23 

73-4 

64 

147.2 

105 

221.0 

I? 

+     1.4 

24 

75-2 

65 

149.0 

106 

222.8 

16 

3-2 

25 

77.0 

66 

150.8 

107 

224.6 

15 

5.0 

26 

78.8 

67 

152.6 

108 

226.4 

14 

6.8 

27 

80.6 

68 

154.4 

109 

228.2 

13 

8.6 

28 

82.4 

69 

156.2 

no 

230.0 

12 

10.4 

29 

84.2 

70 

158.0 

III 

231.8 

II 

12.2 

30 

86.0 

7i 

159-8 

112 

233-6 

10 

I4.O 

31 

87.8 

72 

161.6 

H3 

2354 

9 

I5.8 

32 

89.6 

73 

163.4 

H4 

237.2 

8 

I7.6 

33 

91.4 

74 

165.2 

H5 

239.0 

7 

194 

34 

93-2 

75 

167.0 

116 

240.8 

6 

21.2 

35 

95.0 

76 

168.8 

117 

242.6 

5 

23.0 

36 

96.8 

77 

170.6 

118 

244.4 

4 

24-8 

37 

98.6 

78 

172.4 

119 

246.2 

3 

26.6 

38 

100.4 

79 

174.2 

120 

248.0 

2 

28.4 

39 

102.2 

80 

176.0 

121 

249.8 

I 

30.2 

40 

104.0 

81 

177-8 

122 

251.6 

0 

32.0 

4i 

105.8 

82 

179.6 

I23 

2534 

+     I 

33-8 

42 

107.6 

83 

181.4 

124 

255.2 

2 

35-6 

43 

109.4 

84 

183-2 

125 

257.0 

3 

37-4 

44 

III.  2 

85 

185.0 

126 

258.8 

4 

39-2 

45 

II3.0 

86 

1  86.8 

127 

260.6 

5 

41.0 

46 

II4.8 

87 

1  88.6 

128 

262.4 

6 

42.8 

47 

II6.6 

88 

190.4 

129 

264.2 

7 

44.6 

48 

118.4 

89 

192.2 

130 

266.0 

8 

46.4 

49 

120.2 

90 

194.0 

I3l 

267.8 

9 

48.2 

50 

122.0 

9i 

195.8 

132 

269.6 

10 

50.0 

51  ; 

123.8 

92 

197.6 

133 

271.4 

ii 

51.8 

52 

125.6 

93 

199.4 

134 

273.2 

12 

53.6 

53 

1274 

94 

201.2 

135 

275.0 

13     1      55-4 

54 

129.2 

95 

203.0 

•    136 

276.8 

H 

57-2 

55 

I3I.O 

96 

204.8 

137 

278.6 

15           59.0 

56 

132.8 

97 

206.6 

138 

280.4 

16           60.8 

57 

134.6 

98 

208.4 

139 

282.2 

17 

62.6 

58 

136.4 

99 

210.2 

140 

284.0 

18 
19 

64.4 
66.2 

£ 

138.2 
140.0 

100 
IOI 

2I2.O 
213.8 

141 
142 

285.8 
287.6 

20 

68.0 

61 

I4I.8 

102 

215.6 

H3 

289.4 

324 


HEAT. 


Table  No.  105  (continued}. 
CENTIGRADE  AND  FAHRENHEIT. 


1  K-^rees 
Kal.r. 

IVgivcs 

Cent 

Degrees 

Degrees 
Cent. 

Degrees. 
Fahr. 

Cent 

1  '•  _;rees 
Fahr. 

+  144 

+  291.2 

+  I89 

+  372.2 

+  234 

+  453-2 

+  279 

+  534-2 

145 

293.0 

I90 

374-0 

235 

455-o 

280 

536.0 

146 

294.8 

191 

375-8 

236 

456.8 

28l 

537-8 

U7 

296.6 

192 

377-6 

237 

458.6 

282 

539-6 

148 

298.4 

'93 

379-4 

460.4 

283 

541.4 

149 

300.2 

194 

381.2 

239 

462.2 

284 

543-2 

150 

302.0 

195 

383.0 

240 

464.0 

285 

545.0 

I5I 

303.8 

,96 

384-8 

241 

465.8 

286 

546.8 

152 

305.6 

197 

j86.6 

242 

467.6 

287 

548.6 

i53 

3074 

198 

388.4 

243 

469-4 

288 

550.4 

154 

309.2 

199 

390.2 

244 

471.2 

289 

552.2 

i55 

31  i.o 

200 

392.0 

245 

473-o 

290 

554-o 

156 

312.8 

201 

393-8 

246 

474-8 

291 

555-8 

i57 

3H.6 

202 

5-6 

247 

476.6 

292 

- 

557-6 

158 

3l6.4 

203 

7-4 

248 

478.4 

293 

5594 

i59 

318.2 

204 

399-2 

249 

480.2 

294 

561.2 

100 

320.0 

205 

401.0 

250 

482.0 

295 

563.0 

161 

321.8 

206 

402.8 

251 

483.8 

296 

564.8 

162 

323.6 

207 

404.6 

252 

485.6 

297 

566.6 

163 

325.4 

208 

406.4 

253 

487.4 

298 

568.4 

164 

327.2 

209 

408.2 

254 

489.2 

299 

570.2 

165 

329.0 

210 

410.0 

255 

491.0 

300 

572.o 

166 

330-8 

21  I 

411.8 

256 

492.8 

301 

573-8 

167 

332.6 

212 

413.6 

257 

494-6 

302 

575-6 

168 

3344 

213 

415.4 

•258 

496.4 

303 

577-4 

169 

336.2 

214 

417.2 

259 

498.2 

3°4 

579-2 

170 

338.0 

215 

419.0 

260 

500.0 

3°5 

581.0 

171 

339-8 

216 

420.8 

26l 

501.8 

306 

582.8 

172 

341.6 

217 

422.6 

262 

503.6 

307 

584.6 

173 

343-4 

218 

424.4 

263 

505.4 

308 

586.4 

174 

345.2 

219 

426.2 

264 

507.2 

309 

588.2 

175 
176 

347-0 
348.8 

220 
221 

428.0     265 
429.8     266 

509.0 
510.8 

310 

590.0 
591.8 

177 

350.6 

222 

431.6 

267 

512.6 

312 

593-6 

-  178 

352.4 

223 

433-4 

268 

514.4 

313 

595-4 

179 

354-2 

224 

435-2 

269 

516.2 

314 

597.2 

1  80 

356.0 

225 

437-0 

270 

518.0 

315 

599-o 

181 

357-8 

226 

438.8 

271 

519.8 

316 

600.8 

182 

359.6 

227 

440.6 

272 

521.6 

317 

602.6 

183 

361-4 

228 

442.4 

273 

5234 

318 

604.4 

184 

363.2 

229 

444.2 

274 

525.2 

319 

606.2 

185 

365.0 

230 

446.0 

527.0 

320 

608.0 

1  86 

366.8 

231 

447.8 

276 

528.8 

187 

368.6 

449-6 

277 

530.6 

188 

370-4 

4514 

278 

53-4 

AIR-THERMOMETERS. 


325 


AIR-THERMOMETERS. 

Air-thermometers,  or  gas-thermometers,  though  inconvenient  because 
bulky,  are,  by  reason  of  the  great  expansiveness  of  air,  superior  to  such  as 
depend  upon  the  expansion  of  liquids  or  solids,  in  point  of  delicacy  and 
exactness.  In  any  thermometer,  whether  liquid  or  gas,  the  indications 
depend  jointly  upon  the  expansion  by  heat  of  the  fluid  substance,  and  that 
of  the  tube  which  holds  it.  The  expansion  of  mercury  is  scarcely  seven 
times  that  of  the  glass  tube  within  which  it  expands,  and  the  exactness  of 
its  indications  are  interfered  with  by  the  variation  in  the  expansiveness  of 
glass  of  different  qualities.  In  the  gas-thermometer,  on  the  contrary,  the 
expansiveness  of  the  gas  is  160  times  that  of  the  glass,  and  the  inequalities 
of  the  glass  do  not  sensibly  affect  the  indications  of  the  instrument. 

Gas-thermometers,  or,  as  they  are  commonly  called,  air-thermometers, 
are  designed  either  to  maintain  a  constant  pressure  with  a  varying  volume 
of  air,  or  to  maintain  a  constant  volume  of  air  while  the  pressure  varies. 
In  the  first  case,  Fig.  119,  the  thermometer  consists  of  a  reservoir  A,  to  be 
placed  in  the  substance  of  which  the  temperature 
is  to  be  ascertained;  a  tube  df,  connected  at  a 
suitable  distance  by  a  small  tube  ab  to  the  reservoir; 
a  tube  cd,  open  above,  through  which  mercury  is 
introduced  into  the  instrument;  a  stop-cock  r  to 
open  or  close  a  communication — ist,  between  the 
tube  dfand  the  atmosphere;  2d,  between  the  base 
of  the  tube  cd  and  the  atmosphere;  3d,  between 
the  two  tubes  df,  cd-,  4th,  between  both  these 
tubes  and  the  atmosphere.  The  tube  df,  which  is 
carefully  gauged,  answers  the  purpose  of  the  gradu- 
ated tube  of  the  mercury-thermometer,  and  receives 
the  air  driven  over  by  expansion  from  the  reservoir, 
at  the  same  time  that  it  is  maintained  at  or  near 
the  temperature  of  the  surrounding  atmosphere. 
Thus  the  air  is  divided  between  the  reservoir  A  and 
the  tube  df,  of  which  the  air  in  the  former  is  at  the 


Fig.  119. — Air-Thermometer. 


temperature  of  the  substance  under  observation,  and  that  in  the  latter  is 
at  the  temperature  of  the  atmosphere.  These  two  portions  of  air  support 
the  same  pressure,  which  can  at  all  times  be  approximated  to  that  of  the 
atmosphere  by  means  of  the  cock  r,  through  which  the  mercury  is  allowed 
to  escape  until  it  arrives  at  the  same  level  in  the  two  tubes.  By  means  of 
a  formula  embracing  the  respective  volumes  of  the  two  portions  of  air  and 
the  temperature  of  the  atmosphere,  the  temperature  of  the  substance  under 
observation  is  determined.  But  it  is  apparent  that,  when  applied  as  a 
pyrometer  to  the  measurement  of  high  temperatures — higher,  that  is  to 
say,  than  the  boiling  point  of  mercury  (676°  F.) — the  greater  part  of  the  air 
passes  by  expansion  into  the  tube  df,  leaving  but  a  small  remainder  in  the 
reservoir  A.  A  serious  objection  to  this  is  that  the  proportion  of  air  which 
passes  over  into  the  tube  df  for  a  new  increase  of  temperature  is  very 
small,  and  is  with  difficulty  measured  with  sufficient  precision. 

The  second  form  of  air  thermometer,  in  which  the  pressure  varies  whilst 
the  volume  remains  the  same,  was  used  by  M.  Regnault  in  his  researches. 


326  HEAT. 

The  temperature  is  measured  by  means  of  the  increased  elastic  force  of  the 
inclosed  air,  and  the  instrument  is  both  more  convenient  and  more  precise 
than  that  in  which  the  volume  varies,  for  at  all  temperatures  the  sensibility 
of  the  instrument  is  the  same.  At  high  temperatures  the  apparatus  is  liable 
to  distortion  under  the  pressure  of  the  inclosed  air;  but  this  may  be  pre- 
vented, if  needful,  by  introducing  air  of  a  lower  than  atmospheric  pressure  at 
an  ordinary  temperature,  even  so  low  as  one-fourth  of  an  atmosphere; 
for,  although  the  apparatus  is  less  sensitive  in  proportion  as  the  first  supply 
of  air  is  of  less  density  and  pressure,  yet  withal  it  is  sufficiently  sensitive. 
The  thermometer,  as  employed  by  M.  Regnault,  is  shown  in  Fig.  120.  Two 

glass  tubes,  df,  cd,  about  half-an-inch  bore, 
are  united  at  the  base  by  a  stop-cock  r. 
The  tube  cd  is  open  above,  and  df  is  con- 
nected to  the  reservoir  A  by  a  small  tube  ab. 
The  cover  of  the  boiler  in  which  the  reser- 
voir is  inclosed  is  shown  at  B,  and  the  tubes 
are  protected  from  the  heat  of  the  boiler  by 
the  partition  CD.  By  means  of  a  three-way 
connection,  g,  and  tube  //,  the  connecting 
tube  ab  communicates  with  an  air  pump, 
by  means  of  which  the  apparatus  may  be 
dried,  and  air  or  other  gas  supplied  to  it. 
The  first  thing  to  be  done  is  to  completely 
dry  the  apparatus,  and  for  this  object,  a  little 
mercury  is  passed  into  the  tube  &</,  and  the 
cock  r  is  closed  against  it.  The  exhausting 
pump  is  then  set  to  work  to  exhaust  the 
tube,  which  is  done  several  times,  the  air 
being  slowly  re-admitted  after  each  exhaus- 
tion, after  having  been  passed  through  a 
filter  of  pumice-stone  in  connection  with 
the  pump,  saturated  with  concentrated  sul- 
phuric acid  to  absorb  moisture,  and  thus 
desiccate  the  air.  During  this  part  of  the 
process,  the  reservoir  is  maintained  at  a 
temperature  of  130°  R,  or  140°  R,  to  insure 

complete  desiccation.  Next,  the  reservoir  is  plunged  into  melting  ice, 
the  two  vertical  tubes  bd,  cd,  are  put  into  communication,  and  filled  with 
mercury  up  to  a  suitable  level  /,  marked  on  the  tube  bd.  If  it  is  desired 
to  establish  an  internal  pressure  less  than  that  of  the  atmosphere,  the  air 
is  partially  exhausted  by  means  of  the  pump,  the  degree  of  exhaustion  being 
recorded  by  the  difference  of  level  in  the  two  tubes.  The  exhausting  tube 
h  is  then  hermetically  sealed,  and  the  mercury  adjusted  to  the  level /in  the 
tube  bd. 

PYROMETERS. 

Pyrometers  are  employed  to  measure  temperatures  above  the  boiling 
point  of  mercury,  about  676°  R  They  depend  upon  the  change  of  form  of 
either  solid  or  gaseous  bodies,  liquids  being  necessarily  inadmissible. 
Pyrometric  estimations  are  of  three  classes: — First,  those  of  which  the 


PYROMETERS.  327 

indications  are  based  upon  the  change  of  dimensions  of  a  particular  body, 
solid  or  gaseous — the  pyrometer;  second,  those  based  on  the  heat  imparted 
to  water  by  a  heated  body;  third,  those  which  are  based  upon  the  melting 
points  of  metals  and  metallic  alloys. 

Wedgwood's  pyrometer,  invented  in  1782,  was  founded  on  the  property 
possessed  by  clay  of  contracting  at  high  temperatures,  an  effect  which  is 
due  partly  to  the  dissipation  of  the  water  in  clay,  and  subsequently  to  partial 
vitrification.  The  apparatus  consists  of  a  metallic  groove,  24  inches  long, 
the  sides  of  which  converge,  being  half-an-inch  wide  above  and  three-tenths 
below.  The  clay  is  made  up  into  little  cylinders  or  truncated  cones,  which 
fit  the  commencement  of  the  groove  after  having  been  heated  to  low  red- 
ness; their  subsequent  contraction  by  heat  is  determined  by  allowing  them 
to  slide  from  the  top  of  the  groove  downwards  till  they  arrive  at  a  part  of 
it  through  which  they  cannot  pass.  The  zero  point  is  fixed  at  the  tempera- 
ture of  low  redness,  1077°  F.  The  whole  length  of  the  groove  or  scale  is 
divided  into  240  degrees,  each  of  which  was  supposed  by  Wedgwood 
equivalent  to  130°  F.,  the  other  end  of  the  scale  being  assumed  to  represent 
32,277°  F.  Wedgwood  also  assumed  that  the  contraction  of  the  clay  was 
proportional  to  the  degree  of  heat  to  which  it  might  be  exposed;  but  this 
assumption  is  not  correct,  for  a  long-continued  moderate  heat  is  found  to 
cause  the  same  amount  of  contraction  as  a  more  violent  heat  for  a  shorter 
period.  Wedgwood's  pyrometer  is  not  employed  by  scientific  men,  because 
its  indications  cannot  be  relied  upon  for  the  reason  just  given,  and  also 
because  the  contraction  of  different  clays  under  great  heat  is  not  always  the 
same. 

In  Daniell's  pyrometer  the  temperature  is  measured  by  the  expansion  of 
a  metal  bar  inclosed  in  a  black-lead  earthenware  case,  which  is  drilled  out 
longitudinally  to  ^  inch  in  diameter  and  7^  inches  deep.  A  bar  of 
platinum  or  soft  iron,  a  little  less  in  diameter,  and  an  inch  shorter  than  the 
bore,  is  placed  in  it  and  surmounted  by  a  porcelain  index  i  y2  inches  long, 
kept  in  its  place  by  a  strap  of  platinum  and  an  earthenware  wedge. 
When  the  instrument  is  heated,  the  bar,  by  its  greater  rate  of  expansion 
compared  with  the  black-lead,  presses  forward  the  index,  which  is  kept  in 
its  new  situation  by  the  strap  and  wedge  until  the  instrument  cools,  when 
the  observation  can  be  taken  by  means  of  a  scale. 

The  air-pyrometer.  The  principle  and  construction  of  the  air-thermo- 
meter are  directly  applicable  for  pyrometric  purposes,  substituting  a  platinum 
globe  for  the  glass  reservoir  already  described,  for  resisting  great  heat,  and 
as  large  as  possible.  The  chief  cause  of  uncertainty  is  the  expansion  of 
the  metal  at  high  temperatures. 

The  second  means  of  estimation  is  best  represented  by  the  "pyrometer" 
of  Mr.  Wilson,  of  St.  Helen's.  He  heats  a  given  weight  of  platinum  in  the 
fire  of  which  the  temperature  is  to  be  measured,  and  plunges  it  into  a 
vessel  containing  twice  the  weight  of  water  of  a  known  temperature. 
Observing  the  rise  of  temperature  in  the  water,  he  calculates  the  tempera- 
ture to  which  the  platinum  was  subjected,  in  terms  of  the  rise  of  tempera- 
ture of  the  water,  the  relative  weights  of  the  platinum  and  the  water,  and 
their  specific  heats.  In  fact,  the  elevation  of  the  temperature  of  the 
water  is  to  that  of  the  platinum  above  the  original  temperature  of  the  water 
in  the  compound  ratio  of  the  weights  and  specific  heats  inversely;  that  is 
to  say,  that  the  weights  of  the  platinum  and  the  water  being  as  i  to  2,  and 


328  HEAT. 

their  specific  heats  as  .0314  to  i,  the  rise  of  temperature  of  the  water  is  to 
that  of  the  platinum  as  i  x  .0314  to  2  x  i,  or  as  i  to  63.7,  and  the  rule  for 
finding  the  temperature  of  the  fire  is  to  multiply  the  rise  of  temperature  of 
the  water  by  63.7,  and  add  its  original  temperature  to  the  product.  The 
sum  is  the  temperature  of  the  fire,  subject  to  correction  for  the  heat 
absorbed  by  the  thermometer  in  the  water,  and  by  the  iron  vessel  contain- 
ing the  water,  and  the  heat  retained  by  the  platinum.  The  correction  is 
estimated  by  Mr.  Wilson  at  -frih,  taking  the  weight  of  water  at  2000  grains, 
and  that  of  the  platinum  i  ooo  grains,  and  it  may  be  allowed  for  by  increas- 
ing the  above-named  multiplier  by  TVth,  to  67.45. 

Mr.  Wilson  proposed  that  for  general  practical  purposes  a  small  piece  of 
Stourbridge  clay  be  substituted  for  platinum,  to  lessen  the  cost  of  the 
apparatus.  With  a  piece  of  such  clay,  weighing  200  grains,  and  2000 
grains  of  water,  he  found  that  the  correct  multiplier  was  46. 

The  third  means  of  estimation,  based  on  the  melting  points  of  metals 
and  metallic  alloys,  is  applied  simply  by  suspending  in  the  heated  medium 
a  piece  of  metal  or  alloy  of  which  the  melting  point  is  known,  and,  if 
necessary,  two  or  more  pieces  of  different  melting  points,  so  as  to  ascertain, 
according  to  the  pieces  which  are  melted  and  those  which  continue  in  the 
solid  state,  within  certain  limits  of  temperature,  the  heat  of  the  furnace.  A 
list  of  melting  points  of  metals  and  metallic  alloys  is  given  in  a  subsequent 
chapter. 

LUMINOSITY  AT  HIGH  TEMPERATURES. 

The  luminosity  or  shades  of  temperature  have  been  observed  by  M. 
Pouillet  by  means  of  an  air-pyrometer  to  be  as  follows : — 

SHADE.  TEMPERATURE,        TEMPERATURE, 

Centigrade.  Fahrenheit. 

Nascent  Red 525°  977° 

Dark  Red 700  1292 

Nascent  Cherry  Red 800  1472 

Cherry  Red 900  1652 

Bright  Cherry  Red 1000  1832 

Very  Deep  Orange noo  2012 

Bright  Orange. 1200  2192 

White 1300  2372 

"  Sweating "  White 1400  2552 

Dazzling  White 1500  2732 

A  bright  bar  of  iron,  slowly  heated  in  contact  with  air,  assumes  the 
following  tints  at  annexed  temperatures  (Claudel) : — 

Centigrade.  Fahrenheit. 

1 .  Cold  iron  at  about 12°       or         54° 

2.  Yellow  at 225  437 

3.  Orange  at 243  473 

4.  Red  at 265  509 

5.  Violet  at 277  531 

6.  Indigo  at 288  550 

7.  Blue  at 293  559 

8.  Green  at 332  630 

9.  Oxide  Gray  (gris  d'oxyde)  at 400  752 


MOVEMENTS   OF   HEAT.  329 

MOVEMENTS   OF   HEAT. 

When  two  bodies  in  the  neighbourhood  of  each  other  have  unequal 
temperatures,  there  exists  between  them  a  transfer  of  heat  from  the  hotter 
of  the  two  to  the  other.  The  tendency  to  an  equalization,  or  towards  an 
equilibrium,  of  temperatures  in  this  way  is  universal,  and  the  passage  of 
heat  takes  place  in  three  ways :  by  radiation,  by  conduction,  and  by  con- 
vection or  carriage  from  one  place  to  another  by  heated  currents. 

RADIATION  OF  HEAT  FROM  COMBUSTIBLES. 

It  is  a  common  assumption  that  the  quantity  of  heat  radiated  from  com- 
bustibles is  very  small  in  comparison  with  the  total  quantity  of  heat  evolved. 
Holding  the  hand  near  the  flame  of  a  candle,  laterally,  the  radiant  heat,  which 
is  the  only  heat  thus  experienced,  is  much  less  than  the  heat  experienced 
by  the  hand  when  held  above  the  flame,  which  is  the  heat  by  convection 
of  the  hot  current  of  air  which  rises  from  the  flame.  But  it  is  to  be  noted 
that,  whilst  the  radiant  heat  is  dissipated  all  round  the  flame,  the  diameter 
of  the  upward  current  is  little  more  than  that  of  the  flame,  and  the  conveyed 
heat  is  therefore  concentrated  in  a  narrow  compass. 

M.  Peclet,  by  means  of  a  simple  apparatus,  consisting  of  a  cage  suspend- 
ing the  combustible  within  a  hollow  cylinder  filled  with  water  in  an  annular 
space,  ascertained  that  the  proportion  of  the  total  'heat  radiated  from 
different  combustibles  was  as  follows : — 

Radiant  heat  from  wood nearly  %- 

Do.         do.       wood  charcoal „      y2. 

Do.         do.       oil „      '/5. 

These  values  serve  to  show  that  radiation  of  heat  is  considerable,  and  that 
flameless  carbon  radiates  much  more  than  flame,  though  the  proportion  of 
heat  radiated  from  fuels  depends  very  much  upon  the  disposition  of  the 
material  and  the  extent  of  radiating  surface. 

With  respect  to  heated  bodies,  apart  from  combustibles  as  such,  the 
radiation  or  emission  of  heat  implies  the  reverse  process  of  absorption,  and 
the  best  radiators  are  likewise  the  best  absorbents  of  heat.  All  bodies 
possess  the  property  of  radiating  heat.  The  heat  rays  proceed  in  straight 
lines,  and  the  intensity  of  the  heat  radiated  from  any  one  source  of  heat 
becomes  less  as  the  distance  from  the  source  of  heat  increases,  in  the 
inverse  ratio  of  the  square  of  the  distance.  That  is  to  say,  for  example, 
that  at  any  given  distance  from  the  source  of  radiation,  the  intensity  of  the 
radiant  heat  is  four  times  as  great  as  it  is  at  twice  the  distance,  and  nine 
times  as  great  as  it  is  at  three  times  the  distance. 

The  quantity  of  heat  emitted  by  radiation  increases  in  some  proportion 
with  the  difference  of  temperatures  of  the  radiating  body  and  the  surrounding 
medium,  but  more  rapidly  than  the  simple  proportion  for  the  greater  differ- 
ences; and  the  quantity  of  heat,  greater  or  less,  emitted  by  bodies  by  radiation 
under  the  same  circumstances  is  the  measure  of  their  radiating  power. 

Radiant  heat  traverses  air  without  heating  it. 

When  a  polished  body  is  struck  by  a  ray  of  heat,  it  absorbs  a  part  of  the 
heat  and  reflects  the  rest.  The  greater  or  less  proportion  of  heat  absorbed 
by  the  body  is  the  measure  of  its  absorbing  power,  and  the  reflected  heat  is 
the  measure  of  its  reflecting  power. 


330 


HEAT. 


When  the  temperature  of  a  body  remains  constant  it  indicates  that  the 
quantity  of  heat  emitted  is  equal  to  the  quantity  of  heat  absorbed  by  the 
body.  The  reflecting  power  of  a  body  is  the  complement  of  its  absorbing 
power;  that  is  to  say,  that  the  sum  of  the  absorbing  and  reflecting  powers  of 
all  bodies  is  the  same,  which  amounts  to  this,  that  a  ray  of  heat  striking  a 
body  is  disposed  of  by  absorption  and  reflection  together,  that  which  is  not 
absorbed  being  necessarily  reflected. 

For  example,  the  radiating  power  of  a  body  being  represented  by  90,  the 
reflecting  power  is  also  90,  and  the  absorbing  power  is  10,  supposing  that 


Table  No.    106. — COMPARATIVE   RADIATING  OR  ABSORBENT  AND 
REFLECTING  POWERS  OF  SUBSTANCES. 


SUBSTANCE. 

POWERS. 

Radiating  or 
Absorbing. 

Reflecting. 

Lamp  Black  

100 
100 
100 

98 

93  to  98 

9i 
90 

85 
85 
72 
27 
25 
23 
23 
19 
17 
24 
17 
i7 

15 
1  1 

9 
7 

7 
14 
7 
7 
5 
3 
3 
3 

0 
0 
0 
2 

7  to  2 
9 

10 

15 
15 
28 

73 
75 
77 

77 
81 

83 
76 

83 
83 
85 
89 

9i 
93 
93 
86 

93 
93 
95 
97 
97 
97 

Water  

Carbonate  of  Lead 

Writing  Paper 

Ivory,  Jet,  Marble 

Isinglass  .    ... 

Ordinary  Glass  

China  Ink  ,  

Ice  

Gum  Lac  

Silver  Leaf  on  Glass 

Cast  Iron,  brightly  polished.    . 

Mercury,  about 

Wrought  Iron,  polished    . 

Zinc,  polished  

Steel,  polished  

Platinum,  a  little  polished  

Do.      deposited  on  Copper  

Do.      in  Sheet  

Tin 

Brass,  cast  dead  polished 

Do     hammered,  dead  polished 

Do.    cast,  bright  polished 

Do.    hammered,  bright  polished     

Copper,  varnished  

Do.      deposited  on  iron  

Do       hammered  or  cast 

Gold  plated 

Do     deposited  on  polished  Steel 

Silver,  hammered,  polished  bright.. 

Do.    cast,  polished  bright  

MOVEMENTS   OF   HEAT. 


331 


the  total  quantity  of  heat  which  strikes  the  body  is  represented  by  100. 
The  reflecting  power  of  soot  is  sensibly  nil,  and  its  absorbing  and  radiating 
powers  are  100. 

The  absorbing  power  varies  with  the  nature  of  the  source  of  heat,  with 
the  condition  of  the  substance,  and  with  the  inclination  of  the  direction  of 
the  heat  radiated  upon  the  body.  That  of  a  metallic  surface  is  so  much 
the  less,  and  consequently  the  reflecting  power  is  so  much  the  more,  in 
proportion  as  the  surface  is  better  polished. 

The  reflecting  power  of  metals,  according  to  MM.  de  la  Provostaye  and 
Desains,  is  practically  the  same,  when  the  angle  of  incidence,  that  is  the 
angle  at  which  the  rays  of  heat  strike  the  surface,  is  less  than  70°  of  inclina- 
tion with  the  surface;  but  for  greater  angles,  approaching  more  nearly  to 
90°,  perpendicular  to  the  surface,  it  sensibly  diminishes. 

For  example,  at  angles  of  from  75  to  80  degrees,  the  reflecting  power  is 
only  94  per  cent,  of  what  it  is  under  the  smaller  angles  of  incidence. 

The  table  No.  106  contains  the  radiating  and  absorbing  powers  and  the 
reflecting  powers  of  various  substances.  (Leslie,  De  la  Provostaye  and 
Desains,  and  Melloni.) 

The  reflecting  power  of  glass  has  been  found  to  be  the  same  for  heat  and 
for  light. 

Conduction  of  Heat. — Conduction  is  the  movement  of  heat  through  sub- 
stances, or  from  one  substance  to  another  in  contact  with  it.  The  table 
No.  107  contains  the  relative  internal  conducting  power  of  metals  and  earths, 
according  to  M.  Despretz.  A  body  which  conducts  heat  well  is  called  a 
good  conductor  of  heat;  if  it  conducts  heat  slowly,  it  is  a  bad  conductor  of 
heat.  Bodies  which  are  finely  fibrous,  as  cotton,  wool,  eider-down,  wadding, 
finely  divided  charcoal,  are  the  worst  conductors  of  heat.  Liquids  and 
gases  are  bad  conductors;  but  if  suitable  provision  be  made  for  the  free 
circulation  of  fluids  they  may  abstract  heat  very  quickly  by  contact  with 
heated  surfaces,  acting  by  convection. 

Convection  of  Heat. — Converted  or  carried  heat  is  that  which  is  trans- 
ferred from  one  place  to  another  by  a  current  of  liquid  or  gas :  for  example, 
by  the  products  of  combustion  in  a  furnace  towards  the  heating  surface 
in  the  flues  of  a  boiler. 


Table  No.  107. — RELATIVE  INTERNAL  CONDUCTING  POWER  OF  BODIES. 


Substance. 

Relative  conducting 
power. 

Substance. 

Relative  conducting 
power. 

Gold 

IOOO 

Zinc     

363 

Platinum  

081 

Tin    

304 

Silver 

Q73 

Lead 

1  80 

Copper          

V  /  O 

802 

Marble           

24 

Brass     

74.0 

Porcelain     

12 

Cast  Iron  

(C62 

Terra  Cotta  

II 

Wrought  Iron  

374 

332 


HEAT. 


THE    MECHANICAL   THEORY   OF    HEAT. 

Heat  and  mechanical  force  are  identical  and  convertible.  Independently 
of  the  medium  through  which  heat  may  be  developed  into  mechanical 
action,  the  same  quantity  of  heat  is  resolved  into  the  same  total  quantity 
of  work.  The  English  unit  of  heat  is  that  which  is  required  to  raise  the  tem- 
perature of  i  Ib.  of  water  at  39°.  i,  i  degree  Fahr.  If  2  Ibs.  of  water  be  raised 
i  degree,  or  i  Ib.  be  raised  2  degrees  in  temperature,  the  expenditure  of 
heat  is  the  same  in  amount,  namely,  two  units  of  heat-  and  to  express  the 
mechanical  equivalent  of  heat,  the  comparison  lies  between  the  unit  of 
heat  on  the  one  part,  and  the  unit  of  work,  or  the  foot-pound,  on  the  other 
part.  The  most  precise  determination  yet  made  of  the  numerical  relation 
subsisting  between  heat  and  mechanical  work  was  obtained  by  the  following 
experiment  of  Dr.  Joule.  He  constructed  an  agitator,  Fig.  121,  consisting 

of  a  vertical  shaft  carry- 
ing a  brass  paddle-wheel, 
of  which  the  paddles  re- 
volved between  station- 
ary vanes,  which  served 
to  prevent  the  liquid  in 
the  vessel  from  being 
bodily  whirled  in  the 
direction  of  rotation. 
The  vessel  was  rilled 


Fig.  121. — Dr.  Joule's  Agitator. 


was 

with  water,  and  the  agi- 
tator was  made  to  revolve 
by  means  of  a  cord  wound 
round  the  upper  part  of 
the  shaft,  and  attached 
to  a  weight  which  de- 
scended in  front  of  a 
scale,  by  which  the  work 
done  was  measured. 
When  all  corrections  had  been  applied,  it  was  found  that  the  heat  com- 
municated to  the  water  by  the  agitation  amounted  to  one  pound-degree 
Fahrenheit  for  every  772  foot-pounds  of  work  expended  in  producing  it. 
It  was  deduced,  inversely,  that  one  unit  of  heat  was  capable  of  raising 
772  Ibs.  weight  i  foot  in  height.  The  mechanical  equivalent  of  heat, 
known  as  "Joule's  equivalent,"  is  therefore  taken  as  772  foot-pounds  for 
i  unit  of  heat.  Sperm  oil  was  tried  as  the  fluid  medium,  and  it  yielded  the 
same  results  as  water. 

The  following  are  the  values  of  Joule's  equivalent  for  different  thermo- 
metric  scales,  and  in  English  and  French  units : — 

i   English  thermal   unit,  or  i    degree  )          c 

Fahrenheit  in  i  pound  of  water.  ....  ]  "2  f°°t-POunds. 
i  French  thermal  unit,  or  i  degree  centi-  )  /  x ,  ., 

grade  in  i  kilogramme  of  water }  W-SS  («y  424)Mogrammetres. 

i  degree  Centigrade  in  i  pound  of  water — 1389.60  (say  1390)  foot-pounds, 
i  French  thermal  unit  is  equal  to  3.968  English  thermal  units — about 
4  English  units. 


MECHANICAL  THEORY   OF   HEAT.  333 

According  to  the  mechanical  theory  of  heat,  in  its  general  form,  heat, 
mechanical  force,  electricity,  chemical  affinity,  light,  and  sound,  are  but 
different  manifestations  of  motion.  Dulong  and  Gay  Lussac  proved  by 
their  experiments  on  sound,  that  the  greater  the  specific  heat  of  a  gas,  the 
more  rapid  are  its  atomic  vibrations.  Elevation  of  temperature  does  not 
alter  the  rapidity  but  increases  the  length  of  their  vibrations,  and  in  con- 
sequence produces  "expansion"  of  the  body.  All  gases  and  vapours  are 
assumed  to  consist  of  numerous  small  atoms,  moving  or  vibrating  in  all 
directions  with  great  rapidity;  but  the  average  velocity  of  these  vibrations 
can  be  estimated  when  the  pressure  and  weight  of  any  given  volume  of  the 
gas  is  known,  pressure  being,  as  explained  by  Joule,  the  impact  of  those 
numerous  small  atoms  striking  in  all  directions,  and  against  the  sides  of  the 
vessel  containing  the  gas.  The  greater  the  number  of  these  atoms,  or  the 
greater  their  aggregate  weight,  in  a  given  space,  and  the  higher  the  velocity,, 
the  greater  is  the  pressure.  A  double  weight  of  a  perfect  gas,  when  con- 
fined in  the  same  space,  and  vibrating  with  the  same  velocity — that  is, 
having  the  same  temperature — gives  a  double  pressure ;  but  the  same  weight 
of  gas,  confined  in  the  same  space,  will,  when  the  atoms  vibrate  with  a 
double  velocity,  give  a  quadruple  pressure.  An  increase  or  decrease  of 
temperature  is  simply  an  increase  or  decrease  of  molecular  motion.  When, 
the  piston  in  the  cylinder  yields  to  the  pressure  of  steam,  the  atoms  will 
not  rebound  from  it  with  the  same  velocity  with  which  they  strike,  but  will 
return  after  each  succeeding  blow,  with  a  velocity  continually  decreasing 
as  the  piston  continues  to  recede,  and  the  length  of  the  vibrations  will  be 
diminished.  The  motion  gained  by  the  piston  will  be  precisely  equivalent 
to  the  energy,  heat,  or  molecular  motion  lost  by  the  atoms  of  the  gas; 
and  it  would  be  as  reasonable  to  expect  one  billiard  ball  to  strike  and  give 
motion  to  another  without  losing  any  of  its  own  motion,  as  to  suppose  that 
the  piston  of  a  steam-engine  can  be  set  in  motion  without  a  corresponding 
quantity  of  energy  being  lost  by  some  other  body. 

In  expanding  air  spontaneously  to  a  double  volume,  delivering  it,  say,. 
into  a  vacuous  space,  it  has  been  proved  repeatedly  that  the  air  does  not 
appreciably  fall  in  temperature,  no  external  work  being  performed;  but  that, 
on  the  contrary,  if  the  air  at  a  temperature,  say,  of  230°  F.,  be  expanded 
against  an  opposing  pressure  or  resistance,  as  against  the  piston  of  a  cylinder,, 
giving  motion  to  it  and  raising  a  weight  or  otherwise  doing  work,  the  tem- 
perature will  fall  nearly  170°  F.  when  the  volume  is  doubled,  that  is  from 
230°  F.  to  about  60°  F.,  and,  taking  the  initial  pressure  at  40  Ibs.,. 
the  final  pressure  would  be  15  Ibs.  per  square  inch. 

When  a  pound  weight  of  air,  in  expanding,  at  any  temperature  or  pressure, 
raises  130  Ibs.  one  foot  high,  it  loses  i°  F.  in  temperature;  in  other 
words,  this  pound  of  air  would  lose  as  much  molecular  energy  as  would 
equal  the  energy  acquired  by  a  weight  of  one  pound  falling  through  a 
height  of  1 30  feet.  It  must,  however,  be  remarked  that  but  a  small  portion 
of  this  work — 130  foot-pounds — can  be  had  as  available  work,  as  the  heat 
which  disappears  does  not  depend  on  the  amount  of  work  or  duty  realized, 
but  upon  the  total  of  the  opposing  forces,  including  all  resistance  from  any 
external  source  whatever.  When  air  is  compressed  the  atmosphere  descends 
and  follows  the  piston,  assisting  in  the  operation  with  its  whole  weight;  and 
when  air  is  expanded  the  motion  of  the  piston  is,  on  the  contrary,  opposed 
by  the  whole  weight  of  the  atmosphere,  which  is  again  raised.  Although, 


334  HEAT. 

therefore,  in  expanding  air,  the  heat  which  disappears  is  in  proportion  to 
the  total  opposing  force,  it  is  much  in  excess  of  what  can  be  rendered 
.available;  and,  commonly,  where  air  is  compressed  the  heat  generated  is 
much  greater  than  that  which  is  due  to  the  work  which  is  required  to  be 
expended  in  compressing  it,  the  atmosphere  assisting  in  the  operation. 

Let  a  pound  of  water,  at  a  temperature  of  212°  F.,  be  injected  into  a 
vacuous  space  or  vessel,  having  26.36  cubic  feet  of  capacity — the  volume 
of  one  pound  of  saturated  steam  at  that  temperature — and  let  it  be  evapor- 
ated into  such  steam,  then  893.8  units  of  heat  would  be  expended  in  the 
process.  But  if  a  second  pound  of  water,  at  212°,  be  injected  and  evapor- 
ated at  the  same  temperature,  under  a  uniform  pressure  of  14.7  Ibs.  per 
square  inch,  being  the  pressure  due  to  the  temperature,  the  second  pound 
must  dislodge  the  first,  supposing  the  vessel  to  be  expansible,  by  repelling 
that  pressure;  and  this  involves  an  amount  of  labour  equal  to  55,800  foot- 
pounds (that  is,  14.7  Ibs.  x  144  square  inches  x  26.36  cubic  feet),  and  an 
additional  expenditure  of  72.3  units  of  heat  (that  is,  55, 800  -=-  772),  making 
a  total,  for  the  second  pound,  of  965.1  units. 

Similarly,  when  1408  units  of  heat  are  expended  in  raising  the  tempera- 
ture of  air  under  a  constant  pressure,  1000  of  these  units  increase  the 
velocity  of  the  molecules,  or  produce  a  sensible  increase  of  temperature; 
while  the  remaining  408  units,  which  disappear  as  the  air  expands,  are 
directly  consumed  in  repelling  the  external  pressure  for  the  expansion  of 
volume. 

Again,  if  steam  be  permitted  to  flow  from  a  boiler  into  a  comparatively 
vacuous  space  without  giving  motion  to  another  body,  the  temperature  of 
the  steam  entering  this  space  would  rise  higher  than  that  of  the  steam  in 
the  boiler.  Or,  suppose  two  vessels,  side  by  side,  one  of  them  vacuous  and 
the  other  filled  with  air  at,  say,  two  atmospheres;  if  a  communication  be 
opened  between  them,  the  pressure  becomes  the  same  in  both.  But  the 
temperature  would  fall  in  one  vessel  and  rise  in  the  other;  and  although 
the  air  is  expanded  in  this  manner  to  double  its  first  volume,  there  would 
not,  on  the  whole,  be  any  appreciable  loss  of  heat,  for  if  the  separate  por- 
tions of  air  be  mixed  together,  the  resulting  average  temperature  of  the 
whole  would  be  very  nearly  the  same  as  at  first.  It  has  been  proved 
experimentally,  corroborative  of  this  statement,  that  the  quantity  of  heat 
required  to  raise  the  temperature  of  a  given  weight  of  air,  to  a  given  extent, 
is  the  same,  irrespective  of  the  density  or  the  volume  of  the  air.  Regnault 
and  Joule  found  that  to  raise  the  temperature  of  a  pound  of  air,  whether 
i  cubic  foot  or  10  cubic  feet  in  volume,  the  same  quantity  of  heat  was 
expended. 

In  rising  against  the  force  of  gravity  steam  becomes  colder,  and  it  par- 
tially condenses  while  ascending,  in  the  effort  of  overcoming  the  resistance 
of  gravity.  For  instance,  a  column  of  steam  weighing,  on  a  square  inch  of 
base,  250.3  Ibs.,  that  is  to  say,  having  a  pressure  of  250.3  Ibs.  per  square 
inch,  would,  at  a  height  of  275,000  feet,  be  reduced  to  a  pressure  of  i  Ib. 
per  square  inch,  and,  in  ascending  to  this  height,  the  temperature  would 
fall  from  401°  to  102°  F.,  while,  at  the  same  time,  nearly  25  per  cent, 
of  the  whole  vapour  would  be  precipitated  in  the  form  of  water,  unless  it 
were  supplied  with  additional  heat  while  ascending. 

If  a  body  of  compressed  air  be  allowed  to  rush  freely  into  the  atmosphere, 
the  temperature  falls  in  the  rapid  part  of  the  current,  by  the  conversion  of 


EXPANSION    BY   HEAT. 


335 


heat  into  motion,  but  the  heat  is  almost  all  reproduced  when  the  motion 
has  quite  subsided.  From  recent  experiments,  it  appears  that  nearly 
similar  results  are  obtained  from  the  emission  of  steam  under  pressure. 

When  water  falls  through  a  gaseous  atmosphere,  its  motion  is  constantly 
retarded  as  it  is  brought  into  collision  with  the  particles  of  that  atmosphere, 
and  by  this  collision  it  is  partly  heated  and  partly  converted  into  vapour. 

If  a  body  of  water  descends  freely  through  a  height  of  772  feet,  it  acquires 
from  gravity  a  velocity  of  223  feet  per  second;  and,  if  suddenly  brought  to 
rest  when  moving  with  this  velocity,  it  would  be  violently  agitated,  and 
would  be  raised  one  degree  of  temperature.  But  suppose  a  water-wheel, 
772  feet  in  diameter,  into  the  buckets  of  which  the  water  is  quietly  dropped; 
when  the  water  descends  to  the  foot  of  the  fall,  and  is  delivered  gently  into 
the  tail-race,  it  is  not  sensibly  heated.  The  greatest  amount  of  work  it  is 
possible  to  obtain  from  water  falling  from  a  given  level  to  a  lower  level  is 
expressible  by  the  weight  of  water  multiplied  by  the  height  of  the  fall. 

These  illustrative  exhibitions  of  the  nature  and  reciprocal  action  of  heat 
and  motive  power,  show  that  the  nature  and  extent  of  the  change  of  tem- 
perature of  a  gas  while  expanding  depend  nearly  altogether  upon 
cumstances  under  which  the  change  of  volume  takes  place. 


EXPANSION   BY   HEAT. 

All  bodies  are  expanded  by  the  application  of  heat,  but  in  different 
degrees.  Expansion  is  measurable  in  three  directions : — Length,  breadth, 
and  thickness;  and  it  may  be  measured  as  linear  expansion,  in  one  direc- 
tion; as  superficial  expansion,  in  two  directions;  or  as  cubical  expansion, 
in  three  directions.  Linear  expansion,  or  the  expansion  of  length,  is  that 
which  will  be  exposed  in  the  following  tables  for  solids  and  liquids.  The 
expansion  of  gases  is  measured  cubically,  by  volume. 

Superficial  expansion,  it  may  be  added,  is  twice  the  linear  expansion,  and 
cubical  expansion  is  three  times  the  linear  expansion.  That  is  to  say,  the 

additional  volume  by  expansion  in  two  direc- 
tions, as  in  length  and  breadth,  is  twice  the 
additional  volume  in  one  direction;  and  the 
additional  volume  in  three  directions  is  three 
times  that  in  one  direction.  For  example, 
take  a  solid  cube  abedefg-,  the  expansion  in 
one  direction  ea,  on  the  face  abed,  is,  say, 
equal  to  that  indicated  by  the  dot  lines  pro- 
jected from  that  face,  and  the  volume  by 
expansion  is  equal  to  the  extension  of  the 
surface  abed  thus  projected.  In  each  of  the 
two  other  directions,  da,  upwards,  and  a  b, 
laterally,  the  volume  by  expansion  is  the  same 
as  that  of  the  expansion  on  the  face  abed.  Consequently,  the  total 
increase  of  volume  by  expansion,  as  measured  cubically,  in  the  three 
directions  of  length,  breadth,  and  thickness,  is  three  times  the  increase  of 
volume  in  one  direction  singly;  and,  as  measured  superficially,  in  two  of 
these  directions,  it  is  twice  the  increase  of  volume  in  one  direction. 


Fig.  122. 


336 


HEAT. 


Table  No.  108. — LINEAR  EXPANSION  OF  SOLIDS  BY  HEAT,  BETWEEN 

32°  AND  212°  F. 


METALS. 

Expansion 
between  32° 
and  212°  F. 
in  common 
fractions. 

Expansion 
between  32° 
and  212°  F. 
in  a  length 

=  100. 

Expansion 
between  32° 
and  212°  F. 
in  a  length 
of  10  feet. 

Expansion 
for  i°  F.  in  a 
length  of  100 
feet. 

Zinc,  sheet      

i/ 

length  =  100. 
2QAl6 

inch. 

•2  C-2 

inch. 
OIQ6 

Do.,  forged  . 

/  34° 
«/- 

•3  JOS'? 

•  o  J  o 

-27/1 

O2O7 

Lead  

'/ 

•Oiv^O 

28484 

•O  /  T- 

^42 

OIQO 

Zinc  8+1  tin,  slightly  ham-  ) 
mered,  j 

/3Si 
'/3T. 

.26917 

•O'f'6 

.322 

.0179 

White  Solder:  —  tin  1  +  2  lead. 
Tin,  grain  

I/ 

/399 

J/,,no 

•25053 
.248^^ 

.301 
.208 

.0167 
.Ol66 

Tin  . 

/  4°3 

i/  , 

21  77O 

260 

OT4C 

Silver     . 

7402 
?/*„., 

-  x  /  ow 
IQO7  tj 

22Q 

'  i'+D 
OI  27 

Speculum  metal 

/524 
?/___ 

•  A  yw/  j 

IQ'?^-? 

•*y 

2?2 

OI  1O 

Brass  

15*7 

T/«o 

•  ^yooo 

.18782 

•*o- 

22? 

^XOV 
OI2  ? 

Copper  .  . 

/  S32 

I/eoT 

.  I  7220 

^•^0 
2O7 

.OI  I  ? 

Gun  Metal  :  —  16  copper  +  i  tin 
8  copper  +  i  tin 
Yellow  Brass  :  —  Rod  

/SO1 

V-4 

r/S50 
x/e«8 

.19083 
.18167 

.180^0 

.229 
.218 

227 

.0127 
.0121 
.OI26 

Do.              Trough  form.. 
Gold:— 
Paris  standard,  annealed 
Do.      unannealed 
Bismuth  

'/Srf 
X/66i 

y^ 

«-/___ 

.18945 

•15153 
-155*6 

.  1  1QI  7 

.227 

.l8l 
.186 
l67 

.0126 

.OIOI 
.0103 
00928 

Iron,  forged  

/7!9 
i/8lQ 

.12204 

146 

.00814 

Do.    wire  

«/8l_ 

.i2/?tro 

.148 

.00823 

Steel,  rod,  5  feet  long  

r/8(,. 

.1  14^0 

.137 

.OO763 

Do.    tempered  

/»74 
I/8-, 

.12306 

.I4Q 

.00826 

Do.    not  tempered 

x/«,<? 

IO7Q2 

I3O 

OO7  10 

Cast  Iron,  rod,  5  feet  long  
Antimony  

7920 

7901 
x/__, 

.IIIOO 
.  108"?  3 

.  ±3<-> 

•133 
1  30 

.00740 
.00722 

Palladium  '.  .  .  .  . 

/923 

Z/Trv 

^^oo 
IOOOO 

.  j-j^ 
120 

.00667 

Platinum  

I/TTA, 

.08^70 

.103 

.OO?7l 

/  "O7 

From  o°  to  300°  C. 
(32°  F.  to  572°  F.) 
n                        /  o°  to  100°  C. 
C°PPer  io°to3oo°C. 

f  o°  to  100°  C. 
Iron                   -so              n  s~t 

Vita 

Vssx 
VS46 

.17182 

.18832 

.11821 

.206 
.226 
.142 

.0115 
.00418 
.00788 

\  o°  to  300°  C. 

«—  igs-ssg 

V68i 
I/no3 
Vxo89 

.14684 
.08842 
.09183 

.176 
.106 
.in 

.00326 
.00589 
.OO204 

LINEAR   EXPANSION   OF  SOLIDS   BY   HEAT. 
Table  No.  108  (continued}. 


337 


GLASS. 

Expansion 
between  32° 
and  212°  F. 
in  common 
fractions. 

Expansion 
between  32° 
and  212°  F. 
in  a  length 

=  100. 

Expansion 
between  32° 
and  212°  F. 
in  a  length 
of  10  feet. 

Expansion 
for  i°  F.  in  a 
length  of  zoo 
feet. 

Flint  Glass  

i/ 

inch. 

inch. 

French  Glass,  with  lead  

71248 
i/ 

08*720 

.0974 

•UO541 

Glass  tube,  without  lead  

/i  i47 
i/ 

OOT  *7  C 

.105 

Glass  of  St.  Gobain.. 

71090 
i/ 

•"y1/^ 

Barometer  tubes  (Smeaton).  .  .  . 
Glass  tube  (Roy) 

/II22 

7  1  175 
I/      0 

•08333 

O*7  *7  C  C 

.107 
.IOO 

.OO594 

•°°555 

Glass  rod,  solid  (Roy)  
Glass  (Dulong  and  Petit)  
Do.  (o°  to  200°  C.)  

/I289 
71237 

Vxrfx 

i/ 

•°7755 
.08083 
.08613 
00484 

.0931 
.0970 
.103 
1  1  A 

.00517 

•00539 
.00574 

Oo6  3  2 

Do.  (o°  to  300°  C.)  

71032 

I/Qo, 

10108 

.  j.  J.^. 
121 

006  74 

7907 

Ice..,  ,  

O'Z'Z'Z 

•uooo 

STONES. 

Initial 
Temperature. 

Final 
Temperature. 

Expansion 
in  a  length 
=  100. 

Expansion 
for  i°  F.  in  a 
length  of  loo 
feet. 

Granite  

4C°  F 

220°  F 

length  =  TOO. 
2Ol6 

inch. 
O2OO 

Do.  . 

to    J 

Af 

IOO 

O4l6 

Clay-slate  

T-D 

46 

87 

041  6 

Do  

46 

<J/ 

104 

060? 

OT  A  1 

York  paving  

46 

QC 

•wwyo 

l6cK 

•0143 

OAT  C 

Micaceous  sandstone  

C2 

VD 

2OO 

•  ivyj 
17-26 

•U410 
OI4I 

Do.            do  

D" 
$2 

2OO 

•  x  /  o" 

1041 

OO844 

Do.            do. 

t\2 

I  CO 

0872 

Do.            do. 

D^ 
C2 

*5r7 

IOO 

QC  20 

Do.            do  

0^ 
AC 

IOO 

O4l6 

OOOoS 

Do.            do  

T-0 

AC 

260 

14^8 

008  14 

Carrara  marble  

TO 

•22 

212 

•^^fo0 

08/1  Q 

ooc.66 

Sost          do  

•22 

212 

oc68 

Stock  Brick  

6^ 

t?2 

260 

2  COO 

D" 

22 


333  HEAT. 

Speaking  exactly,  the  cubical  expansion  is  rather  less  than  three  times, 
and  the  superficial  expansion  rather  less  than  twice,  the  linear  expansion; 
for,  in  fact,  the  expanded  corners  of  the  body  are  carried  out  to  the  full 
square  figure,  and  have  not  the  entering  angles  shown  in  the  figure,  and 
there  is,  in  this  way,  a  certain  overlapping  of  the  strata  of  expansion  at  the 
ends,  sides,  and  top. 

The  same  kind  of  demonstration  applies  to  bodies  of  any  other  than  a 
cubical  shape. 

A  hollow  body  expands  by  heat  to  the  same  extent  as  if  it  were  a  solid 
body  having  the  same  exterior  dimensions. 

The  rate  of  expansion  of  solids  from  the  freezing  point  to  the  boiling 
point  of  water,  32°  to  212°  F.,  is  sensibly  uniform. 

The  table,  No.  108,  gives  the  linear  expansion  of  a  number  of  metals, 
and  of  glass,  between  the  freezing  and  boiling  points;  and  of  ice  for  one 
degree,  and  of  stones  for  various  intervals  of  temperature.  Authorities : — 
Laplace  and  Lavoisier,  Smeaton,  Roy,  Troughton,  Wollaston,  Dulong  and 
Petit,  Froment,  Rennie. 

Zinc  is  the  most  expansible  of  the  metals;  it  expands  fully  one-third 
per  cent.,  or  as  much  as  V^iSt  part  of  its  length,  when  heated  from  32°  F. 
to  212°  F.  Iron  expands  about  one-seventh  to  one-eighth  per  cent;  and 
cast-iron  and  platinum  about  one-tenth  per  cent.  The  expansion  of  metals 
proceeds  at  a  less  rate  above  the  boiling  point  than  below  it.  Ice  expands 
at  the  rate  of  Vse.oooth  of  its  length  for  one  degree  Fahrenheit;  which,  for 
1 80°,  would  be  I/2ooth  of  its  length, — greatly  more  than  that  of  any  metal. 

EXPANSION  OF  LIQUIDS. 

The  measurement  of  the  expansion  of  liquids  by  the  application  of  heat 
cannot  well  be  taken  lineally;  that  is,  as  linear  expansion,  in  the  sense 
in  which  the  expansion  of  solids  is  observed.  For  liquids  must  be  con- 
tained in  vessels,  which  only  admit  of  expansion  in  one  direction,  seeing 
that  the  liquid  is  limited  by  the  bottom  and  sides  of  the  vessel,  which 
throw  the  whole  of  the  expansion  or  enlargement  of  volume  upwards.  The 
observations  on  the  expansion  of  liquids,  therefore,  though  measured  in 
one  direction  only,  necessarily  indicate  the  cubical  expansion  or  total 
enlargement  of  volume.  But,  of  course,  it  is  easy  to  reduce  the  expansion 
of  a  liquid  for  comparison  with  the  linear  expansion  of  a  solid  by  taking 
one-third  of  the  observed  measurement. 

When  the  temperature  of  water  at  the  freezing  point,  32°  F.,  is  raised, 
the  water  does  not  at  first  expand,  but,  on  the  contrary,  contracts  in  volume 
until  the  temperature  is  raised  to  39°.  i  F.,  which  is  7.1  degrees  above  the 
freezing  point.  This  is  called  "the  temperature  of  maximum  density." 
From  this  point  water  expands  as  the  temperature  rises,  until,  at  46°  F.,  it 
regains  its  initial  volume,  that  is,  the  volume  at  32°  F.  Thence,  it  con- 
tinues to  expand  until  it  reaches  the  boiling  point,  212°  F.,  under  one 
atmosphere.  Passing  this  point  upwards,  if  the  pressure  be  suitably 
increased,  water  continues  to  expand  with  a  rise  of  temperature. 

The  cubical  expansion  of  water  when  heated  from  32°  to   212°  F.  is 

.0466;  that  is,  the  volume  is  increased  from  i  at  32°  F.  to  1.0466  at  212° 

F.     This  expansion  is  rather  more  than  4^  per  cent.,  or  between  r/2Ist  and 

1/22d  part  of  the  volume  at  32°.     The  expansion  of  water  increases  in  a 


EXPANSION   OF   LIQUIDS   BY   HEAT. 


339 


Table  No.  109. — EXPANSION  AND  DENSITY  OF  PURE  WATER, 
FROM  32°  TO  390°  F. 

(Calculated  by  means  of  Rankine's  approximate  formula. ) 


Tempera- 
ture. 

Comparative 
Volume. 

Comparative 
Density. 

Density, 
or  weight  of 
i  cubic  foot. 

Weight  of 
i  gallon. 

Remarkable  Temperatures- 

Fahr. 

Water  at  32° 

Water  at  32° 

Pounds. 

Pounds. 

=  i. 

=  i. 

32° 

I.OOOOO 

I.OOOOO 

62.418 

IO.OIOI 

Freezing  point. 

35 
39-i 

0.99993 
0.99989 

1.00007 

I.OOOII 

62.422 
62.425 

10.0103 

IO.OII2 

Point  of  maximum  density. 

40 

0.99989 

I.OOOII 

62.425 

IO.OII2 

45 

0.99993 

1.00007 

62.422 

IO.OIO3 

46 

I.OOOOO 

I.OOOOO 

62.418 

IO.OIOI 

(  Same  volume  and  density 
(    as  at  the  freezing  point. 

5° 

1.00.015 

0.99985 

62.409 

IO.OO87 

52-3 

1.00029 

0.99971 

62.400 

IO.OO72 

(  Weight  taken  for  ordi- 
\    nary  calculations. 

55 

1.00038 

0.99961 

62.394 

10.0063 

60 

1.00074 

0.99926 

62.372 

10.0053 

62 

I.OOIOI 

0.99899 

62.355 

10.0000 

Mean  temperature. 

65 

1.00119 

0.99881 

62.344 

9.9982 

70 

1.  00160 

0.99832 

62.313 

9-9933 

75 

1.00239 

0.99771 

62.275 

9.9871 

80 

1.00299 

0.99702 

62.232 

9.980 

85 

1.00379 

0.99622 

62.182 

9.972 

90 

1.00459 

0-99543 

62.133 

9.964 

95 

1.00554 

0.99449 

62.074 

9-955 

100 

1.00639 

0-99365 

62.O22 

9-947 

(  Temperature  of  conden- 
(    ser  water. 

105 

1.00739 

0.99260 

61.960 

9-937 

no 

1.00889 

0.99119 

61.868 

9.922 

"5 

1.00989 

0.99021 

61.807 

9-9I3 

I2O 

1.01139 

0.98874 

61.715 

9.897 

I25 

1.01239 

0.98808 

61.654 

9.887 

130 

1.01390 

0.98630 

61-563 

9-873 

135 

I-OI539 

0.98484 

61.472 

9-859 

140 

1.01690 

0.98339 

61.381 

9.844 

!45 

1.01839 

0.98194 

61.291 

9.829 

150 

1.01989 

0.98050 

61.201 

9.815 

i55 

1.02164 

0.97882 

61.096 

9-799 

160 

1.02340 

0.97714 

60.991 

9-781 

165 

1.02589 

0.97477 

60.843 

9-757 

170 

1.02690 

0.97380 

60.783 

9.748 

i?5 

1.02906 

0.97193 

60.665 

9.728 

180 

1.03100 

0.97006 

60.548 

9.711 

185 

1.03300 

0.96828 

60.430 

9.691 

340  HEAT. 

Table  No.  109  (continued}* 
(Calculated  by  means  of  Rankine's  approximate  formula. ) 


Tempera- 
ture. 

Comparative 
Volume. 

Comparative 
Density. 

Density, 
or  weight  of 
i  cubic  foot. 

Weight  of 
i  gallon. 

Remarkable  Temperatures. 

Fahr. 

Water  at  32° 

Water  at  32° 

Pounds. 

Pounds. 

=  i. 

=  i. 

IQO 

1.03500 

0.96632 

60.314 

9.672 

195 

1.03700 

0.96440 

60.198 

9-654 

200 

1.03889 

0.96256 

6o.o8l 

9-635 

2O5 

1.0414 

0.9602 

59-93 

9.6ll 

210 

1.0434 

0.9584 

59.82 

9-594 

212 

1.0444 

0-9575 

59-76 

9-584 

Boiling  point;  by  formula. 

212 

1.0466 

0-9555 

59-64 

9-565 

(  Boiling  point  ;  by  direct 
(  measurement. 

230 

1.0529 

0.9499 

59-36 

9.520 

250 

1.0628 

0.9411 

58.75 

9.422 

270 

1.0727 

0.9323 

58.18 

9-331 

290 

1.0838 

0.9227 

57-59 

9.236 

(  Temperature  of  steam  of 

298 

1.0899 

0-9*75 

57-27 

9.185 

<     50  Ibs.  effective  pres- 

(    sure  per  square  inch. 

?  Temperature  of  steam  of 

338 

1.1118 

0.8994 

56.14 

9.004 

100  Ibs.  effective  pres- 

(    sure  per  square  inch. 

(  Temperature  of  steam  of 

366 

1.1301 

0.8850 

55-29 

8.867 

150  Ibs.  effective  pres- 

(    sure  per  square  inch. 

(  Temperature  of  steam  of 

39° 

1.1444 

0.8738 

54-54 

8-747 

205  Ibs.  effective  pres- 

(    sure  per  square  inch. 

greater  ratio  than  the  temperature.  The  annexed  table  No.  109  shows 
approximately  the  cubical  expansion,  comparative  density,  and  comparative 
volume  of  water  for  temperatures  between  32°  and  212°  F.,  calculated  by 
means  of  an  approximate  formula  constructed  by  Professor  Rankine  as 
follows : — 


D,  nearly  =  — 

7     t  +  46 1 


500 
500       ^  +  461 

in  which  D0=62.425  Ibs.  per  cubic  foot,  the  maximum  density  of  water, 
and  Dx  =  its  density  at  a  given  temperature  /  F. 

RULE. — To  find  approximately  the  density  of  water  at  a  given  temperature, 
the  maximum  density  being  62.425  Ibs.  per  cubic  foot.  To  the  given  tempera- 
ture in  Fahrenheit  degrees,  add  461,  and  divide  the  sum  by  500.  Again, 
divide  500  by  that  sum.  Add  together  the  two  quotients,  and  divide 
124.85  by  the  sum.  The  final  quotient  is  the  density  nearly. 


EXPANSION   OF   LIQUIDS   BY   HEAT.  341 

The  results  given  by  this  rule  are  very  nearly  exact  for  the  lower  tempera- 
tures, but  for  the  higher  temperatures  they  are  too  great.  For  212°  F.  the 
density  of  water  by  the  rule  is  59.76  Ibs.  per  cubic  foot,  but  it  is  actually 
only  59.64  Ibs.,  showing  an  error  of  about  Vsootii  part  in  excess. 

From  the  table  it  appears  that  the  density  of  water  at  46°  F.,  or  about 
8°  C,  is  the  same  as  at  the  freezing  point,  32°  F.,  and  that  the  temperature 
of  maximum  density,  39°.!  F.,  or  4°  C.,  lies  midway  between  those  tempera- 
tures. The  expansion  of  water  towards  and  down  to  the  freezing  point  is 
i/goooth  part  of  the  volume  at  the  temperature  of  maximum  density.  It 
would  appear  that  in  thus  expanding  from  39°.!  F.  downwards,  the  particles 
of  water  enter  on  a  preparatory  stage  of  separation,  anticipating  the  still 
further  separation  which  ensues  on  the  conversion  of  water  into  the  solid 
state;  for  ice  is  considerably  lighter  than  water  and  floats  on  it,  and  its 
density  is  little  more  than  nine-tenths  that  of  water. 

In  passing  upwards  from  the  freezing  point  towards  higher  temperatures, 
the  increase  of  volume  of  water  by  expansion,  in  parts  of  the  volume  at 
the  freezing  point,  is  as  follows  :— 

Expansion  in 

parts  of  the  volume 

at  32°  F. 

at  5  2°.  3  F.  corresponding  to  the  weight  per  cubic  foot 

(62.4  Ibs.)  usually  taken  for  ordinary  calcu-       per  cent. 

lations .03 

at    62°        the  mean  temperature .10 

at  100°        the  temperature  of  condenser  water .64 

at  212°        the  boiling  point 4.66 

at  298°        the  temperature  of  steam  of  50  Ibs.  effective 

pressure  per  square  inch 9.0 

at  3380        the  temperature  of  steam  of  100  Ibs.  effective 

pressure  per  square  inch 11.2 

at  366°        the  temperature  of  steam  of  150  Ibs.  effective 

pressure  per  square  inch 13.0 

at  390°        the  temperature  of  steam  of  205  Ibs.  effective 

pressure  per  square  inch 14.4 

The  expanded  volume  of  some  liquids  from  32°  to  212°  F.  is  given  in 
table  No.  no;  that  is,  the  apparent  expansion  as  seen  through  glass.  It 
is  shown  that  alcohol  and  nitric  acid  are  the  most  expansible,  and  water 
and  mercury  the  least;  the  former  expand  one-ninth  of  their  initial  volume, 
and  of  the  latter,  water,  as  already  stated,  expands  I/22^-  part,  and  mercury 
x/65th  part  of  their  initial  volumes  respectively.  Observations  on  the 
absolute  expansion  of  mercury  are  added,  and  they  show  that  whereas  the 
apparent  expansion  in  glass  is  z/65th  part,  the  real  expansion  is  I/5sih  part 
of  the  initial  volume. 

No  other  liquid  besides  water  has  a  point  of  maximum  density;  that  is, 
a  point  higher  than  the  freezing  point  of  the  liquid. 


342  HEAT. 

Table  No.  no. — EXPANSION  OF  LIQUIDS  BY  HEAT,  FROM  32°  to  212°  F. 
Apparent  Expansion,  in  Glass. 


LIQUID. 

Volume  at  212°  F. 

Expansion  in 
Vulgar  Fractions. 

Alcohol  

volume  at  32°  F.=i. 
.1100 
.1100 

.0800 
.0800 
.0700 
.0700 
.0600 
.0600 
.0500 
.0466 

1.0154 

volume  at  32°  F.=i. 

v9 

•£. 

'A, 

VM 
VM 

f/f 

•A, 

'/•o 

•A* 
'As 

Nitric  Acid  

Olive  Oil  

Linseed  Oil         .    . 

Turpentine       

Sulphuric  Ether  

Hydrochloric  Acid  (density  1.137) 

Sulphuric  Acid  (density  1.850).  . 

Water  saturated  with  Sea  Salt  

Water  

Mercury       

Absolute  Expansion  of  Mercury. 
Volume  at       Expan- 
212°  F.            sion. 

Mercury,  from    32°  to  212°  F.  (     o°  to  100°  C.),  Dulongand  Petit,  1.0180180     i/s5.s 
Do.      from  212°  to  392°  F.  (100°  to  200°  C.),                do.             1.0184331     */  54.25 
Do.      from  392°  to  572°  F.  (200°  to  300°  C.),                do.             1.0188679     i/S3 
Do.      from    32°  to  212°  F.  (     o°  to  100°  C.),           Regnault,        1.0181530     Vss-" 

EXPANSION  OF  GASES  BY  HEAT. 

Gases  are  divisible  into  two  classes — permanent  gases  and  vapours. 
Gases  for  which  great  pressure  and  extremely  low  temperatures  are  neces- 
sary to  reduce  them  to  the  liquid  form,  are  called  permanent  gases,  and 
those  which  exist  in  the  fluid  state  under  ordinary  temperatures,  are  called 
vapours. 

The  influence  of  heat  in  expanding  a  permanent  gas  maintained  under  a 
constant  pressure,  is  such  that,  for  equal  increments  of  temperature,  the 
increments  of  volume  by  expansion  are  also  equal  or  very  nearly  equal;  in 
other  words,  the  gas  expands  uniformly,  or  very  nearly  uniformly,  in  pro- 
portion to  the  rise  of  temperature. 

Again,  it  has  been  observed  that  when  the  volume  of  permanent  gases  is 
maintained  constant,  the  pressure  increases  uniformly,  or  nearly  uniformly, 
with  an  increase  of  temperature. 

A  perfect  or  ideal  gas  is  one  which,  under  a  constant  pressure,  expands 
with  perfect  uniformity  in  proportion  to  the  rise  of  temperature;  and  of  which, 
also,  when  confined  to  a  constant  volume,  the  pressure  increases  with  per- 
fect uniformity  in  proportion  to  the  rise  of  temperature. 

When  the  temperature  of  atmospheric  air  is  raised  from  32°  to  212°  F., 
the  following  are  the  total  increments  of  volume  or  of  pressure,  according 
to  the  treatment,  as  determined  by  Regnault,  when  the  volume  at  32°  is 
taken  as  i : — 


EXPANSION   OF  GASES   BY   HEAT. 


343 


AIR.  TEMPERATURE.  INCREMENT. 

Pressure  constant 32°  to  212°  F Volume  increased  from  i  to  1.3670. 

Volume  constant 32°  to  212°  F Pressure  increased  from  i  to  1.3665. 

Showing  that  the  increase  of  pressure,  .3665,  with  a  constant  volume,  is 
sensibly  the  same  as,  though  less  than,  the  expansion  or  increase  of  volume, 
.3670,  when  the  pressure  is  constant. 

The  table  No.  1 1 1  gives  the  expansion  and  the  increase  of  pressure,  for 
several  gases,  when  raised  from  32°  to  212°  F.: — 

Table  No.  in. — EXPANSION  AND  PRESSURE  OF  GASES  RAISED  FROM 

32°  to  212°  F. 
(Regnault.) 


GASES. 

Expansion  of  Gases  under  i  Atmosphere. 

Increase  of  Pressure 
of  Gases  under  a  Con- 
tant  Volume. 

Final  Volume  at  212°  F. 

Expansion  at  212°  F., 
in  Common  Fractions. 

Final  Pressure  at  212°. 

Atmospheric  Air  
Hydrogen 

Initial  volume  at  32°=  i. 
1.3670 
1.3661 

1.3669 
I.37IO 

I.37I9 
L3877 
I-39°3 

Initial  volume  at  32°=!. 
'A.7<5 
'/..n 

I/ 
/2.73 

'A.7I 
XA.7. 

x/a.6x 
xA.6o 

Initial  pressure  at  32°=  i. 

1.3665 
1.3667 
1.3668 
1.3667 
1.3688 
1.3676 
1.3829 
1.3843 

Nitrogen  

Carbonic  Oxide  

Carbonic  Acid  

Nitrous  Oxide 

Cyanogen 

Sulphurous  Acid  

Table  No.  112. — EXPANSION  OF  GASES  RAISED  FROM  32°  TO  212°  F., 
UNDER  DIFFERENT  PRESSURES,  THESE  PRESSURES  REMAINING  CON- 
STANT FOR  EACH  OBSERVATION. 

(Regnault.) 


Gas. 

Pressure. 

Volume  at  212°. 

Air 

Millimetres. 
760 

2525 
262O 

Atmospheres. 
I.OO 
3-32 
3'45 

Volume  at  32°  F.  =  i. 
1.36706 
1.36944 
1.36964 

Hydrogen 

760 
2545 

I.OO 

3-35 

1.36613 
1.36616 

Carbonic  Acid 

760 

2520 

I.OO 

3.32 

L37099 

L38455 

Sulphurous  Acid 

760 
980 

I.OO 

1.16 

L3903 
1.3980 

344  HEAT. 

The  first  part  of  the  table,  No.  in,  on  the  expansion  of  gases  by  heat, 
shows  that  the  expansion,  which  is  a  little  more  than  a  third  of  the  initial 
volume,  is  nearly  the  same  for  air,  hydrogen,  and  carbonic  oxide,  which  are 
sensibly  perfect  gases,  and  have  never  been  liquefied.  On  the  contrary, 
carbonic  acid,  cyanogen,  and  sulphurous  acid  have  a  greater  enlargement  of 
volume  than  those  gases,  and  they  are  gases  which  may  easily  be  liquefied. 

The  second  part  of  the  table,  column  4,  shows  that,  when  the  volume 
is  constant,  the  pressure  is  increased  nearly  in  the  same  proportion  as  the 
volume  is  increased,  when  the  pressure  is  constant.  This  nearness  of  the 
proportions  is  particularly  close  in  the  cases  of  the  three  sensibly  perfect 
gases, — air,  hydrogen,  and  carbonic  oxide. 

The  next  table,  No.  112,  contains  the  results  of  Regnault's  experiments 
on  the  expansion  of  gases  from  32°  to  212°  F.,  under  various  constant 
pressures  of  from  i  to  3^  atmospheres.  It  is  shown  that  the  expansions  of 
air  and  of  hydrogen  are  sensibly  the  same,  whether  the  constant  pressure  be 
i  atmosphere  or  between  3  and  4  atmospheres ;  whilst  the  expansions  of 
carbonic  acid  and  sulphurous  acid  are  higher  at  the  higher  pressure. 

The  deductions  of  Regnault,  from  his  experiments,  comprised  the 
following  principles : — 

That  for  air,  and  all  other  gases  except  hydrogen,  the  coefficient  of 
dilatation,  or  the  increment  of  expansion  for  one  degree  rise  of  temperature, 
increases  to  some  extent  with  their  density. 

That  all  gases  possess  the  same  coefficient  of  dilatation  when  in  a  state 
of  extreme  tenuity;  but  that  this  law  is  departed  from  as  gases  become 
dense. 

Adopting,  nevertheless,  the  mean  of  the  results  of  the  experiments  of  M. 
Regnault  and  of  M.  Rudberg,  the  expansion  of  one  volume  of  air  measured 
at  32°  F.,  when  heated  to  212°  F.,  under  a  constant  pressure,  will,  for  future 
calculation,  be  taken  as  equal  to  0.365 ;  the  ratio  of  the  initial  to  the 
expanded  volume  being  as  i  to  1.365.  As  the  expansion  is  uniform  with 
the  rise  of  the  temperature  through  180°,  the  expansion  for  each  degree 
Fahr.  is — 

.365  +  180=—  L_, 
493-2 

the  volume  at  32°  F.  being  =  i.  The  same  uniform  rate  of  expansion  holds 
sensibly  for  temperatures  higher  than  212°;  it  has  been  verified  experi- 
mentally up  to  700°  F.,  under  one  atmosphere.  It  is  inferred  that,  con- 
versely, air  would  contract  uniformly  under  uniform  reductions  of  temperature 
below  32°  F.,  until,  on  arriving  at  493°. 2  below  the  freezing  point,  or 
461°.  2  F.  below  zero,  the  air  would  be  reduced  to  a  state  of  collapse, 
without  elasticity.  This  point  in  the  Fahrenheit  scale  has  thus  been 
adopted  as  that  of  absolute  zero,  standing  at  the  foot  of  the  natural  scale  of 
temperature;  and  the  temperature,  measured  from  absolute  zero,  or 
—  461°, 2  F.,  is  called  the  absolute  temperature. 

Accordingly,  if  a  given  weight  of  air  at  o°  F.  be  raised  in  temperature  to 
4-461°  F.,  under  a  constant  pressure,  it  is  expanded  to  twice  its  original 
volume;  and  if  heated  from  o°  F.  to  twice  461°,  or  922°,  its  original 
volume  is  trebled. 

In  brief^  it  follows  that,  sensibly, 


EXPANSION   OF  GASES   BY  HEAT.  345 

i  st.  The  pressure  of  air  varies  inversely  as  the  volume  when  the  tempera- 
ture is  constant. 

2d.  The  pressure  varies  directly  as  the  absolute  temperature  when  the 
volume  is  constant. 

3d.  The  volume  varies  as  the  absolute  temperature  when  the  pressure  is 
constant. 

4th.  The  product  of  the  pressure  and  volume  is  proportional  to  the 
absolute  temperature. 

The  absolute  zero-point  by  different  thermometrical  scales  is  as  follows  : — 

Reaumur —  219°.  2 

Centigrade -  274° 

Fahrenheit -  46i°.2 

To  simplify  calculation,  the  decimal  is  usually  dropped  from  the  Fahrenheit 
temperature,  which  is  taken  as  -461°. 

The  foregoing  laws  do  not  apply  exactly  to  the  expansion  and  contraction 
of  the  more  easily  condensable  gases,  for  these,  as  they  approach  the  point 
of  liquefaction,  become  sensibly  more  compressible  than  air.  Oxygen, 
nitrogen,  hydrogen,  nitric  oxide,  and  carbonic  oxide  follow  the  same  ratio  of 
compression  as  that  of  air,  being  incondensable  gases,  at  least  as  far  as  100 
atmospheres  of  pressure.  Sulphurous  acid,  ammoniacal  gas,  carbonic  acid, 
and  protoxide  of  nitrogen,  which  have  been  proved,  on  the  contrary,  to  be 
condensable,  become  sensibly  more  compressible  than  air  when  they  are 
reduced  to  one-third  or  one-fourth  of  their  original  volume  at  atmospheric 
pressure.  Carbonic  acid,  under  five  atmospheres,  occupies  only  97  per 
cent,  of  the  volume  which  air  occupies  under  the  same  pressure ;  and  under 
forty  atmospheres,  near  the  condensing  point,  it  occupies  only  74  per  cent., 
or  barely  three-fourths  of  the  volume  of  air  at  the  same  pressure.  It  has, 
nevertheless,  been  established  that  all  gases,  at  some  distance  from  the 
point  of  maximum  density  for  the  pressure,  beyond  which  point  they  must 
condense,  sensibly  follow  the  first  law  above  recited,  according  to  which  the 
pressure  and  the  density  vary  directly  as  each  other,  when  the  temperature 
is  constant.  With  such  limitations,  they  rank  as  perfect  gases. 

The  table  No.  113  contains  examples  of  the  progressive  pressures  required 
to  compress  air,  nitrogen,  carbonic  acid,  and  hydrogen,  into  one-twentieth 
of  their  original  volumes,  founded  on  experiments  made  by  M.  Regnault. 
The  pressures  are  expressed  in  metres  of  mercury,  the  pressure  of  a  column 
of  mercury  one  metre  high  being  equal  to  19.34  Ibs.  per  square  inch.  The 
table  shows  that  hydrogen  is  the  most  perfect  type  of  gaseity.  When 
compressed  to  a  twentieth  of  its  original  volume,  it  supports  something 
more  than  twenty  times  the  original  pressure.  Air,  on  the  contrary,  requires 
a  quarter  of  a  metre  less  than  20  metres  of  pressure ;  nitrogen  requires  a 
fifth  of  a  metre  less;  and  carbonic  acid,  like  an  overloaded  spring,  3^ 
metres  less. 


346 


HEAT. 


Table  No.  113. — COMPRESSION  OF  GASES  BY  PRESSURE  UNDER  A 
CONSTANT  TEMPERATURE. 


Ratio  of  the 
original  volume 
to  the  reduced 
volume. 

Pressure  in  Metres  of  Mercury  for 

Air. 

Nitrogen. 

Carbonic  Acid. 

Hydrogen. 

Metres. 

Metres. 

Metres. 

Metres. 

I 

I.OOO 

I.OOO 

I.OOO 

I.OOO 

2 

1.998 

1.997 

1.983 

2.OOI 

4 

3.987 

3'992 

3.897 

4.007 

6 

5.970 

5.980 

5-743 

6.018 

8 

7.946 

7.964 

7.5*9 

8.034 

10 

9.916 

9-944 

9.226 

10.056 

12 

11.882 

11.919 

10.863 

12.084 

14 

I3.845 

13.891 

12.430 

14.119 

16 

15.804 

15.860 

13.926 

l6.l62 

18 

17.763 

17.825 

IS-35I 

I8.2II 

20 

I9.72O 

19.789 

16.705 

20.269 

Note. — 20  metres  of  mercury  are  equal  to  a  pressure  of  386.8  Ibs.  per  square  inch,  or 
26.3  atmospheres. 

RELATIONS   OF   THE   PRESSURE,   VOLUME,  AND   TEMPERATURE   OF   AIR 

AND  OTHER  GASES. 

In  accordance  with  the  relations  of  pressure,  volume,  and  temperature 
above  stated,  it  is  found  that  air  and  other  perfect  gases,  and,  within 
practical  limits,  the  permanent  gases  generally,  are  expanded  by  heat  at  the 
rate  of  J/46i  Part  of  their  volume  at  o°  F.  for  each  degree  of  temperature, 
under  a  constant  pressure.  If  the  volume  at  the  freezing  point,  32°  F.,  be 
taken  as  the  point  of  departure,  the  denominator  of  the  fraction  is 
(461°  +  32°  =  )  493°,  and  the  expansion  is  at  the  rate  of  I/493  part  of  the 
volume  at  32°  F.  for  each  degree  of  temperature.  In  general,  for  any 
other  initial  temperature  the  denominator  of  the  fraction  showing  the  rate 
of  expansion  for  each  degree  is  found  by  adding  461°  to  the  initial  tempera- 
ture. But,  for  convenience  of  calculation,  the  initial  temperature  is  usually 
taken  at  o°  F. 

Similarly,  the  pressure  of  air  having  a  given  constant  volume,  is  increased 
by  heat  at  the  rate  of  */46l  part  of  the  pressure  at  o°  F. 

The  fraction  of  expansion  when  the  pressure  is  constant,  and  the  fraction 
of  pressure  when  the  volume  is  constant,  for  each  degree  of  temperature  by 
Fahrenheit's  scale  above  o°,  is,  then, 

i 
46? 

and  the  same  fraction  expresses  the  rate  of  contraction  of  volume  for  each 
degree  of  temperature  below  o°  F. 

A  number  of  proportions  and  rules  for  the  relations  of  the  pressure, 
volume,  and  temperature  of  a  constant  weight  of  a  gas  are  readily  deduced 
from  the  above  defined  ratios. 


PRESSURE,   ETC.,   OF  AIR  AND   OTHER  GASES.  347 

i.  When  the  pressure  is  constant,  the  volume  varies  as  the  absolute 
temperature;  or, 

V  :  V  :  :  /  +  461   :  f  +  461,  and 


in  which  V  is  the  volume  of  the  air  or  other  gas  at  the  temperature  /,  and 
V  is  the  volume  at  the  temperature  f.  Whence  the  rule  — 

RULE  i.  To  find  the  volume  of  a  constant  weight  of  air  or  other  permanent 
gas,  at  any  other  temperature,  when  the  volume  at  a  given  temperature 
is  known,  the  pressure  being  constant.  Multiply  the  given  volume  by  the 
new  absolute  temperature,  and  divide  by  the  given  absolute  temperature. 
The  quotient  is  the  new  volume. 

Note.  —  The  absolute  temperature  is  found  by  adding  461°  to  the 
temperature  indicated  by  the  Fahrenheit  thermometer. 

As  a  common  case  of  the  above  rule,  air  may  be  taken  at  the  mean 
temperature,  62°  F.  The  increased  volume,  by  expansion  by  heat,  taking 
the  initial  volume  =  i,  is  found  by  substitution  and  reduction  to  be  as 
follows  :  — 


523 


RULE  2.  To  find  the  increased  volume  of  a  constant  weight  of  air,  of 
which  the  initial  volume  =  i,  taken  at  62°  F.,  heated  under  a  constant 
pressure,  to  a  given  temperature.  To  the  given  temperature  add  461,  and 
divide  the  sum  by  523.  The  quotient  is  the  increased  volume  by  expan- 
sion. 

2.  When  the  temperature  of  a  constant  weight  of  air,  or  other  gas,  is 
constant,  the  volume  varies  inversely  as  the  pressure  ;  or, 

V  :  V  ::/:/,  and 

.....  (3) 

in  which  V  and  V  are  the  volumes  respectively  at  the  pressures/  and/'. 

RULE  3.  To  find  the  volume  of  a  constant  weight  of  air  or  other  permanent 
gas,  for  any  pressure,  when  the  volume  at  a  given  pressure  is  known, 
the  temperature  remaining  constant.  Multiply  the  given  volume  by  the 
given  pressure,  and  divide  by  the  new  pressure.  The  quotient  is  the  new 
volume. 

3.  When  the  pressure  and  temperature  of  a  constant  weight  of  air  or 
other  gas  both  change,  the  volume  varies  in  the  compound  ratio  of  the 
absolute  temperature  directly,  and  the  pressure  inversely  ;  or, 


V  :  V  ::/ 

or  V//(^46i)  =  V/(/'  +  46i),  and 

/    x 
" 


348  HEAT. 

RULE  4.  To  find  the  volume  of  a  constant  weight  of  air  or  other  permanent 
gas  for  any  other  pressure  and  temperature,  when  the  volume  is  known 
at  a  given  pressure  and  temperature.  Multiply  the  given  volume  by  the 
given  pressure,  and  by  the  new  absolute  temperature,  and  divide  by  the 
new  pressure,  and  by  the  given  absolute  temperature.  The  quotient  is  the 
new  volume. 

4.  When  the  volume  and  temperature  of  a  constant  weight  of  air  or 
other  gas  both  change,  the  pressure  varies  in  the  compound  ratio  of  the 
absolute  temperature  directly,  and  the  volume  inversely. 

p  :/  ::  V'(/  +  46i)  :  V 
or  V'/  (/  +  46i)  =  V/  (/'  +  46i),  and 


RULE  5.  To  find  the  pressure  of  a  constant  weight  of  air  or  other  permanent 
gas  for  any  other  volume  and  temperature,  when  the  pressure  is  known  for 
a  given  volume  and  temperature.  Multiply  the  given  pressure  by  the 
given  volume,  and  by  the  new  absolute  temperature,  and  divide  by  the 
new  volume,  and  by  the  given  absolute  temperature.  The  quotient  is  the 
new  pressure. 

For  the  common  case,  when  the  initial  temperature  is  62°  F.,  and  the 
initial  pressure  is  14.7  Ibs.  per  square  inch,  the  formula  (5)  becomes,  by 
substitution  and  reduction, 


35.58  V  ' 

RULE  6.  To  find  the  pressure  of  a  constant  weight  of  air  or  other  gas  taken 
at  62°  F.,  and  at  14.7  Ibs.  pressure  per  square  inch,  with  a  given  volume,  for 
any  other  volume  and  temperature.  Multiply  the  initial  volume  by  the 
final  temperature  plus  461,  and  divide  the  product  by  the  final  volume, 
and  by  35.58.  The  quotient  is  the  new  pressure  in  Ibs.  per  square  inch. 

When  the  volume  is  constant,  with  an  initial  temperature  of  62°  F.,  and 
an  initial  pressure  of  14.7  Ibs.  per  square  inch,  the  above  formula  (6)  is 
simplified  thus  :  — 


RULE  7.  To  find  the  pressure  of  a  constant  weight  of  air  or  other  gas  taken 
at  62°  F.,  and  at  14.7  Ibs.  pressure  per  square  inch,  with  a  constant  volume, 
for  a  given  temperature.  Add  461  to  the  given  temperature,  and  divide 
the  sum  by  35.58.  The  quotient  is  the  pressure  in  Ibs.  per  square  inch. 

5.  The  mutual  relations  of  pressure,  volume,  and  temperature  are  con- 
densed in  the  following  formula  :  — 

V/=/  +  46i,  ......................................  (a) 

the  product  of  the  volume  and  pressure  of  a  constant  weight  of  air  being 
proportional  to  the  absolute  temperature.  And,  as  that  product  bears 
always  the  same  ratio  to  the  absolute  temperature,  an  equation  may  be 


PRESSURE,   ETC.,    OF   AIR   AND   OTHER   GASES. 


349 


formed    between    them   by   multiplying   the   absolute   temperature  by  a 
coefficient,  which  may  be  put  =  a.     Then  — 


(b) 


that  is,  the  product  of  the  volume  and  pressure  of  a  constant  weight  of  air 
or  other  permanent  gas,  is  equal  to  the  absolute  temperature  multiplied  by 
a  constant  coefficient,  which  is  to  be  determined  for.  each  gas  according  to 
its  density. 

Special  Rules  for  One  Pound  Weight  of  a  Gas. 

The  application  of  formula  (b)  to  a  particular  constant  weight  of  gas,  will 
suffice  for  many  purposes.  Let  the  constant  weight  be  one  pound  of  gas. 
To  settle  the  coefficients  for  the  different  gases,  take,  for  example,  the 
temperature  32°  F.,  giving  an  absolute  temperature  of  493°,  and  the  pressure 
one  atmosphere,  or  14.7  Ibs.  per  square  inch.  The  volume  of  one  pound 
of  air  at  this  temperature  and  this  pressure  is  as  before  stated,  12.387  cubic 
feet.  Substitute  these  values  for  V,  /,  /,  in  the  formula  (/>),  then  — 

12.387  x  14.7=0x493, 


whence  the  coefficient,  «,  for  air  is — 

a  =  -36935,  or 

and  the  formula  (b)  becomes,  for  air, 

^  +  461 . 

2.7074'" 


2.7074 


Table  No.  114. — OF  COEFFICIENTS  OR  CONSTANTS,  a,  IN  THE  EQUATION 
(b)  FOR  THE  RELATIONS  OF  THE  VOLUME,  PRESSURE,  AND  TEM- 
PERATURE OF  GASES;  NAMELY,  V  p  =  a  (t  +  461). 


Name  of  gas. 

Volume  of  one  pound  of 
gas,  at  32°  F.,  under 
one  atmosphere. 

Value  of  coefficient  a 

Hydrogen  

cubic  feet. 

178  S-z 

522200   or  */    o 

Gaseous  steam  

A  /u.tj^ 

TQ  QT  7 

•OJ^UUJ  U1     70.1875 
O  ^  Q  2  7  2     Or  x  /    KS 

Nitrogen  

*af»3F*j 

12  722 

o  27o  27    or  */  f. 

Olefiant  gas 

1  2  ^80 

WtO/Vo/J   U1     72.6359 

O27co6   or  J/  &P.P. 

Air              

12  *87 

o  26031?   or  J/o  ~ 

Oxvsen 

**»«S^f 

1  1  2O5 

Wtowyoo>  vi  72.7074 

O    23AO6       Or    X/o  nn-ir 

Carbonic  acid  (ideal)*  

8  1^7 

**TOOV*J  **•     72'9935 
O  24222    Or  IA  TTT>i 

Do.       do.  (actual)  

r**3l 

8  101 

o  241^1;  or  IA  toon 

Ether  vapour*              ... 

A.  7  77 

O    IA2A6      Or    */tmne 

Vapour  of  mercury*  

T-/  /  / 
I  776 

o  0^206  or  I/TQ  Q^S 

i.  /   /V» 

*  The  densities  are  computed  by  Rankine  for  the  ideal  condition  of  perfect  gas. 


35°  HEAT. 

that  is  to  say,  the  volume  of  one  pound  of  air,  multiplied  by  the  pressure 
per  square  inch,  is  equal  to  the  absolute  temperature  divided  by  the 
constant  2.7074. 

To  adapt  the  formula  (&)  for  other  gases,  the  respective  coefficients,  or 
constants,  are  found  in  the  same  manner,  in  terms  of  the  volume  of  one 
pound  of  each  gas,  at  32°  F.,  under  one  atmosphere  of  14.7  Ibs.  per  square 
inch.  They  are  given  in  table  No.  114. 

6.  The  volume  of  one  pound  of  air  at  any  pressure  and  any  temperature 
is  deduced  as  follows: — 

4.      I         .  f.  T 

(8) 


2- 7074  P 

RULE  8. — To  find  the  volume  of  one  pound  of  air,  of  a  given  temperature 
and  pressure.  Divide  the  absolute  temperature  by  the  pressure  in  Ibs.  per 
square  inch,  and  by  2.7074.  The  quotient  is  the  volume  in  cubic  feet. 

For  the  ordinary  case  when  the  pressure  is  constant  at  14.7  Ibs.  per 
square  inch,  the  formula  (8)  becomes,  by  substituting  and  reducing, 


RULE  9.  —  To  find  the  volume  of  one  pound  of  air  under  14.7  Ibs.  pressure 
per  square  inch,  at  a  given  temperature.  Add  46  1  to  the  temperature,  and 
divide  the  sum  by  39.80.  The  quotient  is  the  volume  in  cubic  feet. 

7.  The  pressure  of  one  pound  of  air  of  any  volume,  and  at  any  tempera- 
ture, is  found  as  follows  :  — 

/+  461 

.................................  (10) 


..... 
*      2.7074V 

RULE  10.  —  To  find  the  pressure  of  one  pound  of  air,  of  a  given  temperature 
and  volume.  Divide  the  absolute  temperature  by  the  volume  and  by  2.7074. 
The  quotient  is  the  pressure  in  Ibs.  per  square  inch. 

8.  The  temperature  of  one  pound  of  air  of  any  volume  and  pressure  is 
found  as  follows  :  — 

^2.7074  V/-46i   ...........................  (n) 

RULE  1  1.  —  To  find  the  temperature  of  one  pound  of  air,  of  a  given  volume 
and  pressure.  Multiply  the  volume  by  the  pressure  in  pounds  per  square 
inch,  and  also  by  2.7074;  subtract  461  from  the  product.  The  remainder 
is  the  temperature. 

9.  The  density  of  air  is  inversely  as  the  volume,  and  is  expressed  by  an 
inversion  of  the  formula  (8),  for  the  volume;  thus,  putting  D  for  the  density, 
or  the  weight  in  pounds  of  one  cubic  foot  of  air  — 


RULE  12.  —  To  find  the  density  of  air,  at  a  given  temperature  and  pressure. 
Multiply  the  pressure  in  pounds  per  square  inch  by  2.7074,  and  divide 
by  the  absolute  temperature.  The  quotient  is  the  density,  or  weight  in 
pounds  of  one  cubic  foot. 


VOLUME,   DENSITY,   AND   PRESSURE   OF  AIR. 


351 


Table  No.  115. — VOLUME,  DENSITY,  AND  PRESSURE  OF  AIR  AT  VARIOUS 

TEMPERATURES. 


Temperature. 

Volume  of  one  pound  of  air  at 
constant  atmospheric  pressure, 
14.  7  Ibs.  per  square  inch. 
Datum  —  Volume  at  62°  F.  =  i. 

Density,  or  weight 
of  one  cubic  foot  of 
air  at  atmospheric 
pressure. 

Pressure  of  a  given  weight  of  air 
having  a  constant  volume. 
Datum  —  Atmospheric  pressure  at 
62°  F.  =  i. 

Fahrenheit. 

cubic  feet. 

comparative 
volume. 

pounds. 

pounds  per 
square  inch. 

comparative 
pressure. 

0° 

H.583 

.881 

.086331 

12.96 

.881 

32 

12.387 

•943 

.080728 

13.86 

-943 

40 

12.586 

.958 

•079439 

14.08 

.958 

50 

I2.§4O 

•977 

.077884 

14,36 

•977 

62 

13.141 

I.OOO 

.076097 

14.70 

I.OOO 

70 

13.342 

1.015 

.07495° 

14.92 

1.015 

80 

13.593 

1.034 

.073565 

15.21 

1.034 

90 

13-845 

1.054 

.072230 

15.49 

1.054 

IOO 

14.096 

1.073 

.070942 

15-77 

1.073 

I2O 

14.592 

I.  Ill 

.068500 

16.33 

I.  Ill 

I4O 

15.100 

1.149 

.O6622I. 

16.89 

1.149 

160 

15-603 

1.187 

.064088 

17.50 

1.187 

180 

16.106 

1.226 

.062090 

18.02 

1.226 

200 

16.606 

1.264 

.O602IO 

18.58 

1.264 

210 

16.860 

1.283 

.059313 

18.86 

1.283 

212 

16.910 

1.287 

•°59I35 

18.92 

1.287 

22O 

17.111 

1.302 

.058442 

19.14 

1.302 

230 

17.362 

1.321 

.057596 

19.42 

1.321 

240 

17.612 

1.340 

•056774 

19.70 

1.340 

250 

17.865 

1-359 

•055975 

19.98 

1-359 

260 

18.116 

1-379 

.055200 

20.27 

1.379 

270 

18.367 

I.398 

•054444 

20.55 

1.398 

280 

18.621 

1.417 

.053710 

20.83 

1.417 

290 

18.870 

1.436 

•052994 

21.  II 

1.436 

300 

19.121 

r"455 

.052297 

21.39 

L455 

320 

19.624 

J.493 

•050959 

21.95 

L493 

340 

20.126 

!-532 

.049686 

22.51 

1.532 

360 

20.630 

I-57o 

.048476 

23.08 

1-570 

380 

21.131 

i.  608 

•047323 

23.64 

i.  608 

4OO 

21.634 

1.646 

.046223 

24.2O 

1.646 

425 

22.262 

1.694 

.044920 

24.90 

1.694 

45° 

22.890 

1.742 

.043686 

25.6l 

1.742 

475 

23-518 

1.789 

.042520 

26.31 

1-789 

500 

24.146 

1.837 

.041414 

27.01 

1-837 

525 

24.775 

1.885 

.040364 

27.71 

1.885 

550 

25-403 

I«933 

•039365 

28.42 

1-933 

575 

26.031 

1.981 

.038415 

29.12 

1.981 

600 

26.659 

2.029 

.037510 

29.82 

2.029 

352 


HEAT. 
Table  No.  115  (continued). 


Temperature. 

Volume  of  one  pound  of  air  at 
constant  atmospheric  pressure, 
14.  7  Ibs.  per  square  inch. 
Datum—  Volume  at  62°  F.  =  i. 

Density,  or  weight 
of  one  cubic  foot  of 
air  at  atmospheric 
pressure. 

Pressure  of  a  given  weight  of  air 
having  a  constant  volume. 
Datum  —  Atmospheric  pressure  at 

Fahrenheit. 

cubic  feet. 

comparative 
volume. 

pounds. 

pounds  per 
square  inch. 

comparative 
pressure. 

650 

70O 

750 
800 

27.915 
29.172 

30.428 
3L685 

2.124 

2.22O 

2.315 
2.4II 

.035822 
.034280 
.032865 
.031561 

3I-23 
32.63 

34-04 
35-44 

2.124 
2.22O 

2.315 
2.4II 

850 
QOO 

95° 

IOOO 

32.941 
34.197 

35-453 
36.710 

2.507 
2.6O2 
2.698 
2.793 

.030358 
.029242 
.O282O6 
.027241 

36.85 
38.25 
39.66 
4I.O6 

2.507 
2.6O2 
2.698 
2-793 

1500 

2000 
2500 
3OOO 

49.274 
61.836 
74.400 
86.962 

3-749 
4.705 
5.661 

6.618 

.020295 
.Ol6l72 
.013441 
.OII499 

55-12 
69.17 
83.22 
97.28 

3-749 
4.705 
5.661 
6.618 

Note  to  Rules  8,  10,  1 1,  12. — The  coefficients  or  constants  for  other  gases, 
in  the  application  of  the  preceding  five  formulas  and  rules,  are  given  in 
table  No.  114. 

The  table  No.  115  contains  the  volume,  density,  and  pressure  of  air  at 
various  temperatures  from  o°  to  3000°  F.,  starting  from  62°  F.  and  14.7  Ibs. 
per  square  inch  respectively  as  unity  for  the  proportional  volumes  and  pres- 
sures. The  second  column  of  the  table,  containing  the  volumes  of  one  pound 
of  air  at  different  temperatures,  was  calculated  by  means  of  the  formula  (9), 
page  350.  The  third  column,  of  comparative  volumes,  the  volume  at  62° 
F.  being  =  i,  was  calculated  by  means  of  formula  (2),  page  347.  The  fourth 
column,  of  density,  contains  the  reciprocals  of  the  volumes  in  column  2, 
but  it  is  calculable  independently  by  means  of  formula  (12),  page  350.  The 
fifth  column,  of  pressures,  due  to  the  temperatures,  was  calculated  by  means 
of  formula  (7),  p.  348.  The  sixth  column  contains  these  pressures  expressed 
comparatively,  the  atmospheric  pressure,  14.7  Ibs.  per  square  inch,  being 
taken  as  i. 

SPECIFIC    HEAT. 

The  specific  heat  of  a  body  signifies  its  capacity  for  heat,  or  the  quantity 
of  heat  required  to  raise  the  temperature  of  the  body  one  degree  Fahrenheit, 
compared  with  that  required  to  raise  the  temperature  of  a  quantity  of 
water  of  equal  weight  one  degree.  The  British  unit^of  heat  is  that  which 
is  required  to  raise  the  temperature  of  one  pound  of  water  one  degree,  from 
32°  F.  to  33°  F.;  and  the  specific  heat  of  any  other  body  is  expressed  by 
the  quantity  of  heat,  in  units,  necessary  to  raise  the  temperature  of  one 
pound  weight  of  such  body  one  degree. 

The  specific  heat  of  water  at  32°  F.  is  represented  by  i,  or  unity,  and 
there  are  very  few  bodies  of  which  the  specific  heat  equals  or  exceeds  that 
of  water.  Specific  heats  are,  therefore,  almost  universally  expressible  by 
fractions  of  a  unit. 


SPECIFIC   HEAT   OF   WATER.  353 

It  is  necessary  to  fix  a  standard  of  temperature,  such  as  the  freezing 
point,  for  the  datum  of  specific  heat,  as  the  specific  heat  of  water  is  not 
exactly  the  same  at  different  parts  of  the  scale  of  temperatures,  but  increases 
in  an  appreciable  degree,  as  well  as  in  an  increasing  ratio,  as  the  tempera- 
ture rises.  For  temperatures  not  higher  than  80°  or  90°  F.,  the  quantity  of 
heat  required  to  raise  the  temperature  of  water  one  degree  is  sensibly 
constant ;  at  86°  F.,  it  is  not  above  one-fifth  per  cent,  in  excess  of  that  at 
the  freezing-point.  At  212°  F.,  it  is  about  1^3  per  cent,  in  excess  of  that 
at  32°  F.  Above  212°  F.,  it  increases  more  rapidly;  at  302°,  it  is  2^  per 
cent,  more  than  at  32°,  and  at  402°,  it  is  4.^4  per  cent.  more. 

The  average  specific  heat  of  water  between  the  freezing  and  the  boiling 
points  is  1.005,  or  one-half  per  cent,  more  than  the  specific  heat  at  the 
freezing  point. 

It  follows  from  the  increasing  specific  heat  of  water,  as  the  temperature 
rises,  that  the  consumption  of  heat  in  raising  the  temperature  is  slightly 
greater  expressed  in  units  than  in  degrees  of  temperature.  To  raise,  for 
example,  one  pound  of  water  from  o°  to  100°  C.,  or  from  32°  to  212°  F., 
there  are  required  100.5  ^-  units,  or  180.9  F.  units,  of  heat. 

The  specific  heats  of  water  in  the  solid,  liquid,  and  gaseous  state  are 
grouped  as  follows  : — 

Ice o.  504 

Water i.ooo 

Gaseous  Steam  0.622 

showing  that  in  the  solid  state,  as  ice,  the  specific  heat  of  water  is  only  half 
that  of  liquid  water ;  and  that,  in  the  gaseous  state,  it  is  a  little  more  than 
that  of  ice,  or  barely  five-eighths  of  that  of  liquid  water. 

The  specific  heat  of  all  liquid  and  solid  substances  is  variable,  increasing 
sensibly  as  the  temperature  rises,  and  the  specific  heats  of  such  bodies,  as 
tabulated,  are  not  to  be  taken  as  exact  for  all  temperatures,  but  rather  as 
approximate  average  values,  sufficiently  near  for  practical  purposes.  The 
specific  heat  of  the  same  body  is,  however,  nearly  constant  for  temperatures 
under  212°  F. 

The  specific  heats  of  such  gases,  on  the  contrary,  as  are  perfectly  gaseous, 
or  nearly  so,  do  not  sensibly  vary  with  density  or  with  temperature. 

For  the  same  body,  the  specific  heat  is  greater  in  the  liquid  than  in  the 
solid  state.  For  example  : — 

Liquid.  Solid. 

Water  (specific  heat)  i.ooo  0.504 

Bromine         „  o.m  0.084 

Mercury         „  0.0333         0.0319 

M.  Regnault  has  verified,  by  numerous  experiments,  the  conclusion 
arrived  at  by  previous  experimentalists,  that,  for  metals,  the  specific  heats 
are  in  the  inverse  ratio  of  their  chemical  equivalents.  Consequently  the 
products  of  the  specific  heats  of  metals,  by  their  respective  chemical 
equivalents,  are  a  constant  quantity.  The  same  rule  holds  good  for  other 
groups  of  bodies  of  the  same  composition,  and  of  similar  chemical  constitu- 
tion. The  specific  heat  of  alloys  is  sensibly  equal  to  the  mean  of  those  of 
the  alloyed  metals. 

The  following  are  the  specific  heats  of  water  for  various  tempera- 

23 


354 


HEAT. 


tures  from  o°  to  230°  C.,  or  32°  to  446°  F.,  by  the  air-thermometer,  calculated 
by  means  of  Regnault's  formula  :— 

c=  i  +  0.00004 /+  0.0000009  /2; (i) 

in  which  c  is  the  specific  heat  of  water  at  any  temperature  /,  the  specific 
heat  at  the  freezing  point  being  =  .1. 

Table  No.  116. — SPECIFIC  HEAT  OF  WATER. 


Temperature. 

Units  of  Heat  required  to  raise 
the  temperature  from  the  freezing 
point  to  the  given  temperature. 

Specific  Heat  at 
the  given 
temperature. 

Mean  Specific 
Heat  between 
the  freezing  point 
and  the  given 
temperature. 

Centigrade. 

Fahrenheit. 

Cent,  units. 

Fahr.  units. 

Freezing  point=i. 

0° 

32° 

0.000 

0.000 

I.OOOO 

IO 

50 

10.002 

18.004 

1.0005 

1.0002 

20 

68 

20.010 

36.018 

1.  0012 

1.0005 

3° 

86 

30.026 

54-047 

1.0020 

1.0009 

40 

104 

40.051 

72.090 

1.0030 

I.OOI3 

50 

122 

50.087 

90.157 

1.0042 

I.OOI7 

60 

140 

60.137 

108.247 

1.0056 

1.0023 

70 

158 

70.210 

126.378 

1.0072 

.0030 

80 

I76 

80.282 

144.508 

1.0089 

•0035 

90 

194 

90.381 

162.686 

I.OI09 

.0042 

IOO 

212 

I00.5OO 

180.900 

I.OI30 

.0050 

no 

230 

II0.64I 

199.152 

I-OI53 

.0058 

I2O 

248 

120.806 

217.449 

I.OI77 

.0067 

130 

266 

130.997 

235-791 

1.0204 

.0076 

140 

284 

I4I.2I5 

254.187 

1.0232 

.0087 

150 

302 

151.462 

272.628 

1.0262 

.0097 

160 

320 

161.741 

291.132 

1.0294 

.OIO9 

170 

338 

172.052 

309.690 

1.0328 

.0121 

1  80 

356 

182.398 

328.320 

1.0364 

•°I33 

190 

374 

192.779 

347.004 

I.040I 

.0146 

200 

392 

2O3.200 

365.760 

1.0440 

.0160 

210 

410 

213.660 

384.588 

1.0481 

.0174 

220 

428 

224.162 

403.488 

1.0524 

.0189 

230 

446 

234.708 

422.478 

1.0568 

.0204 

THE  SPECIFIC  HEAT  OF  AIR  AND  OTHER  GASES. 

The  specific  heat,  or  capacity  for  heat,  of  permanent  gases  is  sensibly 
constant  for  all  temperatures,  and  for  all  densities.  That  is  to  say,  the 
capacity  for  heat  of  each  gas  is  the  same  for  each  degree  of  temperature. 
For  air,  M.  Regnault  proved  that  the  capacity  for  heat  was  uniform  for 
temperatures  varying  from  -30°  C.  to  +  225°  C.  ( -  22°  to  437°  F.);  thus 
the  specific  heat  for  equal  weights  of  air,  at  constant  pressure,  were  as 
follows : — 


SPECIFIC   HEAT  OF  AIR,   ETC.  355 

Air  between  -30°    and  +  10° C Specific  heat,  0.2377 

Do.  10°    and  +  100°  C Do.          0.2379 

Do  100°  and  4- 225°  C Do.         0.2376 

Average 0.2377 

The  temperature  is  then  without  any  sensible  influence  on  the  specific 
heat  of  air;  neither  has  the  pressure,  so  far  as  it  has  been  subjected  to 
experiment — from  one  to  ten  atmospheres — any  influence  on  the  magni- 
tude of  the  specific  heat. 

The  specific  heat  of  gases  is  to  be  observed  from  two  points  of  view : — 
ist,  When  the  pressure  remains  the  same,  and  the  gas  expands  by  heat. 
2d,  When  the  volume  remains  the  same,  and  the  pressure  increases  with 
the  temperature.  There  is  a  striking  difference  in  the  specific  heat,  or 
capacity  for  heat,  according  as  it  is  measured  under  an  increasing  volume, 
or  an  increasing  pressure.  When  the  temperature  is  raised  one  degree, 
under  constant  pressure,  with  increasing  volume,  the  gas  not  only  becomes 
hotter  to  the  same  extent  as  when  the  volume  remains  the  same  and  the 
pressure  alone  is  increased,  but  it  also  expands  I/493d  part  of  its  volume 
at  32°  F.,  and  thus  absorbs  an  additional  quantity  of  heat  in  proportion  to 
the  work  done  by  expansion  against  the  pressure.  It  follows  that  the 
specific  heat  of  a  gas  at  constant  pressure  is  greater  than  that  of  the  same 
gas  under  a  constant  volume;  and  though  the  former  alone  has  been  made 
the  subject  of  direct  experiment,  the  latter  being  of  a  difficult  nature  for 
experimenters,  yet  the  latter,  which  is  properly  the  specific  heat,  is  easily 
deducible  from  the  former  on  the  principle  of  the  mechanical  theory  of 
heat. 

When  the  volume  of  a  gas  is  enlarged  by  expansion  against  pressure,  the 
work  thus  done  in  expanding  the  gas  may  be  expressed  in  foot-pounds  by 
multiplying  the  enlargement  of  volume  in  cubic  feet  by  the  resistance  to 
expansion  in  pounds  per  square  foot.  Having  thus  found  the  work  done 
in  foot-pounds,  it  may  be  divided  by  Joule's  equivalent,  772,  and  the 
quotient  will  be  the  expression  of  that  work  in  units  of  heat.  It  becomes 
latent,  or  insensible  to  the  thermometer,  and  is  called  the  latent  heat  of 
expansion.  It  constitutes  an  expenditure  of  heat  in  addition  to  the  heat 
that  is  sensible  to  the  thermometer,  and  that  raises  the  temperature.  The 
sum  of  these  two  quantities  of  heat  is  that  which  has  been  observed  in  the 
gross  by  experimentalists,  and  which  gives  the  specific  heat  at  constant 
pressure. 

It  follows  that,  when  the  specific  heat  at  constant  pressure  is  known,  the 
specific  heat  at  constant  volume  may  be  arrived  at  by  subtracting  the  pro- 
portion of  heat  devoted  to  the  enlargement  of  the  volume  from  the  total 
heat  absorbed  at  constant  pressure.  The  remainder  is  the  proportion  of 
heat  necessary  and  sufficient  to  elevate  the  temperature  when  the  volume 
remains  unaltered,  from  which  the  specific  heat  at  constant  volume  is 
deduced  by  simple  proportion;  thus — 

As  the  total  heat  absorbed  at  constant  pressure, 

Is  to  the  proportion  of  heat  absorbed  at  constant  volume, 

So  is  the  specific  heat  at  constant  pressure 

To  the  specific  heat  at  constant  volume. 

For  example,  the  specific  heat  of  air  at  constant  pressure  and  with  in- 


356  HEAT. 

creasing  volume  has  been  observed  to  be  .2377,  that  of  water  being  i.  Let 
one  pound  of  air  at  atmospheric  pressure,  and  at  32°  F.,  having  a  volume 
equal  to  12.387  cubic  feet,  be  expanded  by  heat  to  twice  its  initial  volume, 
the  pressure  remaining  the  same.  The  absolute  temperature,  which  is  32° 
+  461  =  493°  F.,  will  be  doubled,  and  the  indicated  temperature  will  be 
32  +  493  =  525°  F.  Thus,  493  degrees  of  heat  are  appropriated,  and  if  the 
capacity  for  heat  of  the  air  were  the  same  as  that  of  water,  493  units  of  heat 
would  be  expended  in  the  process  of  doubling  the  volume.  But,  as  the 
specific  heat  is  only  .2377,  or  less  than  a  fourth  of  that  of  water,  the  expen- 
diture of  heat  is  just  493  x  .2377  =  117.18  units,  and  this  quantity  comprises 
the  fraction  of  heat  consumed  in  displacing  the  atmosphere  and  overcoming 
its  resistance  through  a  space  of  12.387  cubic  feet  additional  to  the  original 
or  initial  volume  of  the  same  amount.  Now,  the  work  thus  done  is  equal 
to— 

12.387  cubic  feet  x  2116.4  Ibs.  pressure  per  sq.  foot  =  26,216  foot-pounds; 

and  dividing  this  by  772,  Joule's  equivalent,  the  work  of  enlarging  or  doub- 
ling the  volume  is  found  to  be  equivalent  to  33.96  units  of  heat.  Deduct- 
ing these  33.96  units  from  the  gross  expenditure,  which  is  117.18  units, 
the  remainder,  83.22  units,  is  the  proportion  of  heat  required  to  raise  the 
temperature  through  493  degrees,  under  an  increasing  pressure  simply, 
without  increasing  the  volume;  and  this  remainder  is  the  datum  from  which 
the  proper  specific  heat  of  air  is  to  be  deduced. 

The  distribution  of  heat  thus  detailed  may  be  concisely  exhibited 
thus : — 

Units. 

To  double  the  temperature  without  adding  to  the  volume....     83.22 
To  double  the  volume,  in  addition 33-96 

To  double  the  temperature  and  double  the  volume  at  con- 
stant pressure 117.18 

Now,  as  before  stated,  the  specific  heat  at  constant  volume  bears  the 
same  ratio  to  that  at  constant  pressure,  as  the  respective  quantities,  or  units 
of  heat,  absorbed,  do  to  each  other,  or  as  83.22  and  117.18;  and  it  is  found 
by  simple  proportion  to  be  .1688  ;  thus — 

117.18  :  83.22   :  :  .2377  :  .1688. 

The  proper  specific  heat  of  air  is  then  .1688,  in  raising  the  temperature 
without  enlarging  the  volume,  and  it  bears  to  the  so-called  specific  heat  of 
air,  at  constant  pressure  and  with  expanding  volume,  the  ratio  of  i  to 
1.408. 

This  ratio,  i  to  1.408,  deduced  by  means  of  the  mechanical  theory  of 
heat,  is  practically  identical  with  the  ratio  experimentally  arrived  at  by  M. 
Masson  from  the  fall  of  temperature  of  a  quantity  of  compressed  air,  which 
was  liberated  and  allowed  to  expand  back  until  it  regained  its  initial  pres- 
sure. The  ratio  he  deduced  from  his  inverse  experiment  was  i  to  1.41; 
which  is  the  ratio  of 

i  to 


SPECIFIC   HEAT   OF   AIR,   ETC.  357 

It  may  be  added,  by  way  of  explanation,  and  to  enforce  the  distinction, 
that  though  the  pressure  of  a  gas  under  constant  volume  rises  with  the 
temperature,  —  a  phenomenon  which  is  analogous,  at  first  sight,  to  that  of  the 
volume  of  a  gas  at  constant  pressure  increasing  with  the  temperature,  —  yet 
there  is  no  expenditure  of  work  in  simply  raising  the  pressure  in  the  former 
case,  while  the  volume  remains  unaltered;  whereas,  in  the  latter  case,  there 
is  an  expenditure  in  increasing  the  volume,  as  has  already  been  shown. 

To  generalize  the  foregoing  process,  by  which  the  specific  heat  of  air  at 
constant  volume  has  been  deduced  from  the  specific  heat  for  constant 
pressure;  and  to  show  its  applicability  for  finding  the  specific  heat  of  all 
gases  at  constant  volume  :  — 

Given  /  =  the  initial  temperature  of  the  gas,  in  degrees  Fahrenheit. 

„      t'  =  the  final  temperature  to  which  the  gas  is  raised. 

„      V  =  the  initial  volume  of  the  gas,  under  one  atmosphere  of  pres- 
sure, in  cubic  feet. 

„      v  —  the  final  volume  of  the  gas,  heated  under  constant  pressure. 

„      h  —  the  specific  heat  of  the  gas  under  constant  pressure. 
Put      h'  —  the  specific  heat  of  the  gas  under  constant  volume. 

„    H  =  the  total  heat  expended  at  constant  pressure,  in  units  of  heat. 

„    H'  =  the  total  heat  expended  at  constant  volume. 

,,    H"  =  the  fractional  quantity  of  heat  expended  in  increasing  the 
volume,  at  constant  pressure;  or  the  latent  heat  of  expansion. 

To  find  the  value  of  h';  then  by  proportion, 
H  :  H'  :  :  h  :  h', 


NowH'  =  H-H", 

TT/          TT  _  TT// 

And  g  =      TT     ,  and,  by  substitution, 
H  H 


(a) 


Again,  H  =  (/'-/)x/;, 
And  H"  =  (V-z>)  x  14.7  x  144^-  772 
*(V-«r)x    2.742; 

AndH-ir  h  (/'-/)-  2.  742  (v-p) 
~ir~          *(/'-/) 

Substituting  this  value  in  equation  (a)  above, 

,,,_h  (h(t'-f)-  2.742  (V-g)). 
*(''-') 


or      = 


, 

Whence  the  following  rule  :  — 

RULE  i.  To  find  the  specific  heat  of  a  gas  at  constant  volume,  when  the 
specific  heat  at  constant  pressure  is  given  together  with  the  initial  and  final 
temperatures  due  to  given  initial  and  final  volumes  under  an  atmosphere  of 


358  HEAT. 

pressure.  Multiply  the  difference  of  the  initial  and  final  temperatures  by 
the  specific  heat  at  constant  pressure.  Also,  multiply  the  difference  of  the 
initial  and  final  volumes  by  2.742.  Find  the  difference  of  these  two  pro- 
ducts, and  divide  it  by  the  difference  of  the  temperatures.  The  quotient  is 
the  specific  heat  of  the  gas  at  constant  volume. 

Applying  the  rule  to  the  example  of  one  pound  of  air  at  atmospheric 
pressure,  and  at  32°  F.,  doubled  in  volume  by  heat;  ^  =  .2377,  /'-/=493°, 
and  V-^=i2.38y  cubic  feet.  Then 


h>  _  (-2377  x  493)  ~  (2.742  x  I2-3^7)  _  l68g 
493 

the  specific  heat  of  air  at  constant  volume,  as  already  found. 

The  comparative  volumes  of  other  gases  are  given  in  table  No.  69, 
page  216,  of  the  Weight  and  Specific  Gravity  of  Gases  and  Vapours. 

THE  SPECIFIC  HEAT  OF  GASES  FOR  EQUAL  VOLUMES. 

The  specific  heats  of  equal  volumes  of  gases  are  deducible  from  their 
specific  heats  proper,  —  which  are  for  equal  weights.  The  greater  the 
density,  the  less  is  the  volume,  and  the  greater  the  weight  of  gas  that  is 
necessary  to  equal  in  volume  a  lighter  gas;  it  is  greater,  in  fact,  in  propor- 
tion to  the  density. 

Hence  the  following  rule  :  — 

RULE  2.  To  find  the  specific  heat  of  a  gas  for  equal  volumes  of  the  gas 
and  of  air.  Multiply  the  specific  heat  of  the  gas,  that  is,  the  specific  heat 
for  equal  weights  of  the  gas  and  air,  by  the  specific  gravity  of  the  gas.  The 
product  is  the  specific  heat  for  equal  volumes. 

Note.  —  The  specific  heat  for  equal  volumes  may  be  found  for  constant 
pressure,  and  for  constant  volume,  in  terms  respectively  of  the  specific  heat 
of  equal  weights  at  constant  pressure  and  constant  volume. 

TABLES  OF  THE  SPECIFIC  HEAT  OF  SOLIDS,  LIQUIDS,  AND  GASES. 

The  annexed  table,  No.  117,  contains  the  specific  heats  of  a  number  of 
solids,  classified  for  convenience  of  reference,  into 

Metals, 

Stones, 

Precious  Stones, 

Sundry  Mineral  Substances, 

Woods. 

It  appears  from  the  tables  that  the  metals,  generally  speaking,  have  the 
least  specific  heat:  ranging  from  bismuth,  having  a  specific  heat  of  .031, 
to  iron,  which  has  a  specific  heat  of  from  .11  to  .13,  and  iridium,  which 
has  the  greatest  specific  heat,  namely,  .189. 

Stones  show  a  specific  heat  of  about  .20,  or  a  fifth  of  that  of  water. 
Precious  stones  average  less  than  that. 

Of  the  sundry  mineral  substances,  glass,  sulphur,  and  phosphorus 
average  about  a  fifth  of  the  specific  heat  of  water,  and  coal  and  coke 
a  fourth. 

Woods  average  a  half  of  the  specific  heat  of  water. 


SPECIFIC   HEAT   OF   SOLIDS. 


359 


It  is  a  useful  practical  conclusion,  as  Dr.  Rankine  remarks,  that  the 
average  specific  heat  of  the  non-metallic  materials  and  contents  of  a  furnace, 
whether  bricks,  stones,  or  fuel,  does  not  greatly  differ  from  one-fifth  of  that 
of  water. 

Of  the  liquids  specified  in  the  table  No.  118,  it  appears  that  all,  with  the 
exception  of  bromine,  which  has  a  specific  heat  of  i.m,  have  less  specific 
heat  than  water.  Olive  oil  has  the  lowest, — only  .31 ;  alcohol  averages  .65, 
and  vinegar,  .92. 

The  table  No.  119  of  the  specific  heat  of  gases,  contains,  in  the  second 
column,  their  specific  heat,  for  equal  weights,  at  constant  pressure,  as 
determined  by  M.  Regnault.  The  third  column  contains  the  specific  heat, 
for  equal  weights,  at  constant  volume,  calculated  by  means  of  the  Rule  i, 
above.  The  fourth  and  fifth  columns  contain  the  specific  heat  of  gases, 
for  equal  volumes,  at  constant  pressure,  and  at  constant  volume,  arrived 
at  by  means  of  the  Rule  2,  above. 

There  is  a  remarkable  nearness  to  equality  in  the  specific  heat  for  equal 
volumes  of  air,  oxygen,  hydrogen,  carbonic  oxide,  and  nitrogen.  It  may 
be  noted,  in  particular,  that  hydrogen,  though  it  has  fourteen  times  the 
specific  heat  of  air  for  equal  weights,  and  has  barely  a  fourteenth  of  the 
density  of  air,  has  no  more  specific  heat  than  air,  for  equal  volumes. 

Table  No.   117. — SPECIFIC  HEAT  OF  SOLIDS. 

(Authority,  Regnault,  when  not  otherwise  stated.) 


METALS,  from  32°  to  212°  F. 


Bismuth .03084 

Lead 

Platinum,  sheet .03 243 

Do.   spongy 03293 

Do.       32°  F.  to  212°  F (Petit  and  Dulong) 

Do.       32°  F.  to  572°  F.  (300°  C.)  „  .0355 

Do.       at    212°  F.  (  100°  C.) (Pouillet}  .0335 

Do.       at    572°F.  (  300°  C.) „  .03434 

Do.       at    932°  F.  (  5oo°C.) 

Do.       at  1292°  F.  (  700°  C.) „ 

Do.       at  1832°  F.  (1000°  C.) 

Do.       at  2192°  F.  (1200°  C.) „  .03818 

Gold .03  244 

Mercury,  solid 

Do.        liquid 

Do.       59°  to  68°  F.  (15°  to  20°  C.) 029 

Do.       32°  to  212°  F (Petit  and  Dulong)  .033 

Do.       32°  to  572°  F.  (300°  C.) „  .035 

Tungsten .03636 

Antimony .05077 

Do.      32°  to  212°  F (Petit  and  Dulong)  .0507 

Do.      32°  to  572°  F.  (300°  C.) „  .0547 

Tin,  English . . 

Do.    Indian .05623 


Water  at  32°=  i. 


360 


HEAT. 


METALS   (continued}. 


Cadmium .05669 

Silver .05701 

Do.  32°  to  212°  F (Petit  and  Dulong)  .0557 

Do.  32°  to  572°  F.  (300°  C.) „  .0611 

Palladium -°5927 

Uranium  .06 1 9 

Molybdenum .07218 

Brass -°939i 

Cymbal  metal .086 

Copper .095 1 5 

Do.     32°  to  212°  F (Petit  and  Dttlong)  .094 

Do.    32°  to  572°  F.  (300°  C.) „  .1013 

Zinc -°9555 

Do.  32°  to  212°  F (Petit  and  Dulong)  .0927 

Do.  32°  to  572°  F.  (300°  C.) „  .1015 

Cobalt  .10696 

Do.     carburetted  .11714 

Nickel 10863 

Do.     carburetted .11192 

Wrought  iron .11379 

Do.  32°  to  212°  F (Petit  and  Dulong)  .1098 

Do.          32°  to  392°  F.  (200°  C.)  „  .115 

Do.          32°  to  572°  F.  (300°  C.)  „  .1218 

Do.          32°  to  662°  F.  (350°  C.)  „  .1255 

Steel,  soft .1165 

Do.    tempered .  1 1 7  5 

Do.    Haussman .  1 1 848 

Cast  iron,  white .  1 2983 

Manganese,  highly  carburetted .  1441 1 

Iridium .1887 

STONES. 

Brickwork  and  masonry (Rankine)  about  .  2  o 

Marble,  gray .  20989 

Do.      white .  2 1 585 

Chalk,  white -21485 

Quicklime .2169 

Dolomite  (Magnesian  limestone) .21 743 

PRECIOUS    STONES. 

Sapphire -21732 

Zircon -14558 

Diamond .14687 

SUNDRY   MINERAL   SUBSTANCES. 

Tellurium  .05 155 

Iodine .0541 2 

Selenium -0837 

Bromine  .0840 

Phosphorus,  50°  to    86°  F 1887 


Water  at  32°  =i. 


SPECIFIC   HEAT  OF  SOLIDS. 


361 


SUNDRY  MINERAL  SUBSTANCES  (Continued). 


Phosphorus,  32°  to  212°  F 25034 

Glass 19768 

Do.    flint 19 

Do.    32°  to  2 1 2°  F (Petit  and  Dulong)          .  1 77 

Do.    32°  to  572°  F 19 

Sulphur 20259 

Do.      crystallized,  natural 

Do.      cast  for  two  years 

Do.      cast  for  two  months .1803 

Do.      cast  recently .1844 

Chloride  of  lead 06641 

Do.         zinc I36j8 

Do.        manganese •I4255 

Do.        tin *4759 

Do.        calcium .16420 

Do.        potassium  •I7295 

Do.         magnesium .19460 

Do.         sodium 214  to  .230 

Perchloride  of  tin 10161 

Protochloride  of  mercury .06889 

Nitrate  of  silver *4352 

Do.       barytes 15228 

Do.       potass 23875 

Do.       soda 27821 

Sulphate  of  lead 08 7 23 

Do.        barytes  11285 

Do.        potash .1901 

Carbonaceous : — 

Coal  24111 

Charcoal 2415 

Coke  of  cannel  coal .20307 

Do.     pit  coal 20085 

Coal  and  coke,  average (Rankine)          .20 

Anthracite,  Welsh 20172 

Do.         American .201 

Graphite,  natural .20187 

Do.      of  blast  furnaces -497 

Animal  black 26085 

Sulphate  of  lime • J  965  9 

Magnesia .22159 

Soda 23115 

Ice 5°4 

WOODS. 

Turpentine .467 

I  Pear  tree 500 

i  Oak 570 

Fir 650 


Water  at  32°=  i. 


362 


HEAT. 


Table  No.   118. — SPECIFIC  HEAT  OF  LIQUIDS. 


Mercury 0333 

Olive  oil (Laplace  and  Lavoisier)    . . .      .3096 

Sulphuric  acid,  density  1.87 „  ...      .3346 

Do.  do.      1.30 „  ...      .6614 

Benzine,  59°  to  68°  F 3932 

Turpentine, 4160 

Do.         density  .872 (Despretz]    ...      .4720 

Ether,  oxalic 4554 

Do.,   sulphuric,  density  0.76 (Daltori)    ...      .6600 

Do.          do.  do.     0.715 (Despretz}    ...      .5200 

Essence  of  juniper 477° 

Do.       lemon 4879 

Do.       orange 4886 

Hydrochloric  acid 6000 

Wood  spirit,  59°  to  68°  F 6009 

Chloride  of  calcium,  solution 6448 

Acetic  acid,  concentrated 6581 

Alcohol  6588 

Do.     density  0.793 (Dalian)    ...     .6220 

Do.         do.     0.81    „          ...      .7000 

Vinegar 9200 

Water,  at  32°  F i.oooo 

Do.    at  212°  F 1.0130 

Do.    from  32°  to  212°  F 1.0050 

Bromine  ..  i.ino 


Water  at  32°  =i. 


FUSIBILITY  OR   MELTING    POINTS  OF  SOLIDS. 


363 


Table  No.  119. — SPECIFIC  HEAT  OF  GASES. 
Water  at  32°  F.  =  I. 


GAS. 

SPECIFIC  HEAT  FOR 
EQUAL  WEIGHTS. 

SPECIFIC  HEAT  FOR 
EQUAL  VOLUMES. 

At  constant 
pressure. 

At  constant 
volume. 
(Real  speci- 
fic heat.) 

At  constant 
pressure. 

At  constant 
volume. 

Sulphurous  acid 

water  =  i. 

o.i553 

0.1568 
0.2164 
0.2182 
0.2377 
0.2440 
0.2479 
0.3694 
3.4046 

0-3754 
0.4008 

o.45T3 
0.4750 
0.5061 
0.5080 
0.5929 

water  =  i. 

0.1246 
0.1438 
0.1714 
0-1559 

0.1688 
0.1740 
0.1768 
0.2992 
2.4096 

0-3499 
0.3781 
0.4124 

0.3643 
0.4915 
0.3911 
0.4683 

air  =  .2377, 
as  in  col.  2 

0.3489 
0.8310 
0.3308 
0.2412 
0.2377 
0.2370 
0.2399 
0.3572 
0.2356 
I.OII4 
1.2184 
0.7171 
0.2950 
2.3776 
0.2994 
0.3277 

air  =  .1688, 
as  in  col.  3. 

0.2799 
0.7621 
O.262O 
0.1723 

0.1688 
0.1690 
0.1711 
0.2893 
0.1667 
0.9427 
1.1490 

0.6553 
0.2262 
2.3090 
0.2305 
0.2588 

Vapour  of  chloroform        

Carbonic  acid              

Oxygen 

Air  

Nitrogen 

Carbonic  oxide 

Olefiant  gas     

Hydrogen  

Vapour  of  Benzine  

Acetic  ether 

Vapour  of  alcohol 

Gaseous  steam     ....          

Vapour  of  turpentine  

Ammoniacal  gas  

Light  carburetted  hydrogen 

FUSIBILITY   OR   MELTING   POINTS   OF   SOLIDS. 

The  metals  are  solid  at  ordinary  temperatures,  with  the  exception  of 
mercury,  which  is  liquid  down  to  -  39°  F.  Hydrogen,  it  is  believed,  is  a 
metal  in  a  gaseous  form. 

All  the  metals  are  liquid  at  temperatures  more  or  less  elevated,  and  they 
probably  vaporize  at  very  high  temperatures.  Their  melting  points  range 
from  39  degrees  below  zero  of  Fahrenheit's  scale,  the  melting,  or  rather  the 
freezing,  point  of  mercury,  up  to  more  than  3000  degrees,  beyond  the 
limits  of  measurement  by  any  known  pyrometer.  Certain  of  the  metals, 
as  potassium,  sodium,  iron,  platinum,  become  pasty  and  adhesive  at 
temperatures  much  below  their  melting  points.  Potassium  and  sodium, 
which  melt  at  temperatures  between  130°  and  200°  F.,  can  be  moulded 
like  wax  at  62°  F.  Two  pieces  of  iron  raised  to  a  welding  heat,  are 
softened,  and  readily  unite  under  the  hammer;  and  pieces  of  platinum 
unite  at  a  white  heat. 

The  melting  points  of  alloys  do  not  follow  the  ratios  of  those  of  their 
constituent  metals,  so  that  it  is  impossible  to  infer  their  melting  points  from 
these  data.  A  remarkable  instance  of  the  absence  of  this  relation  is  afforded 
in  the  fusible  metal  consisting  of  five  parts  of  lead,  three  of  tin,  and  eight 
of  bismuth,  which  melts  at  212°  F.,  the  heat  of  boiling  water,  though  the 


3^4 


HEAT. 


melting  point,  if  it  were  an  average  of  those  of  the  component  metals, 
would  be  about  520°  F.  The  addition  of  bismuth  to  mixtures  of  lead  and 
tin  has  the  effect  of  lowering  the  melting  points. 

According  to  Professor  Rankine,  the  melting  point  of  ice  is  lowered  by 
pressure,  at  the  rate  of  0.0000063°  F.  for  each  pound  of  pressure  on  the 
square  foot.  An  atmosphere  of  pressure  being  2116  Ibs.  per  square  foot, 
the  lowering  of  the  melting  point  per  atmosphere  of  pressure,  is — 

o°.ooooo63  x  2116  — o°.oi33  Fahrenheit. 

To  lower  the  melting  point  one  degree,  a  pressure  of  75  atmospheres  would 
be  required. 

In  the  case  of  water,  antimony,  and  cast  iron,  and  probably  other  sub- 
stances, the  bulk  of  the  substance  in  the  solid  state  exceeds  that  in  the 
liquid  state,  as  is  evidenced  by  the  floating  of  ice  on  water,  and  of  solid 
iron  on  molten  iron.  The  volume  of  water  is  to  that  of  ice  at  32°  F.,  as 
i  to  i. 088;  that  is  to  say,  that  water,  in  freezing  at  32°  F.,  expands  nearly 
9  per  cent. 

The  following  table,  No.  120,  contains  the  melting  points  of  metals, 
metallic  alloys,  and  other  substances: — 


Table  No.  120. — MELTING  POINTS  OF  SOLIDS. 


VARIOUS   SUBSTANCES  (Pouillet,  Claudel,  &c.) 


MELTING  POINTS. 


Sulphurous  acid -  148°  F. 

Carbonic  acid . . » -  i  o 8 

Bromine +9-5 

Turpentine 14 

Hyponitric  acid 1 6 

Ice 32 

Nitro-glycerine 45 

Tallow 92 

Phosphorus 112 

Acetic  acid 113 

Stearine 109  to  120 

Spermaceti 120 

Margaric  acid 131  to  140 

Wax,  rough 142 

,,      bleached 154 

Stearic  acid 158 

Iodine 225 

Sulphur 239 


MELTING   POINTS   OF   SOLIDS. 
Table  No.  120  (continued']. 


365 


METALS. 

MELTING  POINTS. 

Pouillet,  Claudel. 

Wilson. 

Mercury  

Fahrenheit  degrees. 

-39° 

+  136 
194 

446 

5°4 

608 
680 
810 
1692 

(very  pure)  1832 

2156 
(very  pure)  2282 
1922  to  2012 

2OI2 
2192 
2282 

2372  to  2552 

2732 
2912 

Fahrenheit  degrees. 

101° 
144 

208 

356 

442 

442 
507 
561 
6l7 

773 
1150 

full  red  heat, 
full  red  heat. 

1873 
1996 

2016 
2786 

[Fusible  in  highest 
heat  of  forge. 

I  Not  fusible  in  forge 
>  fire,  but  soften  and 
1     agglomerate. 

Only  fusible  before 
•    the     oxyhydrogen 
blow-pipe. 

Rubidium 

Potassium     .            

Sodium          

Lithium      .        

Tin             

Cadmium    

Bismuth 

Thallium                           

Lead                       

Zinc                    

Antimony  

Bronze 

Aluminium                      .      .  , 

Calcium                      .        ... 

Silver                 

Copper     .           

Gold,  standard  

Gold 

Cast  Iron  white 

srav 

„        „    2d  melting... 
„        „     with  manganese... 
Steel  

Wrought  Iron   French 

Hammered  Iron,  English  
Malleable  Iron  

Cobalt  

Nickel  

Manganese 

Palladium 

Molybdenum  

Tungsten  

Chromium 

Platinum 

Rhodium      

Iridium         

Ruthenium 

Osmium                      

366 


HEAT. 
Table  No.  120  (continued}. 


ALLOYS   OF 

LEAD,  TIN,  AND   BISMUTH. 

MELTING  POINTS. 

No.     i. 

2. 

3- 

6. 

7-   i 

8.     2 

9-  3 
10.  4 
ii.  5 

12.    6 

14-  3 
15.   2 
16.   i 

17.     2 

18.  3 

Tin 

33 

33 
3) 

33 
33 
33 
33 

Lead 

33 
33 

33 
33 
33 

,  25 
10 

5 
3 

2 

3    4 

3 

2 
I 
I 

5 

Lead  

Holtzapffel. 

5" 

482 

44  1 
340 

378 
38i 
320 
310 
292 

254 
236 

202 

Claudel. 

466° 

367 
372 

33         

33    

Tin    i  Bismuth 

i 

0 

SUNDRY 

ALLOYS   OF  TIN,  LEAD,  AND 

BISMUTH. 

MELTING  POINTS. 

Lead    2  Tin 

5  Bismuth 

Ure 

IQQ° 

I 

2 

x  yy 
2OI 

I 

4. 

Claudel 

2OI 

c 

5 

8 

Ure 

212 

2 

c 

.  .  .      Claudel 

212 

T 

c 

246 

i 

I 

286 

I 

1  -7  A 

2 

I 

OOT- 

774 

T 

r  Holtzapffel 

360 

a 

I 

'  {        Claudel 

385 
7Q2 

1. 

I 

CC2 

ALLOYS   FOR  FUSIBLE  PLUGS. 

Softens  at 

Melts  at 

2  Tin 

2  Lead  .  .  . 

^6q°F 

772°  Y 

2 

6 

o^j    •*-  • 
11  2 

o  i  *     -1  • 

7.8?. 

2 

7 

01* 

1111A 

O^O 

^88 

2 

8 

7.QC  */£ 

406  to  410 

oyj  /a 

LATENT   HEAT  OF  FUSION  OF  SOLID  BODIES. 


367 


LATENT  HEAT  OF  FUSION  OF  SOLID  BODIES. 

When  a  solid  body  is  exposed  to  heat,  and  ultimately  passes  into  the 
liquid  state  under  the  influence  of  the  heat,  the  temperature  of  the  body 
rises  until  it  attains  the  point  of  fusion,  or  melting  point.  The  temperature 
of  the  body  remains  stationary  at  this  point  until  the  whole  of  it  is  melted; 
and  the  heat  meantime  absorbed,  without  affecting  the  temperature,  is  said 
to  become  latent,  as  it  is  not  sensible  to  the  thermometer.  It  is,  in  fact, 
the  latent  heat  of  fusion,  or  the  latent  heat  of  liquidity,  and  its  function  is  to 
separate  the  particles  of  the  body,  hitherto  solid,  and  change  their  condition 
into  that  of  a  liquid.  When,  on  the  contrary,  the  liquid  is  solidified,  the 
latent  heat  is  disengaged. 

M.  Person  gave  the  following  law  as  the  result  of  his  experiments  on  the 
latent  heat  of  fusion  of  non-metallic  substances : — Let  c  be  the  specific  heat 
of  the  substance  in  the  solid  state,  and  c'  its  specific  heat  in  the  liquid  state; 
/  the  temperature  of  fusion,  or  melting  point,  by  Fahrenheit's  scale,  and  / 
the  latent  heat.  Then  the  latent  heat  of  fusion  of  one  pound,  in  British 
thermal  units,  is 

/^'-.(/+256°) (i) 

RULE. — To  find  the  latent  heat  of  fusion  of  a  non-metallic  substance. 
Subtract  the  specific  heat  of  the  substance  in  the  solid  state  from  its 
specific  heat  in  the  liquid  state,  and  multiply  the  remainder  by  the  tempera- 
ture of  fusion  or  melting  point  by  the  Fahrenheit  scale,  plus  256.  The 
product  is  the  latent  heat  of  fusion  in  heat-units. 

Table  No.  121. — LATENT  HEAT  OF  FUSION  OF  SOLID  BODIES. 

Person. 


Non-metallic. 

Melting  Point. 

Specific  Heat. 

Latent  Heat 
in  heat-units, 
of  i  pound. 

Liquid. 

Solid. 

Ice  

32°  F. 
83 
97 

112 
120 
142 
239 
591 
642 

I.OOOO 

•5550 
.7467 

.2045 

.2340 
.4130 

•3319 

.5040 

•345° 
.4077 
.1788 

.2026 
.2782 
.2388 

142.6 

73 

120 

9 
148 

175 
17 
H3 
85 

Chloride  of  calcium.    . 

Phosphate  of  soda  

Phosphorus 

Spermaceti  

Wax  

Sulphur  

Nitrate  of  soda 

Nitrate  of  potass  .... 

Metallic. 

Tin                       

442 
442 

507 
6l7 

773 
1873 

.0637 
.0642 

.0363 

.0402 

.0562 
.0567 
.0308 
.0314 
.0956 
.0570 

25.6 
25-6 
22.7 
9.86 
50.6 

37-9 

Cadmium          

Bismuth     

Lead  

Zinc                               .  .    . 

Silver             

368  HEAT. 

EXAMPLE. — To  find  the  latent  heat  of  fusion  of  ice,  the  specific  heat  of 
ice,  c=  0.504,  and  that  of  water  c'  =  i ;  /=  32°  F.  Then 

the  latent  heat  •-=  (i  -  0.504)  (32°  +  256)  =  0.496  x  288  =  142.86 
Do.,  by  M.  Person's  experiment =  142.65 

Difference o.  2 1 

showing  that  the  latent  heat  effusion  of  one  pound  of  ice  is  142.86  units. 

The  table  No.  121  contains  the  latent  heat  of  fusion  of  several  metals 
and  other  bodies,  according  to  M.  Person.  On  an  inspection  of  the  table, 
it  appears  generally  that  the  latent  heat  of  fusion  of  non-metallic  bodies  is 
greater  for  those  which  have  the  lower  melting  points,  and  that,  for  metals, 
the  proportion  lies  rather  the  other  way.  The  greatest  latent  heat  of  fusion 
belongs  to  wax,  which  has  175  units  per  pound,  and  the  least  to  phos- 
phorus and  lead,  which  have  only  9  and  9.86  units  respectively  per  pound 
weight. 

BOILING  POINTS  OF  LIQUIDS. 

When  a  cold  liquid,  contained  in  a  vessel  open  to  the  air,  is  subjected  to 
heat,  the  temperature  of  the  liquid  is  raised,  and  a  quantity  of  vapour  is 
emitted  from  the  surface  of  the  fluid,  the  pressure  of  which  gradually 
increases  until  it  becomes  equal  to  the  pressure  of  the  atmosphere.  When 
this  pressure  is  reached,  the  aggregation  of  the  vapour  becomes  visible  in 
the  interior  of  the  liquid,  and  the  vapour  rises  to  the  surface  and  escapes. 
This  is  ebullition^  or  evaporation,  or  vaporization.  When  the  liquid  has 
attained  to  the  state  of  ebullition,  the  temperature  ceases  to  rise,  and 
remains  stationary,  and  it  so  remains  until  the  whole  of  the  liquid  is  evapo- 
rated. This  phenomenon  of  stationary  temperature  is  analogous  to  that 
which  attends  the  fusion  of  solids  into  liquids. 

The  proper  boiling  point  of  water,  under  one  atmosphere  of  pressure,  is 
100°  C.,  or  212°  F.  It  is  affected  to  some  extent  by  the  nature  of  the  vessel 
which  contains  it  and  the  presence  of  other  objects  in  the  vessel.  In  a 
glass  retort,  for  example,  water  boils  with  jolts  and  small  explosions,  and 
the  temperature  of  the  water  at  which  ebullition  takes  place  is  from  two 
to  three  degrees  higher  than  when  it  is  evaporated  in  an  iron  vessel. 
Sulphuric  acid  behaves  similarly  under  ebullition,  and  the  explosions  are  as 
much  more  violent  as  the  liquid  has  greater  cohesiveness,  and  as  it  acts 
chemically  upon  the  matter  of  the  vessel  in  which  it  boils.  A  few  pieces  of 
metal  thrown  into  the  glass  vessel  arrest  the  explosive  ebullition,  and  the 
temperature  of  the  liquid  falls  to  the  same  level  as  in  the  metallic  vessel. 

The  boiling  point  of  liquids  is  not  altered  by  the  presence  of  foreign 
bodies  mechanically  in  mixture  with  them,  such  as  sand,  sulphate  of  lime, 
and  carbonate  of  lime.  But  it  is  always  greater  when  matters  are  present 
in  chemical  combination  with  the  liquids.  All  the  soluble  salts  have  this 
effect  when  dissolved  in  water;  but,  on  the  contrary,  it  has  been  proved 
experimentally,— 

That  the  vapour  produced  at  the  surface  of  saline  solutions  is  the  steam 
of  pure  water, 

And  that  at  atmospheric  pressure  the  temperature  of  the  steam  formed 
is  invariably  212°  F.,  whatever  be  the  nature  of  the  dissolved  salt,  or  of  the 
vessel  containing  the  solution.  It  further  appears  that  at  higher  pressures 


BOILING  POINTS   OF   LIQUIDS. 


369 


Table  No.  122. — BOILING  POINTS  OF  LIQUIDS  UNDER  ONE  ATMOSPHERE 

OF  PRESSURE. 


Fahrenheit. 

Sulphuric  ether  

IOO° 

Sulphuret  of  carbon  

118  4 

Ammonia  

1  4O 

Chloroform         

T  A  O 

Bromine  

I4C 

Wood  spirit  

*-*TJ 
I  ^O 

Alcohol  

ADW 

17-2 

Benzine...  

176 

Water  

x  i  v 
212 

Average  sea-  water  

211  2 

Saturated  brine  

*  ^O"6 

226 

Nitric  acid  

248 

Oil  of  turpentine  

•2  I  C 

Phosphorus  

3*-J 

C  CA 

Sulphur  

DJT- 

C.7O 

Sulphuric  acid  

0  /  w 

CQO 

Linseed  oil    .  . 

Oy^ 
CQ7 

Mercury  

DV  / 
648 

Table  No.  123. — BOILING  POINTS  OF  SATURATED  SOLUTIONS  OF  SALTS 
UNDER  ONE  ATMOSPHERE. 


NAME  OF  SALT. 

BOILINC 

,  POINT. 

Quantity  of  salt 
which  saturates  100 
parts  of  water. 

Chlorate  of  potash  

Centigrade. 
1  04°.  2 

Fahrenheit. 

2iq°.6 

per  cent. 

61  c 

Chloride  of  barium 

I  04  A 

220  o 

"  AO 
60  I 

Carbonate  of  soda  .  . 

I  O4  6 

220  3 

48  c 

Phosphate  of  soda  

ICK.C. 

•w-o 

222  O 

V**3 

1112 

Chloride  of  potassium  

AW3O 
108.3 

227  O 

i  ±3.4 
CQ  A 

Chloride  of  sodium  (common  salt)  .  .  . 
Hydrochlorate  of  ammonia 

108.4 
I  14  2 

227.2 
237  6 

OV'^r 
41.2 

88  9 

Neutral  tartrate  of  potash  

I  14  67 

2^8  A 

2Q6  2 

Nitrate  of  potash  

1  1  C  Q 

2AO  6 

•}•}  C.I 

Chloride  of  strontium  

A  *  O'y 

117.  Q 

2AA.  2 

•JOD'  * 

H7.  c 

Nitrate  of  soda     

121  O 

2  ZO  O 

224  8 

Acetate  of  soda  

I2A  37 

.^3^.  w 
2«?5  8 

2OQ.O 

Carbonate  of  potash  

I  -3C  O 

•jj*" 

27C  O 

20^.0 

Nitrate  of  lime 

XOJ'W 

I  r  j  o 

•  /  0>w 
•2Q4  O 

362  2 

Acetate  of  potash  

160  o 

•2-26  O 

70S  2 

Chloride  of  calcium  

I  7Q  5 

OO^' 
•7CC.I 

fjr**' 

32?.  O 

Nitrate  of  ammonia 

x  /  VO 
1  80  O 

JJJ'  A 
•2  r6  O 

unlimited 

O  J^' 

3/0  HEAT. 

the  temperatures  of  steam  formed  from  sea-water  are  the  same  as  those  of 
steam  generated  from  fresh-water  under  equal  pressures. 

Table  No.  122  contains  the  boiling  points  of  liquids  under  one  atmos- 
phere of  pressure.  It  shows  that  the  boiling  points  vary  from  100°  F.  for 
sulphuric  ether  to  648°  F.  for  mercury.  Linseed  oil  boils  at  597°  F.,  and 
the  great  elevation  of  this  temperature,  as  representing  more  or  less  approxi- 
mately the  boiling  points  of  oils  and  fats  generally,  explains  the  capacity  of 
these  substances  when  heated  for  cooking  meat  immersed  in  them. 

It  is  shown  that  sea-water  boils  at  2 13°.  2  F.  under  one  atmosphere,  and 
that  saturated  brine  does  not  boil  until  the  temperature  rises  to  226°  F. 
The  boiling  point  of  sea-water  is  raised  in  proportion  to  its  concentration 
as  brine. 

Table  No.  123  contains  the  boiling  points  of  saturated  solutions  of 
various  salts  in  water,  under  one  atmosphere,  according  to  the  experiments 
of  M.  Legrand.  They  vary  from  220°  to  356°  F.  They  present  a  striking 
diversity,  even  among  salts  having  the  same  base. 

BOILING  POINTS  OF  LIQUIDS  AT  VARIOUS  PRESSURES. 

The  boiling  points  of  liquids  rise  as  the  pressure  increases  under  which 
they  are  evaporated,  and  they  contrast  strikingly  in  this  respect  with  the 
melting  points  of  solids,  which  are  practically  constant  under  all  pressures. 
It  has  already  been  stated  that,  to  lower  the  melting  point  of  ice  only  one 
degree  Fahrenheit,  75  atmospheres  of  pressure  were  required.  On  the 
contrary  the  boiling  point  of  water  is  raised  75  degrees  by  an  augmentation 
of  less  than  three  atmospheres  above  the  atmospheric  pressure. 

Table  No.  124  contains  a  comparative  statement  of  the  pressures  of  the 
vapours  of  water  and  other  liquids,  at  temperatures  varying  from  o°  C.,  or 
32°  F.,  to  222°  C.,  or  432°  F. — in  fact,  their  boiling  points  for  various  pres- 
sures— the  results  of  experiments  by  Regnault. 

The  table  No.  124  shows  a  great  diversity  of  pressure  of  saturated 
vapours  for  given  temperatures.  At  the  temperature  of  2 12°  F.,  for  example, 
at  which  water  boils  under  one  atmosphere  of  pressure;  in  other  words, 
at  which  the  pressure  of  the  vapour  of  boiling  water  is  14.7  Ibs.  per  square 
inch,  the  pressures  of  the  saturated  vapours  of  the  several  liquids  are  as 
follows : — 

per  square  inch. 

Vapour  of  water  at  212°  F pressure,  14.7  Ibs. 

Do.       alcohol        „ „        32.6  Ibs. 

Do.       ether  „       „        95.17  Ibs. 

Do.       chloroform,,       „        45.54  Ibs. 

Do.       turpentine   „       „          2.61  Ibs. 

The  relations  of  the  vapour  of  water  or  steam  are  fully  considered  in  a 
subsequent  section. 

LATENT  HEAT  AND  TOTAL  HEAT  OF  EVAPORATION  OF  LIQUIDS. 

Liquids,  in  the  course  of  being  transformed  into  vapour  on  the  applica- 
tion of  heat,  absorb  a  certain  quantity  of  heat  which  remains  latent  in  the 
vapour,  and  is,  on  the  contrary,  restored  to  sensibility,  and  communicated  to 
other  bodies  when  the  vapours  are  condensed  into  liquids.  The  following 


BOILING   POINTS   OF   SATURATED  VAPOURS. 


371 


Table  No.  124. — BOILING  POINTS  OF  SATURATED  VAPOURS  UNDER 
VARIOUS  PRESSURES,  OR  THEIR  CORRESPONDING  TEMPERATURES 
AND  PRESSURES. 

Regnault. 


Pressure  per  square  inch  of  the  vapour  of  the  following  liquids:— 

TEMPERATURE. 

Water. 

Alcohol. 

Ether. 

Chloroform. 

Turpentine. 

Centigrade. 

Fahrenheit. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

0° 

32° 

.089 

.246 

3-53 



.041 

10 

5° 

.178 

.466 

5-54 

2.52 

•045 

20 

68 

-337 

-85I 

8.60 

3-68 

.083 

30 

86 

.609 

1.52 

12.32 

5-34 

•135 

40 

104 

i.  06 

2-59 

17.67 

7.04 

.217 

50 

122 

1.78 

4.26 

24-53 

10.14 

•333 

60 

I4O 

2.88 

6-77 

33-47 

14.27 

.520 

70 

158 

4-51 

10.43 

44.67 

18.88 

.810 

80 

I76 

6.86 

15-72 

57-01 

26.46 

1.18 

90 

194 

10.16 

23.02 

75-41 

35-03 

1.74 

100 

212 

14.70 

32.60 

95-17 

45-54 

2.61 

no 

230 

20.80 

45-50 

120.9 

58.42 

3-62 

116 

240.8 

25-37 

— 

137.0 

— 

— 

I2O 

248 

29.88 

62.05 

— 

4-97 

130 

266 

39-27 

83.80 

— 

— 

6.71 

I36 

276.8 

46.87 

— 

— 

— 

I4O 

284 

52-56 

109.1 

— 

— 

8.94 

150 

3O2 

69.27 

140.4 

— 

— 

11.70 

I52 

305-6 

73-07 

147-3 

— 

— 

— 

160 

320 

89.97 

147.3 

— 

— 

13.10 

170 

338 

II5-3 

— 

— 

I9-I3 

1  80 

356 

146.0 

— 

— 

— 

23.70 

190 

374 

182.6 

— 

— 

— 

29.30 

200 

392 

226.1 

— 

— 

— 

36.09 

2IO 

410 

277.1 

— 

— 

— 

43-54 

22O 

428 

336.4 

— 

— 

— 

52.04 

222 

431.6 

349-3 

— 

— 

— 

53-74 

BOILING  POINTS  UNDER  ONE  ATMOSPHERE. 

i  st.  According  to  table  No.  122. 

Water.                     Alcohol.               Sulphuric  ether.          Chloroform.              Turpentine. 

212°  F  173°  F  100°  F. 

140°  ] 

F.  3JS°  F. 

2d.  By  interpolation  in  the  above  table. 

212°       173°       94°       142°        335° 

table,  No.  125,  gives,  on  the  authority  of  Despretz,  Favre  and  Silbermann, 
and  Regnault,  the  latent  heat  of  evaporation  of  several  vapours  under  one 
atmosphere.  The  total  heat  of  evaporation,  reckoned  from  32°  F.,  is  added 
in  the  last  column.  It  is  calculated  for  each  liquid  by  multiplying  its 


372 


HEAT. 


boiling  point  less  32°,  by  its  specific  heat,  to  find  the  quantity  of  heat  in 
units  required  to  raise  it  to  the  boiling  point,  and  adding  this  product  to 
the  latent  heat. 

Table  No.   125. — LATENT  HEAT  AND  TOTAL  HEAT  OF  EVAPORATION  OF 
LIQUIDS  UNDER  ONE  ATMOSPHERE. 


Liquid. 

Boiling  Point. 

Latent  Heat  of 
Evaporation. 

Total  Heat, 
reckoned  from 
32°  F. 

Sulphuric  ether  

Fahrenheit. 
100° 

Units  of  heat. 
175 

Units  of  heat. 
2IO.4 

Wood  spirit 

I  ^O 

A.1  ^ 

CAC    Q 

Acetic  ether 

-1  Jw 

i6c 

T-/  J 

IQI 

OT-J'? 

Alcohol  pure         .         .... 

A  ^0 

17-2 

j.y  A 

374 

4.6l    7 

Water         .         

x  /  o 
212 

O  1  T- 

o6c  2 

ij.WA.    ^ 

I  146    I 

Essence  of  lemon     

2Q7 

yw  j"6 
126 

2CC    ? 

Oil  of  turpentine  

31  e 

124. 

J  J'O 

256  6 

LIQUEFACTION  AND  SOLIDIFICATION  OF  GASES. 

Professor  Faraday  succeeded  in  liquefying,  and  even  solidifying,  many 
gases,  and  it  is  probable  that  all  gases  are  susceptible  of  being  solidified, 
and  that  they  might  be  so  condensed  if  sufficiently  low  temperatures  and 
sufficiently  strong  vessels  could  be  produced. 

At  -  112°  F.,  and  under  a  pressure  less  than  one  atmosphere,  Faraday 
reduced  the  following  gases  to  the  liquid  or  the  solid  state : — Chlorine, 
cyanogen,  ammonia,  hydrosulphuric  acid,  arseniated  hydrogen,  hydriodic 
acid,  hydrobromic  acid,  and  carbonic  acid. 

The  following  gases  were  solidified  at  the  annexed  temperatures : — 

Cyanogen -31°  F. 

Hydriodic  acid -  60 

Carbonic  acid -  72 

Oxide  of  chlorine -  76 

Ammonia -  103 

Sulphurous  acid -  105 

Sulphuretted  hydrogen -123 

Hydrobromic  acid -  126 

Protoxide  of  nitrogen -  148 

The  following  gases  were  not  solidified,  even  at  a  temperature  of 
-i66°F.:— 

Olefiant  gas. 
Fluosilicic  acid. 
Protophosphuretted  hydrogen. 
Fluoboric  acid. 
Hydrochloric  acid. 
Arseniated  hydrogen. 

The  following  gases  gave  no  sign  of  even  approaching  liquefaction,  even 
at  -  166°  F.,  and  with  many  atmospheres  of  pressure: — 


LIQUEFACTION   AND    SOLIDIFICATION    OF   GASES.  373 

Hydrogen at  -  166°  F.,  and  27  atmospheres. 

Oxygen „  -  166  and  27  „ 

Do „  -140  and  58  „ 

Nitrogen „  -  166  and  50  „ 

Nitric  oxide „  -  166  and  50  „ 

Carbonic  oxide „  -  166  and  40  „ 

Coal  gas „  -  166  and  32  „ 

The  greatest  known  degree  of  cold,  -  166°  F.,  or  -  110°  C.,  was  pro- 
duced for  these  experiments  by  Professor  Faraday.  As  to  the  method  of 
production,  see  SOURCES  OF  COLD. 

According  to  the  results  of  more  recent  experiments,  hydrogen  has  been 
subjected  to  a  pressure  of  8000  atmospheres  without  making  any  sign  of 
condensation. 

SOURCES   OF   COLD. 

The  production  of  cold — the  abstraction  of  heat — is  a  curious  subject  of 
inquiry.  When  a  salt  is  dissolved  in  water,  cold  is  produced.  When  a 
liquid  vaporizes,  the  heat,  latent  and  sensible,  necessary  for  the  production 
of  the  vapour,  is  abstracted  from  some  other  body  in  contact  with  the 
liquid,  and  cold  is  produced.  When  spirits  of  wine  or  aromatic  vinegar, 
for  example,  is  thrown  on  the  body,  a  sense  of  cold  immediately  results 
from  the  vaporization  of  the  liquid  which  draws  heat  from  the  body.  If 
air  is  allowed  to  expand,  there  is  a  reduction  of  temperature,  and  a  transla- 
tion of  heat  from  neighbouring  bodies.  Again,  in  hot  climates,  water  is 
successfully  cooled  in  porous  vessels,  through  the  pores  of  which  the  water 
passes  to  the  exterior,  and  is  vaporized,  and  the  cooling  process  is  accele- 
rated by  a  current  of  air  directed  upon  the  vessel,  which  quickens  the 
vaporization. 

Siebe's  ice-making  machine,  invented  originally  by  Jacob  Perkins  in  1834, 
is  based  on  the  principle  of  producing  cold  by  the  evaporation  of  a  volatile 
fluid — ether  by  preference.  The  fluid  is  placed  in  an  air-tight  vessel,  and 
evaporated  in  vacuo,  the  vacuum  being  formed  by  means  of  a  pump,  which, 
in  its  continued  efforts  to  reduce  the  pressure,  promotes  the  evaporation  of 
the  fluid  at  a  low  temperature.  A  temperature  50°  below  the  freezing 
point  may  be  effected;  but  in  place  of  an  unprofitably  low  temperature,  the 
cooling  action  is  distributed  through  the  mass  of  salt  water  employed  as  the 
freezing  medium,  the  salt  water  retaining  its  fluidity  below  32°  F.,  and 
circulating  in  the  refrigerator  around  and  between  the  ice-moulds,  which 
are  filled  with  fresh  water.  The  water  in  the  moulds  is  successively  frozen, 
and  replaced  by  fresh  moulds  filled  with  water. 

Carre's  cooling  apparatus  is  based  on  the  fact  that  water,  when  cold, 
absorbs  a  large  quantity  of  ammoniacal  gas,  which,  when  the  water  is 
heated,  escapes,  and  is  condensed  in  a  cold  vessel.  On  the  contrary,  when 
the  water  just  heated  becomes  cold,  a  vacuum  is  formed,  and  excites  a 
rapid  evaporation  of  the  ammonia  into  the  vessel  of  cooled  water,  when  it 
is  again  absorbed.  The  heat  necessary  for  the  evaporation  of  the  ammonia 
is  extracted  from  the  water  surrounding  the  vessel  in  which  the  liquid 
ammonia  is  contained,  and  the  water  consequently  is  frozen. 

FRIGORIFIC  MIXTURES. 
For  the  production  of  intense  cold,  mixtures  of  various  salts  and  acids  in 


374 


HEAT. 


various  proportions  with  water  are  very  effective.  But  more  intense  degrees 
of  cold  are  produced  with  snow  or  ice. 

Table  No.  126  contains  the  ordinary  mixtures  for  the  artificial  produc- 
tion of  cold,  known  as  freezing  mixtures.  The  first  part  of  the  table  com- 
prises mixtures  of  salts  and  acids  with  each  other  and  with  water;  the 
second  part,  mixtures  of  salts  and  acids  with  snow  or  ice. 

The  blanks  in  the  third  column  of  the  table  indicate  that  the  thermo- 
meter sinks  to  the  degrees  named  in  the  second  column,  but  never  lower, 
whatever  may  be  the  initial  temperature  of  the  materials  when  mixed. 

The  vessels  containing  the  mixtures  should  be  cooled  before  the  elements 
are  put  into  them. 

If  the  materials  of  the  mixtures  enumerated  in  the  first  part  of  the  table 
be  mixed  at  a  higher  temperature  than  that  given  in  the  table,  namely, 
50°  F.,  the  fall  of  temperature  is  greater.  Thus,  if  the  most  powerful  of  these 
mixtures,  No.  n,  be  made  at  the  temperature  80°  F.,  it  will  sink  the 
thermometer  to  +  2°,  making  a  fall  of  78  degrees,  as  against  71  degrees  in 
the  table. 

The  third  part  of  the  table  contains  frigorific  mixtures  partly  selected 
from  the  other  parts,  and  combined  so  as  to  extend  the  cold  to  the  extreme 
degree,  -91°  F.  The  materials  should  be  cooled  previously  to  being 
mixed  to  the  initial  temperature,  by  mixtures  taken  from  previous  parts  of 
the  table. 

Table  No.  126. — FRIGORIFIC  MIXTURES. 
FIRST  PART. — Proportional  mixtures  of  Salts  and  Acids  with  Water. 


Mixtures. 

Fall  of  Temperature. 

Degrees 

of  cold 
produced. 

i    Nitrate  of  ammonia 

i  ) 

Fahrenheit. 

Fahr. 

Water         

T| 

from  +50°  to  +  4°  

46° 

2.  Muriate  of  ammonia     

5 

Nitrate  of  potash  

5> 

from  +  50°  to  +  10°  

AO 

Water  

T6J 

3    Muriate  of  ammonia 

c  \ 

Nitrate  of  potash 

5 

Sulphate  of  soda 

if 

from  +  50°  to  +  4°  

46 

Water      

T6 

4.  Sulphate  of  soda  

3  ) 

Diluted  nitric  acid  

2} 

from  +  50°  to  -  3°  

53 

5.  Nitrate  of  ammonia  
Carbonate  of  soda 

±4 

from  +  50°  to  —  7°  

^7 

Water 

)] 

6    Phosphate  of  soda  

•  •  •  •     A  / 
Q  ) 

Diluted  nitric  acid  
7    Sulphate  of  soda  

...4} 

.  8  ) 

from  +  50°  to-  12°  

62 

Hvdrochlonc  acid         . 

,} 

from  +  50  to     o   

5° 

8    Sulphate  of  soda              .  . 

•  •  •  •      J    ) 
C    } 

Dilute  sulphuric  acid  

47 

from  +  50°  to  +  3°  

47 

FRIGORIFIC  MIXTURES. 
Table  No.  126  (continued}. 


375 


Mixtures. 

Fall  of  Temperature. 

Degrees 

of  cold 
produced. 

9.  Sulphate  of  soda 

6  \ 

Fahrenheit. 

Fahr. 

Muriate  of  ammonia 

U 

Nitrate  of  potash  

•   *\ 

from  +  50°  to-  10°  

60° 

Dilute  nitric  acid  

4J 

10    Sulphate  of  soda 

6  i 

Nitrate  of  ammonia 

sl 

from  +  5  o°  to  —  14° 

64 

Dilute  nitric  acid         

•   4J 

1  1    Phosphate  of  soda  

T^  / 
.      Q    ) 

Nitrate  of  ammonia 

6\ 

from  +  50°  to  —  2  1° 

71 

Dilute  nitric  acid 

'  4) 

1       T-  / 

SECOND  PART. — Proportional  mixtures  of  Salts  and  Acids  with  Snow  or  Ice. 


Mixtures. 

Fall  of  Temperature. 

Degrees 
of  cold 
produced. 

12.  Muriate  of  soda  (common  salt). 
Snow,  or  pounded  ice  

*} 

Fahrenheit. 

from  any  temp,  to  -  5° 

Fahr. 

1  3.  Muriate  of  soda  

2   ) 

Muriate  of  ammonia  

do.       do.    to-  12° 

Snow  or  pounded  ice 

5 

14    Muriate  of  soda 

0  / 
IO  i 

Muriate  of  ammonia  

5 

Nitrate  of  potash  

sf 

do.       do.    to  -  1  8° 



Snow,  or  pounded  ice  

4J 

15.  Muriate  of  soda  

jfl 

Nitrate  of  ammonia  . 

B 

do        do.    to  —  25° 

Snow,  or  pounded  ice  

'') 

1  6.  Dilute  sulphuric  acid  

1  } 

Snow  

3} 

from  +  32°  to  -23°  

55° 

1  7    Muriatic  acid 

tr  ) 

Snow     

1) 

from  +  32°  to-  27°  

59 

1  8    Dilute  nitric  acid  

1  ) 

Snow 

77 

from  +  3  2°  to  -30°  

62 

i  o    Muriate  of  lime  .  ... 

/  ) 
r  ) 

Snow      

I/ 

from  +  3  2°  to  -40°  

72 

20.  Crystallized  muriate  of  lime  
Snow                                   . 

1} 

from  +  3  2°  to  -50°  

82 

2  1    Potash                

A) 

Snow              

3J 

from  +  32°  to  -51°  

83 

O  J 

376 


HEAT. 


Table  No.  126  (continued}. 

THIRD  PART. — Mixtures  partly  selected  from  the  foregoing  series,  and  combined  so  as 
to  increase  or  extend  the  cold  to  the  greatest  extremes. 


Mixtures. 

Fall  of  Temperature. 

Degrees 
of  cold 
produced. 

22    Sea  salt              .    . 

r\ 

Fahrenheit. 

Fahr. 

Muriate  of  ammonia  ) 

I 

from-  5°  to-  1  8° 

Q 

Nitrate  of  potash        j 
Snow,  or  pounded  ice  

23    Sea  salt 

5  "I 

Nitrate  of  ammonia 

i 

t  > 

from-  1  8°  to-  25° 

Snow  or  pounded  ice 

« 

24    Phosphate  of  soda 

X^    J 

c  } 

Nitrate  of  ammonia  .    . 

31- 

from  o°  to  —  34° 

Dilute  nitric  acid  

l) 

34 

25.  Phosphate  of  soda  

*r  / 

<2    } 

Nitrate  of  ammonia  

n 

from  —  34°  to  —  50° 

16 

Dilute  mixed  acids  

1 

26.  Snow 

3  ) 

0  > 

from  o°  to  —  46°  

46 

Dilute  nitric  acid 

2    I 

27.  Snow  

X) 

Dilute  sulphuric  acid 

31- 

from  —  10°  to  —  ^6° 

46 

Dilute  nitric  acid  

3J 

q.\j 

28.  Snow  

O  J 

T   ) 

Dilute  sulphuric  acid  

T| 

from  -10°  to  -60°  

5o 

2  Q    Snow 

Muriate  of  lime 

1 

from  +  20°  to  -  48°  

68 

30.  Snow  

1  ) 

Muriate  of  lime  . 

4/ 

from+  10°  to-  54°  

64 

31.  Snow  

T-   ) 

2   \ 

Muriate  of  lime  

3f 

from  -i  5°  to  -68°  

S3 

32.   Snow... 

O   ) 
T   ) 

Crystallized  muriate  of  lime  
33.   Snow  

1} 

from  o°to-66°  

66 

Crystallized  muriate  of  lime  
M.  Snow... 

3} 

8  ) 

from  -40°  to-  73°  

33 

Dilute  sulphuric  acid  .  .  . 

TO} 

from  -68°  to  -91°  

23 

>W   J 

COLD  BY  EVAPORATION. 

M.  Gay-Lussac  directed  a  current  of  air,  dried  or  desiccated  by  being 
passed  through  chloride  of  calcium,  upon  the  bulb  of  a  thermometer 
wrapped  in  moist  cambric.  The  temperature  was  lowered  from  10°  to 
26°  F.,  according  to  the  temperature  of  the  current  of  air,  which  varied 
from  32°  to  77°  F.  It  is  presumed  that  the  surrounding  temperature  was 
the  same  as  that  of  the  current.  The  following  are  the  falls  of  temperature 
for  currents  of  air  of  given  temperatures : — 


COLD   BY  EVAPORATION.  377 

Temperature  of  current,  Fahrenheit,  32°,     41°,  50°,  59°,     68°,  77°. 
Fall  of  temperature,  do.         io°.5,  13°,  16°,  i9°.5,  23°,  26°.$. 

The  most  intense  cold  as  yet  known  was  produced  by  Professor  Faraday 
in  the  course  of  his  experiments  on  the  liquefaction  and  solidification  of 
gases,  from  the  evaporation  of  a  mixture  of  solid  carbonic  acid  and 
sulphuric  ether  under  the  receiver  of  an  air-pump.  For  the  following  pres- 
sures, measured  in  inches  of  mercury,  and  given  also  in  pounds  per  square 
inch,  he  obtained  the  corresponding  temperatures  subjoined : — 

Inches  of  mercury 28.4,     19.4,       9.6,        7.4,       5.4,       3.4,       2.4,        1.4,       1.2. 

Lbs.  per  square  inch .     14.0,       9.5,       4.6,       3.6,       2.7,        1.7,       1.2,       0.7,       0.6. 
Temperatures,  Fahr. .  -  107°,  -  112°,  -  121°,  -  125°,  -  132°,  -  139°,  -  146°,  -  161°,  -  166°. 

Showing  that  when  a  perfect  vacuum  was  nearly  approached,  an  intense 
cold,  measured  by-  166°  F.,  was  attained  by  the  evaporation  of  a  mixture 
of  solid  carbonic  acid  and  sulphuric  ether. 


STEAM. 


When  steam  is  generated  in  a  boiler,  the  water  is  heated  till  it  arrives  at 
the  temperature  of  ebullition,  and  the  elevation  of  temperature  is  sensible  to 
the  thermometer;  next,  the  water  is  converted  into  steam  by  an  additional 
absorption  of  heat,  which  is  not  measured  by  the  thermometer,  and  is 
therefore  called  latent  heat.  The  heat  is  not,  in  fact,  latent,  but  is  appro- 
priated in  converting  water  into  steam,  of  the  same  temperature. 

The  pressure,  as  well  as  the  density,  of  steam  which  is  generated  over 
water  in  a  boiler  rises  with  the  temperature;  and,  reciprocally,  the  tempera- 
ture rises  with  the  pressure  and  density.  There  is  only  one  pressure  and 
one  density  for  each  temperature ;  and  thus  it  is  that  steam,  produced  in  a 
boiler  over  water,  is  always  generated  at  the  maximum  density  and  maxi- 
mum pressure  corresponding  to  its  temperature.  In  such  condition  steam 
is  said  to  be  saturated,  being  incapable  of  vaporizing  more  water  into  the 
same  space,  unless  the  temperature  be  raised.  Saturation  is  therefore  the 
normal  condition  of  steam  generated  in  contact  with  a  store  of  water,  and 
the  same  density  and  the  same  pressure  are  always  to  be  found  in  conjunc- 
tion with  the  same  temperature. 

In  consequence,  saturated  steam  over  water  stands  both  at  the  condensing 
point  and  at  the  generating  point;  that  is,  it  is  condensed  if  the  temperature 
falls,  and  more  water  is  evaporated  if  the  temperature  rises. 

But,  supposing  the  whole  of  the  water  to  be  evaporated,  or  that  a  body 
of  saturated  steam  is  isolated  from  water,  in  a  space  of  fixed  dimensions,  if 
an  additional  quantity  of  heat  be  supplied  to  the  steam,  the  state  of  satura- 
tion ceases,  the  steam  becomes  superheated,  and  the  temperature  and  the 
pressure  are  increased,  whilst  the  density  is  not  increased.  Steam,  thus 
surcharged  with  heat,  approaches  to  the  condition  of  a  perfect  gas. 


PHYSICAL  PROPERTIES  OF  STEAM. 
RELATION  OF  THE  TEMPERATURE  AND  PRESSURE  OF  SATURATED  STEAM. 

The  results  of  the  experimental  observations  of  M.  Regnault  on  the 
temperature  and  pressure  of  saturated  steam,  whose  observations  have 
superseded  in  practice  those  of  previous  experimentalists,  show  that  the 
temperature  rises  more  slowly  than  the  pressure.  For  example,  the 
pressures  being  advanced  at  equal  intervals  of  5  Ibs.  per  square  inch,  thus : — 

i  Ib.     6  Ibs.     ii  Ibs.     16  Ibs.     21  Ibs.     26  Ibs.     31  Ibs.     36  Ibs., 
the  temperatures  in  Fahrenheit  degrees  are — 

102°.  I,        I70°.2,        I970.8,       2l6°.3,       230°.6,       242°.3,       2S2°.2,       26o°.9, 


PHYSICAL   PROPERTIES  OF  STEAM.  379 

which  advance  by  the  following  diminishing  differences,  — 

68°.i,     27°.6,     i8°.s,     i4°.3,     "°.7,     9°-9>     8°-7, 

Without  quoting  the  formula  employed  by  M.  Regnault  for  calculating  the 
pressures  due  to  the  temperatures  in  French  measures,  it  will  suffice  to  give, 
in  a  subsequent  table  (No.  127),  the  relative  pressures  in  inches  of  mercury 
and  in  pounds  per  square  inch,  based  on  his  formula,  for  low  temperatures 
ranging  from  32°  F.  to  212°  F.,  as  given  by  Claudel. 

To  define  the  relation  of  the  temperature  and  pressure  of  saturated  steam 
for  the  higher  temperatures  comprised  in  the  observations  of  M.  Regnault. 
the  late  Mr.  W.  M.  Buchanan  arranged  a  simple  formula  which  applies  with 
accuracy  to  temperatures  ranging  from  120°  F.  to  446°  F.,  the  higher  limit 
of  the  range  of  Regnault's  observations.  These  limits  correspond  to 
pressures  of  from  1.68  Ibs.  to  445  Ibs.  per  square  inch.  The  formula  is  as 
follows:- 


in  which  /  is  the  pressure  in  Ibs.  per  square  inch,  and  /  is  the  temperature 
of  saturated  steam  in  degrees  Fahrenheit,  as  observed  by  means  of  an  air- 
thermometer. 

TOTAL  HEAT  OF  SATURATED  STEAM. 

The  constituent  or  total  heat  of  steam  consists  of  its  latent  heat,  in 
addition  to  its  sensible  heat.  The  latent  heat  of  saturated  steam  at  o°  C, 
the  freezing  point,  was  experimentally  determined  by  Regnault  to  be  equal 
to  606°.  5  C.;  or  such  that  the  total  heat  of  one  pound  of  saturated  steam 
at  o°  C.  would  be  capable  of  raising  the  temperature  of  606.5  IDS-  °f 
water  one  degree.  At  higher  temperatures,  the  total  heat  of  saturated  steam 
was  found  to  increase  uniformly  between  the  temperatures  o°  C.  and  230°  C., 
at  the  rate  of  .305°  C.  for  each  increment  of  temperature  of  i°;  and, 
therefore,  if  the  temperature  in  Centigrade  degrees  be  multiplied  by  .305, 
and  606.5  be  added  to  the  product,  the  sum  will  express  the  total  heat  of 
saturated  steam  at  the  given  temperature  measured  from  o°  C.  ;  or 

H  =  606.  5  +  .305  /(Centigrade),  ...............  (  2  ) 

in  which  H  =  the  total  heat  of  saturated  steam  of  any  temperature  f  C. 

This  formula  is  adapted  to  the  Fahrenheit  scale,  by  taking  the  total  heat 
at  32°  F.  equal  to  6o6°.5  C.  x  9/s  =  1091°.  7  F.  For  any  other  temperature 
/°  F.,  the  total  heat  is  equal  to  1091°.  7  F.  +  .305  (^-32).  The  first 
quantity  in  this  expression,  namely  1091°.  7,  is  slightly  too  much,  for  whilst 
Regnault  found  that  the  total  heat  of  steam  at  100°,  his  starting  point,  was 
636°.67  C.,  it  was  calculated  by  his  formula  (No.  2  above)  to  be  637°  C. 
The  above-named  quantity  should  therefore  be  reduced  to  1091.16,  and 
the  formula  for  the  total  heat  of  steam  in  terms  of  Fahrenheit  degrees  will 
stand  thus:-  H  =  1091.16  +  .305  (/-  32),  or 

H=  1081.4  +.305  *;  ..............................  (3) 

that  is,  the  total  heat  of  saturated  steam  of  any  given  temperature  in 
Fahrenheit  degrees  is  equal  to  io8i°.4  plus  the  product  of  the  temperature 
by  .305,  supposing  that  the  water  from  which  the  steam  is  generated  is 
supplied  at  the  temperature  32°  F. 


380  STEAM. 

The  expression  of  the  total  heat  represents  units  of  heat  when  the  weight 
of  the  steam  is  one  pound. 

Supposing  that  the  water  to  be  evaporated  is  supplied  at  any  higher 
temperature  than  32°  F.,  the  total  heat  to  be  expended  in  evaporating  it  is 
found  by  deducting  the  difference  of  temperature  from  the  total  heat  as 
found  by  the  formula  (  3  ).  Or,  the  formula  may  be  modified  by  deducting 
the  difference  of  temperature  from  the  first  quantity,  1081.4. 

For  example,  if  water  be  supplied  at  the  ordinary  temperature,  62°  F., 
which  is  30  degrees  above  32°  F.,  then  1081.4-30=1051.4  will  be  the 
proper  first  quantity.  For  these  and  the  other  cardinal  temperatures  of 
water,  100°  F.  and  212°  F.,  the  four  equations  for  the  total  heat  of  steam 
raised  from  water  at  the  respective  temperatures  are  as  follow: — 

(Initial  temperature  32°  F.),  H=  1081.4  +  .305^° (3) 

(  Do.  62°  R),  H  =  io5i.4  +  .3o5/0 (4) 

(  Do.  100°  F.),  H=ioi3.4  +  .3o5/° (5) 

(  Do.  212°  R),  H=    900.5  +  . 305^° (6) 

In  the  reduction  for  the  last  equation  (  6  ),  32°  has  been  deducted  from 
2i2°.9,  and  not  from  212°,  in  order  to  take  into  account  the  item  .9°,  being 
the  extra  specific  heat  of  water  at  212°  F.,  compared  with  that  of  water  at 
32°  F. 

LATENT  HEAT  OF  SATURATED  STEAM. 

As  the  total  heat  is  increased  .305°,  which  is  less  than  a  third  of  a  degree, 
whilst  the  sensible  heat,  or  temperature,  rises  i°,  and  the  sensible  heat  thus 
rises  faster  than  the  total  heat,  the  latent  heat  must  be  reduced  as  the 
temperature  rises,  by  as  much  as  .305°  is  less  than  i°,  or  by  i°  -  .305°  =  .695°, 
for  each  degree  of  temperature,  and  the  latent  heat  for  any  temperature 
t°  C.  is  expressed  by  the  quantity  6o6°.5  -  .6g$t. 

There  is  a  modifying  element,  namely,  the  specific  heat  of  the  water, 
which  slightly  increases  with  the  temperature,  and  which  requires  the 
fraction  .695  /  to  be  proportionally  increased.  The  equation  of  Clausius, 
in  which  this  slight  variation  is  allowed  for,  is 

L=6o7  -.708  /  (Centigrade), (  7  ) 

where  L  =  the  latent  heat  due  to  the  temperature  /  C. 

To  adapt  this  formula  to  the  Fahrenheit  scale,  take  9/stns  °f  607  = 
io92°.6  F.,  and  substitute  (t°  -  32°)  F.  for  t°  C.;  then  the  formula  becomes 

L=  1092.6  -  .708  (t  —  32);  or 

L=  1 1 15. 2 -.708  /; (8) 

that  is,  the  latent  heat  of  saturated  steam  at  any  given  temperature  in 
Fahrenheit  degrees  is  equal  to  1115.2  less  the  product  of  the  temperature 
by  .708,  supposing  that  the  water  which  is  converted  into  steam  is  supplied 
at  32°  F. 

APPROPRIATION  OF  THE  CONSTITUENT  HEAT  OF  SATURATED 
STEAM  AT  212°  F. 

To  trace  the  appropriation  of  all  the  heat  that  goes  to  the  formation 
of  a  pound  of  steam,  in  the  sensible  and  the  latent  state,  in  terms  of 
thermal  units,  as  well  as  of  dynamic  units,  or  foot-pounds,  take  one  pound 
of  water  at  32°  F.,  and  convert  it  into  saturated  steam  at  212°  F.,  the  first 


CONSTITUENT   HEAT   OF   SATURATED   STEAM.  381 

instalment  of  heat  is  the  sensible  heat,  and  it  is  required  for  elevating  the 
temperature  of  the  water  to  2 1 2°,  through  180°;  in  other  words,  to  increase  the 
molecular  velocity,  and  slightly  expand  the  liquid,  which  appropriates  180.9 
units  of  heat,  equivalent  to  i8o.9x  772,  or  139,655  foot-pounds.  Secondly, 
latent  heat  is  applied  in  overcoming  the  molecular  attraction,  and  separating 
the  particles;  that  is  to  say,  in  the  formation  of  steam,  which  appropriates 
892.9  units  of  heat,  equal  to  689,318  foot-pounds.  Thirdly,  latent  heat  is 
applied  in  repelling  the  incumbent  pressure,  whether  of  the  atmosphere  or 
of  the  surrounding  steam;  that  is  to  say,  in  raising  a  load  of  14.7  Ibs.  per 
square  inch,  or  2116.4  Ibs.,  on  a  square  foot,  through  a  cubic  space  of  26.36 
cubic  feet,  being  the  volume  of  one  pound  of  saturated  steam.  The  work 
thus  done  is  equal  to  2116.4.  x  26.36,  or  55,788  foot-pounds,  or  its  equivalent, 
72.3  units  of  heat.  In  strictness,  there  is  the  initial  volume  of  the  pound 
of  water  to  be  deducted  from  this  total  volume,  to  show  the  exact  volume 
generated ;  but  it  is  relatively  very  small,  and  is  inconsiderable. 

The  second  of  the  above  appropriations  of  the  heat  was  found  by  sub- 
tracting the  sum  of  the  first  and  third,  which  are  both  arrived  at  by  direct 
observation,  from  the  total  heat. 

The  first  appropriation  of  heat  is  thus  seen  to  be  the  sensible  heat,  and 
the  second  and  third  together  constitute  the  latent  heat.  The  third,  it  may 
be  added,  is  simply  an  expression  of  the  mechanical  labour  necessary  to 
disengage  26.36  cubic  feet  of  steam,  and  force  it  into  space  against  an 
atmospheric  pressure  of  2116.4  Ibs.  per  square  foot. 

The  appropriation  of  the  heat  expended  in  the  generation  of  one  pound 
of  saturated  steam  at  212°  F.,  from  water  supplied  at  32°  R,  may  be 
exhibited  thus: — 

To  GENERATE  ONE  POUND  OF  STEAM  AT  212°  F. 


Units  of  heat. 


The  sensible  heat: — 

1.  To  raise  the  temperature  of 

the  water  from  3 2° to  2 1 2°  F.,  180.9 

The  latent  heat : — 

2.  In  the  formation  of  steam ...   892. 93^ 

3.  In  resisting  the   incumbent 

atmospheric  pressure  of  1 4. 7 
Ibs.  per  square  inch,  or 
2116.4  Ibs.  per  square  foot,  72.265 


965.2 


Total  or  constituent  heat, 1 1 46.  i 


Mechanical  equivalent 
in  foot-pounds. 


139^55 


689,346 


55,788 


745»I34 
884,789 


VOLUME  AND  DENSITY  OF  SATURATED  STEAM. 

The  density  of  steam  is  expressed  by  the  weight  of  a  given  constant 
volume,  say,  one  cubic  foot;  and  the  volume  is  expressed  by  the  number 
of  cubic  feet  in  one  pound  of  steam.  The  density  and  volume,  which  are 
the  reciprocals  of  each  other,  have  not  yet  been  accurately  ascertained  by 
direct  experiment.  They  are,  however,  determinable  in  terms  of  the 
pressure,  temperature,  and  latent  heat  of  steam,  all  of  which  have  been 
experimentally  ascertained,  by  means  of  the  mechanical  theory  of  heat. 


382  STEAM. 

Mr.  Brownlee  has  deduced  a  simple  expression  for  the  density  of  saturated 
steam  in  terms  of  the  pressure,  as  follows  :  — 


or,  log  D-.94I  Iog/-2.5i9,  .........................  (10) 

in  which  D  is  the  density,  and  p  the  pressure  in  Ibs.  per  square  inch.     In 


this  expression,  p-w  is  the  equivalent  of  p,  as  employed  by  Dr.  Rankine  ; 
and  it  is  simpler  to  handle.  The  equation  signifies  that  the  logarithm  of 
the  pressure  is  to  be  multiplied  by  .941,  and  that  2.519  is  to  be  subtracted 
from  the  product  ;  the  remainder  is  the  logarithm  of  the  density,  from  which 
the  density  is  found  by  means  of  a  table  of  logarithms. 

The  results  presented  by  the  above  formula  are  very  accurate;  they  do 
not  differ  from  those  obtained  in  terms  of  the  temperature  and  the  latent 
heat,  for  pressures  of  from  i  Ib.  to  250  Ibs.  per  square  inch,  by  more  than 
one-seventh  per  cent. 

The  volume  being  the  reciprocal  of  the  density,  then,  putting  V  for  the 
volume, 


or  log  ¥  =  2.519  -.941  log/;  .........................  (12) 

that  is,  that  if  the  logarithm  of  the  pressure  in  Ibs.  per  square  inch  be 
multiplied  by  .941,  and  the  product  be  deducted  from  2.519,  the  remainder 
is  the  logarithm  of  the  volume,  in  cubic  feet,  of  one  pound  of  saturated 
steam.  The  nearness  of  the  power,  .941,  to  unity,  indicates  that  the  density 
of  saturated  steam  varies  nearly  as  the  pressure,  but  in  a  lower  ratio;  and 
that  the  volume  of  saturated  steam  varies,  for  short  intervals,  nearly  in  the 
inverse  ratio  of  the  pressure.  For  example,  the  pressures  per  square  inch 
being  — 

i,  2,  4,  8,          16,          32,          64,         128  Ibs., 

the  densities,  or  weights  per  cubic  foot,  which  are  inversely  as  the  volumes, 
are  — 

.0030,     .0058,     .0112,     .0214,     .0411,     .0789,     .1516,     .2911  Ibs., 
being  in  the  ratios  of  — 

i,          1-93,       3-73,       7-13,       13-7,       29.3,       50.5,        97. 
RELATIVE  VOLUME  OF  SATURATED  STEAM. 

The  relative  volume  of  saturated  steam  is  expressed  by  the  number  of 
volumes  of  steam  produced  from  one  volume  of  water,  the  volume  of 
water  being  measured  at  the  temperature  62°  F.  The  relative  volume  is 
found  by  multiplying  the  volume,  in  cubic  feet,  of  one  pound  of  steam  by 
the  weight  of  a  cubic  foot  of  water  at  62°  F.,  which  is  62.355  Ibs. 

Or,  it  may  be  found  directly  in  terms  of  the  pressure,  by  multiplying  the 
second  member  of  the  formula  (n)  by  62.355.  Thus,  putting  n  for  the 
relative  volume, 


GASEOUS  STEAM.  383 

=  62.355  x  33°.36 


or,  log  «  =  4.31388  -(.941  *log/);  ..................  (14) 

that  is,  if  the  logarithm  of  the  pressure  in  Ibs.  per  square  inch  be  multiplied 
by  .941,  and  the  product  be  deducted  from  4.31388,  the  remainder  is  the 
logarithm  of  the  relative  volume. 

GASEOUS  STEAM. 

When  saturated  steam  is  superheated,  or  surcharged  with  heat,  it  advances 
from  the  condition  of  saturation  into  that  of  gaseity.  The  gaseous  state  is 
only  arrived  at  by  considerably  elevating  the  temperature,  supposing  the 
pressure  remains  the  same.  Steam  thus  sufficiently  superheated  is  known 
as  gaseous  steam,  or  "  steam-gas,"  as  Dr.  Rankine  has  named  it. 

The  test  of  perfect  gaseity  is  the  uniformity  of  the  rate  of  expansion  with 
the  rise  of  temperature;  and,  whereas,  during  the  first  few  degrees  which 
follow  the  temperature  of  saturation,  the  rate  of  expansion  is  notably 
greater  than  that  of  air,  the  rate  diminishes  at  still  higher  temperatures,  and 
ultimately  becomes  uniform,  like  that  of  the  expansion  of  permanent  gases. 

Dr.  C.  W.  Siemens,  experimenting  on  the  expansion  of  isolated  steam, 
generated  at  212°,  and  superheated  and  maintained  at  atmospheric  pressure, 
found  that  expansion  proceeded  rapidly  until  the  temperature  rose  to  220°, 
and  less  rapidly  up  to  230°,  or  18°  above  the  saturation  point;  above  which 
it  expanded  uniformly,  as  a  permanent  gas.  Up  to  230°,  the  expansion 
was  five  times  as  much  as  that  of  air. 

Messrs.  Fairbairn  and  Tate  found  that  for  steam  generated  at  low 
temperatures  of  saturation,  —  under  150°  F.,  —  the  rate  of  expansion  when 
the  steam  was  heated  was  nearly  uniform.  At  175°  F.,  the  expansion  for 
the  first  five  degrees  averaged  more  than  three  times  that  of  air;  above 
that  point,  it  was  nearly  the  same.  For  steam  generated  at  the  high 
temperature  of  324°  F.,  for  a  total  pressure  of  95  Ibs.  per  square  inch,  the 
rate  of  expansion  up  to  331°  was  nearly  three  times  that  of  air;  and  for  the 
next  25  degrees,  one-sixth  greater. 

M.  Regnault  concluded  from  his  experiments  that  saturated  steam  was 
nearly  gaseous  at  temperatures  below  60°  F. 

It  may  be  gathered  from  these  observations  that  saturated  steam  of 
ordinary  temperatures  may  be  made  gaseous  by  superheating  it  to  the 
extent  of  from  10  to  20  degrees.  It  is  thought  that  the  rapidity  of  expan- 
sion by  heat,  near  the  boiling  point,  is  to  be  accounted  for  by  the  supposed 
insensible  moisture  of  steam  in  the  saturated  condition,  as  generated  from 
water,  being  evaporated  and  contributing  to  increase  the  quantity  of  steam 
without  raising  the  temperature.  This  argument  is  plausible;  but  it  might 
be  argued,  on  the  contrary,  that  in  the  converse  process,  of  abstracting  heat 
from  superheated  steam,  the  accelerated  reduction  of  volume  when  it  ap- 
proaches the  point  of  saturation,  is  due  to  incipient  condensation,  which 
would  be  absurd. 

It  may  be  inferred,  further,  that  saturated  steam  of  very  low  temperatures, 
under  150°  or  100°  F.,  is  gaseous. 


384  STEAM. 

TOTAL  HEAT  OF  GASEOUS  STEAM. 

Regnault  found  that  the  total  heat  of  gaseous  steam  increased,  like  that 
of  saturated  steam,  uniformly  with  the  temperature;  and  at  the  rate  of 
.475°  for  each  degree  of  temperature,  under  a  constant  pressure.  A  formula 
for  the  total  heat  of  gaseous  steam  may  be  constructed  on  the  basis  of  that 
for  saturated  steam,  by  a  modification  of  the  constants;  and  for  the  adjust- 
ment of  these,  take  the  two  steams  at  a  low  temperature,  as  40°  F.,  where 
they  are  identical  in  constitution,  both  being  gaseous.  Then,  by  formula 
(  3 )  for  saturated  steam,  the  total  heat  at  this  temperature  is 

1081.4 +  (.305  x  40°)=  io93°.6  F. 

Substituting  for  the  second  quantity  in  this  equation,  the  quantity  (.475  x  40°), 
and  reducing,  then 

1074.6  +  (.475  x  40°)  =  io93°.6  F. 

Whence  the  general  formula  for  the  total  heat  of  gaseous  steam,  produced 
from  water  at  32°  F., 

H'  =  1074.6  +  .475  /, (15) 

FT  being  the  total  heat,  in  Fahrenheit  degrees,  and  /  the  temperature ;  that 
is,  that  to  the  constant  1074.6,  is  to  be  added  the  product  of  the  tempera- 
ture by  .475,  to  find  the  total  heat. 

By  this  formula  it  is  found  that  the  total  heat  of  gaseous  steam  at  212°  F., 
and  at  atmospheric  pressure,  is  1175.3°  F.,  which  is  29.2  degrees,  or  2% 
per  cent,  more  than  that  of  saturated  steam. 

SPECIFIC  HEAT  OF  STEAM. 

The  specific  heat  of  saturated  steam  is  .305,  that  of  water  being  unity; 
or,  it  is  1.281,  that  of  air  being  unity.  It  may  be  noted  that  .305°  is  the 
quantity  by  which  the  total  heat  of  saturated  steam  is  increased  for  each 
degree  of  temperature  (see  formula  3);  so  that  equal  intervals  of  tempera- 
ture correspond  to  equal  quantities  of  heat.  The  expression,  .305,  for 
specific  heat,  is  taken  in  a  compound  sense,  comprising  the  changes  both 
of  volume  and  of  pressure  which  take  place  in  the  production  of  saturated 
steam. 

The  specific  heat  of  gaseous  steam  is  .475,  under  constant  pressure,  as 
found  by  Regnault.  It  is  upwards  of  a  half  more  than  that  of  saturated 
steam.  It  is  identical  with  the  increase  of  total  heat  for  each  degree  of 
temperature  (formula  15). 

THE  SPECIFIC  DENSITY  OF  STEAM. 

The  specific  density  of  gaseous  steam  has  been  found  by  M.  Regnault 
to  be  .622,  that  of  air  being  i.  That  is  to  say,  that  the  weight  of  a  cubic 
foot  of  gaseous  steam  is  about  five-eighths  of  that  of  a  cubic  foot  of  air,  of 
the  same  pressure  and  temperature. 

The  specific  density  of  saturated  steam  is  usually  taken  at  the  same 
value  as  that  of  gaseous  steam,  as  an  approximation  to  the  actual  value. 
Thus  approximated,  it  is  only  correct  at  very  low  temperatures,  for  the 
specific  density  increases,  though  not  rapidly,  with  the  temperature,  inso- 
much that  though  it  is  practically  the  same  as  that  of  gaseous  steam  at 
100°  F.,  it  becomes  .643  at  212°  F.;  and  at  303°  F.,  with  70  Ibs.  absolute 


PROPERTIES   OF    SATURATED   STEAM.  385 

pressure  per  square  inch,  it  becomes  .664,  or  two-thirds  of  that  of  air.     At 
358.3°  R,  with  150  Ibs.  pressure,  it  is  .681.     (See  table  No.  129.) 

DENSITY  OF  GASEOUS  STEAM. 

The  density  or  weight  of  a  cubic  foot  of  gaseous  steam  is  expressible  by 
the  same  formula  as  for  that  of  air  (page  350),  except  that  the  multiplier  or 
coefficient  is  less  in  proportion  to  the  less  specific  density,  thus  :  — 

TV  _  2-7Q74/  x  -622  _  1.684  /  /T6x 

~  ~'' 


in  which  D'  is  the  weight  of  a  cubic  foot  of  gaseous  steam,/  the  total  pressure 
per  square  inch,  and  t  the  temperature  by  Fahrenheit. 

TABLES  OF  THE  PROPERTIES  OF  SATURATED  STEAM. 

The  first  table,  No.  127,  of  the  properties  of  saturated  steam  of  tempera- 
tures ranging  from  32°  to  2  1  2°  F.  is  adapted  from  a  table  prepared  by  Claudel, 
partly  based  on  Regnault's  formulas,  and  partly  on  the  assumption  that  the 
specific  density  of  saturated  steam  is  uniformly  .622,  or  about  five-eighths 
that  of  air  at  the  same  temperature.  As  already  mentioned,  the  specific 
density  increases,  in  fact,  slightly  with  the  temperature,  and  this  deviation 
from  uniformity  explains  the  small  discrepancies  between  the  weights  of 
steam  as  given  in  table  No.  127,  and  those  as  given  for  temperatures  below 
212°  in  the  next  table. 

The  table  No.  128  gives  the  properties  of  saturated  steam  for  pressures 
of  from  i  Ib.  per  square  inch  to  400  Ibs.  per  square  inch,  the  temperatures 
ranging  from  102°  to  445°  F.  The  first  column  contains  the  ascending 
total  pressures  in  Ibs.  per  square  inch.  The  second  column,  of  tempera- 
tures, was  calculated  from  the  pressures  by  means  of  the  formula  (  i  )  :  — 

/  =  -      2938-16  _  -371.85. 
6-  1993544  -log/ 

The  third  column,  of  the  total  heat  of  saturated  steam,  by  formula  (  3  )  :  — 

H=  1081.4  +  .  305  /. 
The  fourth  column,  of  the  latent  heat  of  saturated  steam,  by  formula  (  8  )  :  — 

L=  1115.2  -  708  /. 

The  fifth  column,  of  the  density  of  saturated  steam,  by  formula  (10):  — 
log  D  =  .94i  log/-  2.519. 

The  sixth  column,  of  the  volume  of  saturated  steam,  was  calculable  by 
finding  the  reciprocals  of  the  densities,  or  by  formula  (12)  :  — 

log  ¥-2.519  -.941  log/. 

The  seventh  column,  of  the  relative  volume  of  saturated  steam,  by  the 
formula  (14):  — 

log  n  =  4.31388  -  (.941  x  log/). 

The  table  No.  129  contains  the  comparative  densities  and  volumes  of 
air  and  saturated  steam  for  pressures  up  to  300  Ibs.  total  pressure  per  square 
inch,  and  temperatures  to  4  17°.  5  F.,  together  with  the  specific  density  of 
saturated  steam. 

25 


386 


STEAM. 


Table  No.  127. — PROPERTIES  OF  SATURATED  STEAM  FROM  32°  TO  212°  F. 


PRES 

5URE. 

Total  heat 

• 

Volume  of  one 

TEMPERATURE. 

Inches  of 

Lbs.  per 

reckoned  from 
water  at  32°  F. 

VV  eight  of  100 
cubic  feet. 

pound  of 
vapour. 

mercury. 

square  inch. 

Fahrenheit. 

inches. 

Ibs. 

Fahrenheit. 

Ibs. 

cubic  feet. 

32°    ... 

...        .l8l... 

...        .089... 

...     I09I.2     ... 

...      .031   ... 

...3226 

35 

.204 

.100 

IO92.I 

•034 

2941 

40 

.248 

.122 

1093.6 

.041 

2439 

45    ••• 

...        .299... 

...        .147... 

...     I095.I     ... 

...      .049... 

...2041 

5o 

.362 

.178 

1096.6 

•059 

1695 

55 

.426 

.214 

1098.2 

.070 

1429 

60    ... 

...        .517... 

...        .254... 

...     1099.7     ... 

...      .082... 

...1220 

65 

.619 

•304 

IIOI.2 

.097 

1031 

70 

•733 

.360 

II02.8 

.114 

877.2 

75    ••• 

...     .869... 

...        .427... 

...     1104.3     ••• 

...      .134... 

...     746.3 

80 

1.024 

•503 

II05.8 

.156 

641.0 

35 

1.205 

•592 

II07.3 

.182 

549-5 

QO 

.    1.410... 

607 

1  108.9 

.212  ... 

471  7 

y 

T^ 

•wyo'  •  • 

T-  /      / 

95 

1.647 

.809 

IIIO-4 

.245 

408.2 

100 

1.917 

.942 

IIII.9 

•283 

353-4 

105     ... 

...    2.229... 

...    1.095... 

...     III3.4    -.. 

...      .325   ... 

...   307.7 

no 

2-579 

1.267 

III5.0 

"373 

268.1 

115 

2.976 

1.462 

IIl6.5 

.426 

234-7 

120      ... 

...  3.430... 

...    1.685... 

...     IIlS.O     ... 

...      .488... 

...   204.9 

I25 

3-933 

1.932 

III9.5 

•554 

180.5 

130 

4-5°9 

2.215 

II2I.I 

.630 

158.7 

135  ••• 

.   5.174... 

...    2.542... 

...     II22.6     ... 

...    .714... 

...    1  40.  i 

140 

5.860 

2.879 

II24.I 

.806 

124.1 

145 

6.662 

3-273 

II25.6 

.909 

IIO.O 

150  ... 

...   7-548... 

...   3.708... 

...     II27.2     ... 

...    .022  ... 

...     97.8 

J55 

8-535 

4.193 

II28.7 

•145 

87-3 

1  60 

9.630 

4-731 

II30.2 

•333 

75-o 

165    ... 

...10.843... 

...    5.327... 

II3I.7     . 

...    .432... 

...     69.8 

170 

12.183 

5.985 

H33.3 

.602 

62.4 

175 

13-654 

6.708 

II34.8 

•774 

56-4 

180    ... 

...15.291... 

...    7.511... 

...     1136.3     ... 

...  1.970  ... 

...     50.8 

185 

17.044 

8.375 

II37.8 

2.181 

45-9 

190 

19.001 

9-335 

H39.4 

2.411 

41-5 

195    ... 

...21.139... 

...10.385... 

...     II40.9     ... 

...  2.662  ... 

...     37.6 

200 

23.461 

11.526 

II42.4 

2-933 

34-i 

205 

25-994 

12.770 

H43-9 

3-225 

31.0 

210      ... 

...28.753... 

...14.126... 

...     1145.5     ••• 

•••  3-543  ••. 

...     28.2 

212 

29.922 

14.700 

II46.I 

3-683 

27.2 

PROPERTIES   OF   SATURATED   STEAM. 
Table  No.  128. — PROPERTIES  OF  SATURATED  STEAM. 


387 


Total 
pressure 
per  square 
inch. 

Temperature 
in  Fahrenheit 
degrees. 

Total  heat,  in 
Fahr.  degrees, 
from  water 
at  32°  F. 

Latent  heat, 
Fahrenheit 
degrees. 

Density, 
or  weight  of 
one  cubic  foot. 

Volume  of 
one  pound 
of  steam. 

Relative  volume, 
or  cubic  feet  of 
steam  from  one 
cubic  ft.  of  water. 

Ibs. 

Fahr. 

Fahr. 

Fahr. 

Ibs. 

cubic  feet. 

Rel.  vol. 

I 

1  02.  1 

III2.5 

1042.9 

.0030 

330.36 

20600 

2 

126.3 

III9.7 

1025.8 

.0058 

172.08 

10730 

3"- 

...  I4I.6  ... 

...  II24.6  ... 

...  IOI5.O  ... 

...  .0085  ... 

...  117.52... 

...     7327 

4 

I53.I 

II28.I 

1006.8 

.0112 

89.62 

5589 

5 

162.3 

II30.9 

1000.3 

•0138 

72.66 

4530 

6... 

...  170.2  ... 

...  1133.3  .•• 

...     994-7... 

...  .0163  ... 

...     6l.2I  ... 

...     3816 

7 

176.9 

H35-3 

990.0 

.0189 

52.94 

3301 

8 

182.9 

1137.2 

985.7 

.0214 

46.69 

2911 

9... 

...  188.3... 

...  1138.8  ... 

...     981.9... 

....0239... 

...     41-79- 

...     2606 

10 

193.3 

1140.3 

978.4 

.0264 

37.84 

2360 

ii 

197.8 

1141.7 

975-2 

.0289 

34.63 

2157 

12 

...  2O2.O 

Q72  2 

O^T  A. 

31  88 

IQ88 

13 

205.9 

1144.2 

969.4 

.0338 

29.57 

1844 

14 

209.6 

H45-3 

966.8 

.0362 

27.61 

1721 

IA  7  .. 

...  2I2.O  ... 

..ii  46.  i 

n6  c,  2 

...  .0380  ... 

2636 

1642 

*+•  /    *  •  • 

15 

2I3.I 

1146.4 

...      yv^^.^i 
964.3 

.0387 

25.85 

1611 

16 

216.3 

1  147.4 

962.1 

.0411 

24.32 

1516 

17 

2IQ  6 

1148.3  ... 

QCQ  8 

O/13C 

22  06 

.    1432 

*  »y«v*  ..  . 

...     V3V-"  ••> 

...     ^^,.yw  ... 

T^  O 

18 

222.4 

1149-2 

957-7 

.0459 

21.78 

1357 

19 

225.3 

1150.1 

955-7 

.0483 

20.70 

1290 

20  ... 

...  228.0... 

...  1150.9... 

...   953.8... 

....0507... 

...      1972... 

...     1229 

21 

230.6 

1151.7 

95L9 

•0531 

18.84 

1174 

22 

233.1 

1152.5 

950.2 

.0555 

18.03 

1123 

23.. 

-.235.5... 

...  1153.2... 

...    948.5  ... 

...  .0580... 

...      17.26... 

...     1075 

24 

237.8 

II53-9 

946.9 

.0601 

16.64 

1036 

25 

240.1 

1154.6 

945-3 

.0625 

15.99 

996 

26.. 

...242.3... 

...  1155.3... 

...    943-7  ••• 

...  .0650  ... 

...     15.38... 

...       958 

27 

2444 

1155.8 

942.2 

.0673 

14.86 

926 

28 

246.4 

1156.4 

940.8 

.0696 

14-37 

895 

29.. 

...  248.4... 

...  1157.1  ... 

...    939.4... 

....0719... 

...      13.90... 

...     866 

30 

250.4 

1157.8 

937-9 

.0743 

13.46 

838 

31 

252.2 

1158.4 

936.7 

-.0766 

I3.05 

813 

Art 

2C/L  I 

...i  158.9  ... 

0780  . 

12.67  '•• 

780 

33 

255.9 

1159.5 

934-0 

..  .  ,\J  1  <_>y  ..  . 

.0812 

12.31 

l^-y 
767 

34 

257.6 

1  1  60.0 

932.8 

-0835 

11.97 

746 

•2C 

2CQ  7 

...  1  1  60.  5  ... 

031.6 

...  .0858  ... 

1  1.65 

726 

3 

260.9 

1161.0 

930-5 

.0881 

11-34 

*  *   "                      /    ~W 

707 

37 

262.6 

1161.5 

.0905 

11.04 

688 

38 

264.2  . 

...i  162.0  ... 

928.2  ... 

10  76 

671 

39 

265.8 

1162.5 

927.1 

.0952 

...      A  >-».  /  \j  .  .  . 
10.51 

w/  * 

655 

40 

267.3 

1162.9 

926.0 

.0974 

10.27 

640 

41  .. 

...268.7... 

...  1163.4... 

...    924.9... 

....0996... 

...      10.03... 

...  625 

42 

270.2 

1163.8 

923-9 

.1020 

9.8l 

6n 

43 

271.6 

1164.2 

922.9 

.1042 

9-59 

598 

44. 

273  O 

1164.6  ... 

.1065  ... 

585 

45 

274.4 

1165.1 

920.9 

.1089 

99.?s" 

•  •  •          j     ^ 

572 

46 

275.8 

1165.5 

919.9 

.1111 

9.00 

561 

47  .. 

...  277.1  ... 

...  1165.9... 

...    919.0... 

....1133... 

...     8.82... 

...     550 

48 

278.4 

1166.3 

918.1 

.1156 

8.65 

539 

49 

279.7 

1166.7 

917.2 

.1179 

8.48 

529 

50... 

...  281.0... 

...  1167.1  ... 

...  .1202  ... 

...     8.31  ... 

...      518 

388 


STEAM. 


Table  No.  128  (continued). 


Total 
pressure 
per  square 
inch. 

Temperature 
in  Fahrenheit 
degrees. 

Total  heat,  in 
Fahr.  degrees, 
from  water 
at  32°  F. 

Latent  heat, 
Fahrenheit 
degrees. 

Density, 
or  weight  of 
one  cubic  foot. 

Volume  of 
one  pound 
of  steam. 

Relative  volume, 
or  cubic  feet  of 
steam  from  one 
cubic  ft.  of  water. 

Ibs. 

Fahr. 

Fahr. 

Fahr. 

Ibs. 

cubic  feet. 

Rel.  vol. 

51 

282.3 

1167.5 

9154 

.1224 

8.17 

509 

52 

283.5 

1167.9 

9H.5 

.1246 

8.04 

500 

53." 

...284.7... 

...II68.3... 

...    913.6    ... 

...  .1269  ... 

..     7.88     ... 

...      491 

54 

285.9 

II68.6 

912.8 

.1291 

774 

482 

55 

287.1 

1169.0 

912.0 

•I3H 

7.6l 

474 

56... 

...288.2... 

...  1169.3  ... 

...    9II.2    ... 

....1336... 

..     748     ... 

...     466 

57 

289.3 

1169.7 

910.4 

.1364 

7.36 

458 

58 

290.4 

II70.0 

909.6 

.1380 

7-24 

45  l 

59- 

...  291.6  ... 

...  1170.4... 

...    908.8    ... 

....1403  ... 

..     7.12     ... 

...     444 

60 

292.7 

II70.7 

908.0 

.1425 

7.01 

437 

61 

293.8 

II7I.I 

907.2 

.1447 

6.90 

43° 

62... 

...294.8... 

...  II7I.4... 

...    906.4    ... 

...  .1469... 

..    6.81    ... 

...     424 

63 

295.9 

II7I.7 

905.6 

•H93 

6.70 

417 

64 

296.9 

II72.0 

904.9 

.1516 

6.60 

411 

65... 

...  298.0... 

...  II72.3  ... 

...    904.2    ... 

....1538... 

...    6.49    ... 

...     405 

66 

299.0 

II72.6 

903.5 

.1560 

6.41 

399 

67 

300.0 

II72.9 

902.8 

.1583 

6.32 

393 

68 

3OO.Q  .    , 

1  17"?  2 

..    QO2.I     . 

...  .1605  ... 

...    6.23    ... 

...     388 

69 

301.9 

II73-5 

;/ 

901.4 

.1627 

v.  j 

6.15 

383 

70 

302.9 

II73.8 

900.8 

.1648 

6.07 

378 

71... 

•••  303.9.  " 

...  II74.I  ... 

...    900.3    ... 

...  .1670  ... 

...    5.99    ... 

.••     373 

72 

304.8 

H74.3 

899.6 

.1692 

368 

73 

305.7 

II74.6 

898.9 

.1714 

5-83 

363 

74-.. 

...306.6... 

...  1174.9... 

...    898.2    ... 

...  .1736  ••• 

...    5.76    ... 

•••     359 

75 

307.5 

II75.2 

897.5 

•1759 

5.68 

353 

76 

308.4 

II754 

896.8 

.1782 

5.61 

349 

77... 

...309.3... 

...  II75.7  ... 

...    896.1    ... 

...  .1804... 

...    5.54    ... 

.-•     345 

78 

310.2 

II76.0 

895.5 

.1826 

5.48 

34  r 

79 

3II.I 

II76.3 

894-9 

.1848 

5.41 

337 

80 

3I2O... 

1  176.  t\ 

SQA  i 

...  .1869  ... 

c  ->c 

333 

O\J.  .  * 

...   0^4.^)    ... 

81 

312^8 

II76.8 

.    893.7 

.1891 

5.29 

329 

82 

3I3.6 

II77.I 

893.1 

•1913 

5.23 

325 

83,.. 

...  3J4-5  — 

...  II774... 

...  892.5  ... 

....I935". 

...  5.17  ... 

...      321 

84 

3I5.3 

II77.6 

892.0 

•1957 

5.11 

318 

IL 

316.1 
...  316.9... 

II77.9 
...  II78.I   ... 

891.4 
...  890.8  ... 

.1980 

...  .2002  ... 

5.05 
...  5.00  ... 

3H 

87 

317.8 

II78.4 

890.2 

.2O24 

4.94 

308 

88 

318.6 

II78.6 

889.6 

.2044 

4.89 

305 

80 

•2TO  A 

1  178  Q 

889.0  ... 

2O67  .  . 

.    4.84    . 

•2QJ 

oy... 

...   i  i  /  o.y  .  . 

....  4*\~>\j  j   •  . 

9° 

320.2 

II79.I 

"  888.5 

.2089 

4-79 

298 

9i 

321.0 

II79-3 

887.9 

.2111 

4-74 

295 

92... 

...321.7... 

...  II79.5  .-" 

...  887.3  ... 

....2133... 

...    4.69    ... 

...     292 

93 

322.5 

II79.8 

886.8 

.2155 

4.64 

289 

94 

323.3 

IlSo.O 

886.3 

.2176 

4.60 

286 

9f 

1180.3  ... 

885.8 

...  .2198  ... 

A  er 

283 

5... 
96 

324*8 

II80.5 

885.2 

.2219 

4.51 

281 

97 

325.6 

1  1  80.8 

884.6 

.2241 

4.46 

278 

98... 

...   326.3  ... 

...  IlSl.O  ... 

...  884.1   ... 

...  .2263  ... 

...    4.42     ... 

...  275 

99 

327.1 

1181.2 

883.6 

.2285 

4-37 

272 

100 

327.9 

Il8l.4 

883.1 

.2307 

4-33 

270 

IOI... 

...328.5... 

...  1181.6... 

...  882.6  ... 

....2329... 

...    4.29     ... 

...  267 

PROPERTIES   OF   SATURATED   STEAM. 


389 


Table  No.  128  (continued}. 


Total 
pressure 
per  square 
inch. 

Temperature 
in  Fahrenheit 
degrees. 

Total  heat,  in 
Fahr.  degrees, 
from  water 
at  32°  F. 

Latent  heat, 
Fahrenheit 
degrees. 

Density, 
or  weight  of 
one  cubic  foot. 

Volume  of 
one  pound 
of  steam. 

Relative  volume, 
or  cubic  feet  of 
steam  from  one 
cubic  ft.  of  water. 

Ibs. 

Fahr. 

Fahr. 

Fahr. 

Ibs. 

cubic  feet. 

Rel.  vol. 

102 

329.1 

1181.8 

882.1 

.2351 

4.25 

265 

103 

329-9 

1182.0 

881.6 

•2373 

4.21 

262 

104... 

...330.6... 

..  1182.2  ... 

..    88I.I    ... 

...2393... 

..     4.18     ... 

..       260 

105 

331-3 

1182.4 

880.7 

.2414 

4.14 

257 

1  06 

331-9 

1182.6 

880.2 

•2435 

4.11 

255 

I  O7 

332  6 

..  1182.8  ... 

..    87Q.7    .. 

.  .  24.^6  .  . 

4.  O7 

2C3 

j.\-Y  •  •  • 

108 

•••  jj^"*- 

333-3 

1183.0 

T/y/ 

879-2 

•^TOW 
.2477 

4.04 

251 

109 

334-0 

1183-3 

878.7 

.2499 

4.00 

249 

I  IO 

334.  6  . 

1183  ;  .. 

..  878.3  .. 

.  .2S2I  . 

3  Q7 

247 

in 

•  •  jjf"*' 

335-3 

..ii  '-'JO  • 

1183.7 

*     *•*/  *-'*  J     •  ' 

877.8 

"»3«  *      •  •  • 

-2543 

3-93 

245 

112 

336.o 

1183.9 

877-3 

.2564 

3-90 

243 

I  I  -3 

336  7 

1  184.1  ... 

..  876.8  .. 

.  2;86  .  . 

386 

24.1 

i  i  j.  .  . 
114 

•  •  jjv-'-/  •  • 

337-4 

1184.3 

•  •     *•*/  ^•^-'     •  •  • 

876.3 

2 
.2607 

...          J.«JV/ 

3.83 

239 

H5 

338.0 

1184.5 

875.9 

.2628 

3.80 

237 

116... 

...338.6... 

..  1184.7  ... 

..  875.5  ». 

...  .2649... 

...  3.77  ••• 

-..      235 

117 

339-3 

1184.9 

875.0 

.2652 

3.74 

233 

118 

339-9 

1185.1 

874-5 

.2674 

3.71 

231 

119,.. 

...340.5  ... 

..1185.3... 

..  874.1  ... 

....2696... 

...  3.68  ... 

...      229 

120 

34I-I 

1185.4 

873.7 

.2738 

3.65 

227 

121 

341.8 

1185.6 

873.2 

.2759 

3.62 

225 

122... 

...  342.4... 

..  1185.8  ... 

...  872.8  ... 

....2780... 

...  3.59  ... 

...      224 

123 

343-0 

1186.0 

872.3 

.2801 

3.56 

222 

124 

343-6 

1186.2 

871.9 

.2822 

3-54 

221 

125... 

...344.2  ... 

..  1186.4... 

...  871.5  ... 

....2845... 

...    3.51    ... 

...      219 

126 

344-8 

1186.6 

871.1 

.2867 

3-49 

217 

127 

345-4 

1186.8 

870.7 

.2889 

3'46 

215 

128... 

...  346.0... 

..  1186.9... 

...  870.2  ... 

...  .2911  ... 

...    3.44    ... 

...      214 

I29 

346.6 

1187.1 

869.8 

-2933 

3-41 

212 

I30 

347-2 

1187.3 

869.4 

-2955 

3.38 

211 

131... 

...347-8... 

...1187.5... 

...  869.0  ... 

....2977... 

...    3.35    ... 

209 

I32 

348.3 

1187.6 

868.6 

.2999 

3-33 

2o8 

133 

348.9 

1187.8 

868.2 

.3020 

3-31 

206 

I  34. 

34.Q.H  .. 

...  1188.0  ... 

867  8 

3O4.O  . 

3  2Q 

2o  S 

135 

350.1 

1188.2 

8674 

....  jw-pj  .  . 
.3060 

...     j«^y 
3-27 

•&w  3 
203 

136 

350.6 

1188.3 

867.0 

.3080 

3-25 

202 

177.. 

..  3<n.2  .. 

...  1188.5  ••• 

...  866.6  ... 

...  .3IOI  ... 

3  22 

2OO 

1  J/ 
I38 

jj*"" 
351-8 

1188.7 

8662 

.3121 

3-20 

199 

139 

352.4 

1188.9 

865.8 

.3142 

3.18 

198 

140... 

...352.9... 

...  1189.0  ... 

...  865.4  .. 

...  .3162  ... 

...  3.16  .. 

...       197 

141 

353-5 

1189.2 

865.0 

.3184 

3.14 

195 

143 

354-0 

1189.4 

864.6 

.3206 

3.12 

194 

14.3... 

..  3^4..^ 

11896  ... 

...  864.2  .. 

...  .3228  ... 

3  IO     . 

103 

1Lt-j*  '  • 
144 

•       Jj^T'J 

355-o 

1189.7 

863.9 

.3250 

...          J.*'-' 

3.08 

yj 
192 

H5 

355-6 

1189.9 

863.5 

•3273 

3.06 

190 

I46.. 

...356.1  ... 

...  1190.0  ... 

...  863.1  .. 

....3294... 

...  3.04  .. 

...     189 

H7 

356.7 

1190.2 

862.7 

-3315 

3-02 

1  88 

148 

357-2 

1190.3 

862.3 

.3336 

3-00 

i87 

149.. 

-357-8... 

...  1190.5.. 

...  861.9  •• 

•••.3357" 

...    2.98    .. 

...     1  86 

150 

358.3 

1190.7 

861.5 

-3377 

2.96 

184 

155 

361.0 

1191.5 

859.7 

.3484 

2.87 

179 

160.. 

...363.4... 

...  1192.2  .. 

...  857.9  •• 

....3590.. 

...    2.79    .. 

...     174 

390 


STEAM. 


Table  No.  128  (continued}. 


Total 
pressure 
per  square 
inch. 

Temperature 
in  Fahrenheit 
degrees. 

Total  heat,  in 
Fahr.  degrees, 
from  water 
at  32°  F. 

Latent  heat, 
Fahrenheit 
degrees. 

Density, 
or  weight  of 
one  cubic  foot. 

Volume  of 
one  pound 
of  steam. 

Relative  volume, 
or  cubic  feet  of 
steam  from  one 
cubic  ft.  of  water. 

Ibs. 

Fahr. 

Fahr. 

Fahr. 

Ibs. 

cubic  feet. 

Rel.  vol. 

I65 

366.0 

II92.9 

856.2 

.3695 

2.71 

169 

I/O 

368.2 

II93-7 

854-5 

-3798 

2.63 

164 

175... 

...370.8... 

...  1194.4... 

...    852.9    ... 

...    .3899... 

...     2.56     ... 

...       159 

1  80 

372.9 

II95.I 

85L3 

.4009 

2.49 

155 

I85 

375-3 

II95.8 

849.6 

.4117 

2-43 

151 

190... 

...377.5... 

...  1196.5  ... 

...    848.0    ... 

...    .4222... 

...     2.37     ... 

...       148 

195 

379-7 

II97.2 

846.5 

•4327 

2.31 

144 

200 

38i.7 

II97.8 

845.0 

•4431 

2.26 

141 

210... 

...  386.0... 

...  II99.I   ... 

...    841.9    ... 

...    .4634... 

...     2.16     ... 

•••       135 

220 

389.9 

1200.3 

839.2 

.4842 

2.06 

129 

230 

393-8 

I20I.5 

836.4 

.5052 

1.98 

123 

240.  .  . 

...397.5  ... 

...  1202.6  ... 

...    833.8    ... 

...    .5248... 

...      1.90     ... 

...       119 

250 

401.1 

1203.7 

831.2 

.5464 

1.83 

114 

260 

404.5 

1204.8 

828.8 

.5669 

I.76 

I  10 

270... 

...407.9... 

...  1205.8  ... 

...    826.4    ... 

...    .5868... 

...      1.70     ... 

...     106 

280 

411.2 

1  206.8 

824.1 

.6081 

1.64 

102 

290 

414.4 

1207.8 

821.8 

.6273 

1.59 

99 

300... 

...417.5... 

...  1208.7  ... 

...    819.6    ... 

...    .6486... 

...      1.54     ... 

...      96 

35° 

430.1 

I2I2.6 

810.7 

.7498 

1-33 

83 

400 

444.9 

I2I7.I 

800.2 

.8502 

1.18 

73 

Hypothetical  values,  calculated  by  means  of  the  same  formulas,  for  pressures 

beyond  the  range  of  Regnaulfs  observations:  — 

450... 

...456.7.-. 

...  1220.7  ... 

...    791.9    ... 

...    -9499-  •• 

...    1.05    ... 

...       66 

500 

467.5 

1224.0 

784.2 

1  .0490 

•95 

59 

600 

487.0 

1229.9 

770.4 

1.2450 

.80 

50 

700... 

...504.1  ... 

...  I235.I  ... 

...    758.3    ... 

...1.4395... 

...     .69    ... 

...      43 

800 

5i9'5 

1239.8 

7474 

1.6322 

.61 

38 

900 

533-6 

1244.2 

737-4 

1.8235 

•55 

34 

1000... 

...546.5... 

...  I248.I  ... 

...  728.3  ... 

...2.0140... 

...     .50   ... 

...       31 

Note  to  Table. — This  table  was  originally  published  in  the  article  "Steam,"  contributed 
by  the  author  to  the  Encyclopedia  Britannica,  8th  edition. 


AIR   AND   SATURATED   STEAM. 


391 


Table   No.    129. — COMPARATIVE   DENSITY   AND  VOLUME    OF   AIR    AND 

SATURATED  STEAM. 


Total 
presssure 
per  square 
inch. 

Temperature 
in  Fahrenheit 
degrees. 

Density,  or  weight  of 
one  cubic  foot. 

Volume  of  one  pound. 

Specific  density 
of  saturated 
steam. 

Air. 

Steam. 

Air. 

Steam. 

Air  =  i. 

Ibs. 

Fahrenheit. 

Ib. 

Ib. 

cubic  feet. 

cubic  feet. 

I 

IO2.I 

.0048 

.0030 

2O8.0I 

33°-36 

.622 

5 

162.3 

.O2I7 

.0138 

46.04 

72.66 

.635 

10 

193-3 

.0414 

.0264 

24.17 

37.84 

.638 

14.7 

212.0 

.0591 

.0380 

16.91 

26.36 

•643 

20 

228.0 

.0786 

.0507 

12.72 

19.72 

•645 

30 

250.4 

.1142 

-0743 

8.76 

13.46 

.651 

40 

267.3 

.1487 

.0974 

6-73 

10.27 

.655 

50 

28l.O 

.1824 

.1202 

5.48 

8-31 

•659 

60 

292.7 

•2155 

.1425 

4.64 

7.01 

.661 

70 

302.9 

.2481 

.1648 

4-03 

6.07 

.664 

80 

312.0 

.2802 

.1869 

3-57 

5-35 

.667 

90 

320.2 

•3119 

.2089 

3.21 

4-79 

.670 

IOO 

327.9 

-3432 

.2307 

2.91 

4-33 

.672 

no 

334-6 

•3743 

.2521 

2.67 

3-97 

.673 

120 

34I-I 

.4051 

.2738 

2.17 

3-65 

.676 

I30 

347-2 

•4355 

•2955 

2.30 

3.38 

.678 

I4O 

352.9 

•4657 

.3162 

2.15 

3.16 

.679 

150 

358.3 

-4957 

•3377 

2.02 

2.96 

.68l 

160 

363-4 

•5255 

•3590 

I.9O 

2.79 

.683 

170 

368.2 

•5551 

.3798 

1.80 

2.63 

.684 

1  80 

372.9 

•5844 

.4009 

1.71 

2.49 

.686 

190 

377-5 

.6i35 

.4222 

1.63 

2-37 

.688 

200 

38i.7 

.6425 

•4431 

1.56 

2.26 

.690 

220 

389.9 

.7000 

.4842 

1.43 

2.06 

.692 

240 

397-5 

•7569 

.5248 

1.32 

1.90 

.694 

260 

404-5 

•8i33 

.5669 

1.23 

1.76 

.697 

280 

411.2 

.8691 

.6081 

MS 

1.64 

.700 

300 

417.5 

.9246 

.6486 

i.  08 

1-54 

.702 

MIXTURE  OF  GASES  AND  VAPOURS. 


If  two  or  more  gases  or  vapours,  not  having  any  power  of  chemical 
action  one  upon  another,  be  introduced  into  the  same  space,  each  gas  will, 
after  a  certain  interval,  be  diffused  equally  throughout  the  whole  of  the 
space,  and  will  occupy  the  space  exactly  as  if  no  other  gas  were  present. 
The  gases  thus  become  intimately  mixed. 

Moreover,  the  elastic  force  or  pressure  of  each  gas  is  the  same  as  if  it 
alone  occupied  the  given  space,  and  the  total  or  resulting  pressure  of  the 
mixture  is  equal  to  the  sum  of  the  pressures  of  the  individual  gases. 

If  a  vessel  be  filled  with  dry  air,  and  a  sufficient  quantity  of  water  be 
introduced  into  the  vessel,  the  water  is  evaporated,  and  the  vapour  occupies 
the  vessel  just  as  if  the  vessel  had  been  empty,  and  had  previously  contained 
a  vacuum.  The  evaporation  proceeds  until  the  vapour  becomes  saturated; 
that  is  to  say,  until  the  pressure  and  density  of  the  vapour  arrive  at  the 
maximum  due  to  the  temperature  of  the  mixture. 

And  the  final  pressure  of  the  mixture  of  air  and  vapour  is  equal  to  the 
pressure  of  the  contained  air  plus  the  pressure  of  the  vapour. 

These  two  propositions,  with  respect  to  the  mixture  of  air  and  vapour, 
hold  with  respect  to  the  mixture  of  vapour  with  gases  generally.  They 
have  been  practically  verified  by  the  results  of  direct  experiment  made  by 
M.  Regnault;  though  he  found  a  very  slight  inferiority  of  the  pressure  of 
vapour  to  that  due  to  saturation,  which  he  attributed  to  the  hygroscopic 
affinity  of  the  walls  of  the  vessel. 

The  process  of  evaporation  is  much  less  rapid  in  presence  of  a  gas,  than 
when  it  takes  place  in  a  vacuum;  owing  to  the  resistance  opposed  by  the 
pressure  of  the  gas  to  the  disengagement  of  vapour. 

The  same  law  applies  for  determining  the  pressure  of  a  mixture  of  gas  and 
vapour,  when  the  quantity  of  vapour  falls  short  of  the  condition  of  saturation. 

Air  is  said  to  be  saturated  with  moisture  when  the  moisture  or  vapour  it 
contains  is  itself  in  the  condition  of  saturation,  or  of  maximum  density  due 
to  the  temperature  of  the  air. 

HYGROMETRY. 

The  condition  of  the  air  with  respect  to  moisture  is  called  its  humidity, 
or  its  relative  humidity.  The  degree  of  humidity  is  expressed  as  a  per- 
centage of  that  due  to  the  state  of  saturation  for  the  temperature.  For 
example,  if  the  proportion  of  moisture  in  the  atmosphere  is  just  half  that 
which  it  contains  when  saturated,  the  relative  humidity  is  50  per  cent.,  or  50. 

Dew  Point. — When  atmospheric  air  containing  aqueous  vapour  is 
gradually  cooled,  the  temperature  of  the  vapour  is  lowered  whilst  its  density 


HYGROMETERS. 


393 


is  increased,  until  the  vapour  arrives  at  its  maximum  density  for  the  corre- 
sponding temperature.  If  cooled  below  this  temperature,  a  part  of  the 
vapour  is  condensed  and  precipitated  as  dew.  Hence  this  particular 
temperature  is  called  the  dew-point;  and  it  is  different  for  different 
degrees  of  humidity.  The  dew-point  is  no  other  than  the  temperature  of 
saturated  steam,  which  has  arrived  at  its  maximum  density  in  the  course 
of  contraction  by  cooling,  whilst,  reversely,  its  temperature  has  been 
lowered. 

HYGROMETERS. 

Daniell's  hygrometer,  Fig.  123,  is  an  instrument  of  great  precision  for 
ascertaining  the  dew-point.  It  consists  of  a  bent 
tube  with  a  globe  at  each  end,  and  it  is  partly  filled 
with  ether.  The  rest  of  the  space  is  occupied  with 
vapour  of  ether,  the  air  having  been  expelled.  One 
of  the  globes,  A,  contains  a  thermometer  t.  This 
globe  is  generally  made  of  black  glass,  which  pre- 
sents a  brilliant  surface.  To  use  the  instrument, 
the  whole  of  the  liquid  is  first  passed  into  the  globe 
A,  and  then  the  other  globe  B,  which  is  covered 
with  muslin,  is  moistened  externally  with  ether. 
The  evaporation  of  this  ether  from  the  muslin 
causes  a  partial  condensation  of  vapour  of  ether  in 
the  interior  of  the  globe,  which  produces  a  fresh 
evaporation  from  the  surface  of  the  liquid  in  A,  thus 
lowering  the  temperature  of  that  part  of  the  instru- 
ment. By  carefully  watching  the  surface  of  the  globe,  the  exact  moment 
of  the  deposition  of  dew  may  be  ascertained,  and  then  the  temperature  is 


Fig.  124.  —  Regnault's  Hygrometer. 


read  on  the  inclosed  thermometer.     This  temperature  is  a  little  lower  than 
the  dew-point.     If  the  instrument  be  now  left  to  itself,  the  exact  moment 


394 


MIXTURE  OF  GASES  AND  VAPOURS. 


I 


of  the  disappearance  of  the  dew  may  be  observed,  when  the  thermometer 
shows  a  temperature  a  little  above  the  dew-point.  The  mean  of  the  two 
observed  temperatures  is  taken  as  the  dew-point,  and  the  temperature  of 
the  external  air  is  shown  by  a  thermometer  t'  attached  to  the  stand. 

Regnault's  hygrometer,  Fig.  124,  consists  of  a  glass  tube  closed  at  the 
bottom  by  a  very  thin  silver  cup  D,  and  at  the  top  by  a  cork,  through 
which  the  stem  of  a  thermometer  T  is  passed,  and  a  glass  tube  t'  open  at 
both  ends.  The  lower  end  of  the  tube  and  the  bulb  of  the  thermometer 
dip  into  ether  contained  in  the  silver  cup.  The  tube  D  is  connected  by 
the  tube  uv  with  the  inspirator  A,  which  contains  water.  When  the  water  is 
allowed  to  escape  from  the  bottom,  a  current  of  air  is  drawn  through  the 
ether,  by  agitating  which  the  current  maintains 
a  uniformity  of  temperature  in  it.  The  cold  pro- 
duced by  evaporation  speedily  causes  a  deposition 
of  dew;  and  by  the  inverse  action  the  dew  dis- 
appears. The  mean  of  the  temperatures  observed 
at  the  same  times  is  the  dew-point.  The  other 
tube  D'  is  not  in  connection  with  the  aspirator, 
and  it  contains  a  thermometer  to  measure  the 
temperature  of  the  external  air.  Alcohol  may  be 
used  instead  of  ether. 

The  wet  and  dry  bulb  thermometer,  also  known 
as  Mason's  thermometer,  Fig.  125,  is  in  general 
use.  It  consists  of  two  thermometers  precisely 
alike.  The  bulb  of  one  is  covered  with  muslin, 
which  is  kept  moist  by  means  of  a  cotton  wick 
leading  from  a  glass  of  water.  The  evaporation 
from  the  moistened  bulb  lowers  the  temperature 
of  it,  and  the  difference  of  the  temperatures  read 
on  the  two  thermometers  increases  with  the  dry- 
ness  of  the  air.  The  indications  of  the  two 
thermometers  are  interpreted  by  means  of  tables 
specially  composed.1 

The  pressure  and  density  of  saturated  vapour, 
at  temperatures  ranging  from  the  freezing  point 
to  the  boiling  point,  under  atmospheric  pressure, 
have  been  given  in  table  No.  127,  page  386. 
The  temperatures  may  be  called  the  dew-points  of  the  steams  of  the 
corresponding  densities. 

PROPERTIES  OF  SATURATED  MIXTURES  OF  AIR  AND  AQUEOUS  VAPOUR. 

The  leading  properties  of  saturated  mixtures  of  air  and  aqueous  vapour, 
under  a  constant  pressure  of  one  atmosphere,  or  14.7  Ibs.  per  square  inch, 
at  final  temperatures  or  dew-points  ranging  from  32°  to  212°  F.,  are  given 
in  table  No.  130. 

The  second  and  third  columns  give  the  total  pressures  of  the  vapour  and 
the  air,  in  saturated  mixtures,  at  the  temperatures  given  in  the  first  column. 
The  sum  of  any  pair  of  these  pressures  is  equal  to  14.7  Ibs.,  or  one  atmos- 

1  These  notices  of  hygrometers  are  derived  from  Deschanel's  Natural  Philosophy, 
English  edition.  Blackie  &  Son. 


Wet  and  Dry  Thermometers. 


AIR   AND  AQUEOUS   VAPOUR.  395 

phere:  they  are  complementary  to  each  other.  At  32°,  for  example,  the 
pressure  of  the  vapour,  .089  Ibs. +  14.611  Ibs.,  the  pressure  of  the  air 
mixed  with  it,  =  14.7  Ibs.;  and  at  210°,  the  sum  of  the  pressures,  14.126  + 
.574,  is  also  equal  to  14.7  Ibs. 

Columns  4  and  5  give  the  respective  weights  of  vapour  and  air  in  100 
cubic  feet  of  the  saturated  mixture,  the  sum  of  which,  or  the  total  weight  of 
the  mixture,  is  given  in  column  6. 

Columns  7,  8,  9,  give  the  quantities  of  heat  reckoned  from  32°  F.  in  100 
cubic  feet  of  the  vapour,  the  air,  and  the  mixture,  respectively.  The  quan- 
tity of  heat  in  the  vapour  is  found  by  multiplying  the  quantity  of  heat  in 
one  pound  of  vapour,  as  given  in  column  10,  by  the  weight  of  the  vapour 
in  column  4,  of  the  present  table. 

For  example,  the  total  heat  of  one  pound  of  saturated  vapour  at  32°  F. 
is  1091.2  units,  and  the  weight  of  100  cubic  feet  of  the  vapour  is  .031  Ib. : 
then 

1091.2  x  .031  =33.8  units  of  heat, 

which  is  the  quantity  of  heat  in  the  vapour  given  in  the  fourth  column. 

The  quantities  of  heat,  column  10,  are  copies  of  the  fourth  column  of 
table  No.  127,  page  386,  which  are  expressions  of  the  quantity  of  heat  in 
one  pound  of  vapour,  reckoned  from  32°  as  the  initial  temperature  of  the 
water  converted  into  vapour. 

The  quantity  of  heat  in  the  air,  column  8,  is  found  by  multiplying  the 
specific  heat  of  air,  .2377,  by  the  number  of  degrees  of  the  temperature  in 
excess  of  32°,  and  by  the  weight  of  100  cubic  feet  given  in  the  fifth  column. 

The  total  heat  in  the  mixture,  column  9,  is  the  sum  of  the  heats  in 
columns  7  and  8. 

The  quantity  of  dry  air,  column  n,  required  for  one  pound  of  vapour, 
in  saturated  mixture  with  it,  is  found  by  dividing  the  weight  of  air,  column  5, 
by  the  relative  weight  of  vapour,  column  4.  The  volume  at  62°  F.  of 
the  air,  column  12,  is  found  by  multiplying  the  weight  in  column  n,  by 
13.141  feet,  the  volume  of  one  pound  of  air  at  62°. 

The  thirteenth  column  of  the  table  gives  the  initial  temperature  to  which 
the  quantities  of  dry  air  given  in  columns  n  and  12,  would  require  to  be 
raised,  in  order  to  provide  a  sufficient  quantity  of  heat,  if  applied  to  the 
water  at  32°,  to  evaporate  it,  and  to  form  a  saturated  mixture  at  each  of  the 
temperatures,  given  in  column  i.  The  product  of  the  weight  of  air,  col.  1 1, 
by  the  specific  heat,  is  equal  to  the  number  of  units  of  heat  absorbed 
by  the  air  for  one  degree  elevation  of  temperature.  If,  therefore,  the 
quantity  of  heat  in  one  pound  of  vapour,  column  10,  be  divided  by  the 
weight  of  air,  column  u,  and  by  the  specific  heat  of  air,  .2377,  the  quotient, 
plus  the  final  temperature  of  the  saturated  mixture,  as  given  in  column  i,  is 
the  required  initial  temperature. 

By  following  the  same  directions,  the  various  values  may  be  found  for 
any  other  final  temperature  of  saturated  mixture;  and,  inversely,  the  final 
temperature  may  be  found  for  any  given  initial  temperature. 


396 


MIXTURE   OF   GASES   AND  VAPOURS. 


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397 


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COMBUSTION. 


The  combustible  elements  of  fuel  are  carbon,  hydrogen,  and  sulphur. 
There  are  other  elements  in  fuel — nitrogen,  water,  and  solid  incombustible 
matter — which  do  not  take  part  in  combustion.  Fuel  is  burned  with 
atmospheric  air,  of  which  the  oxygen  combines  with  the  combustible 
matter,  whilst  the  nitrogen  remains  neutral.  The  combining  proportions 
of  the  elements  concerned  in  or  about  combustion  are  given  in  table 
No.  131 : — 

Table  No.  131. — COMPOSITION  AND  COMBINING  EQUIVALENTS  OF  GASES 

CONCERNED  IN  THE  COMBUSTION  OF  FUEL.      (OLD  NOMENCLATURE.) 


Gases. 

Elements  of  the 
Gases. 

Combining  Equivalents. 

By  Weight. 

By  Measure. 

ELEMENTS  :— 
Oxvsren 

Equivalents. 

Oxygen,        i 
Hydrogen,    I 
Carbon,         i 
Sulphur,        i 
Nitrogen,      i 

Carbon,        2 
Hydrogen,    4 
Carbon,        4 
Hydrogen,    4 
Oxygen,      23 
Nitrogen,    77 
Oxygen,        i 
Carbon,         i 
Oxygen,        2 
Carbon,         i 
Oxygen,        i 
Hydrogen,    i 
Oxygen,        2 
Sulphur,        i 

8 

One  Volume  =  Q 

Hvdrosren 

Carbon,  

6 

Sulphur 

16 

Q 

Nitrogen  . 

14. 

n 

COMPOUNDS  :  — 
Light    Carburetted    Hy-  ( 
drogen,  ( 

12 

4 
24 

4 
8 
26.8 
8 

6 

16 

6 

8 

,6 

16 

=     16 

-    28 

•  -  -  1  —  '  —  i 

Olefiant  Gas,  \ 

Atmospheric    Air     (me-  \ 
chanical  mixture),  ) 

Carbonic  Oxide,  

—  34-8 
=     14 

=      22 

=       9 
=     32 

0          ) 

m  5  ~; 

a 
D  (ideal) 

B 
D  (ideal) 
a 

| 

D  (ideal) 

i  i  i 

CEP 

ipproximately 

-  n 

Carbonic  Acid 

Aqueous       Vapour       or  ( 
Water                              ( 

-  n 
=  a 

Sulphurous  Acid,  j 

The  volume  of  one  pound  of  the  principal  gases  at  62°  F.,  under  one 
atmosphere  is  as  follows: — 


CHEMISTRY  OF   COMBUSTION.  399 

GAS  AT  62°  F.  ONE  POUND. 

Cubic  feet. 

Oxygen,  ................................................  11.887 

Hydrogen,  .............................................  1  90.000 

Nitrogen,  ..............................................  J  3-  5  o  i 

Air......  ......  .  .........................................  13-141 

Carbonic  Acid,  .......................................  8.594 

Aqueous  Vapour,  as  Gaseous  Steam,  .............  21.125 

Sulphurous  Acid,  ....................................  5.848 

The  source  of  oxygen,  as  the  supporter  of  the  combustion  of  fuel,  is 
atmospheric  air,  which  consists  of  oxygen  and  nitrogen  in  mechanical 
combination,  in  the  proportion  of  8  to  26.8;  or  i  Ib.  of  oxygen  to  3.35 
Ibs.  of  nitrogen;  or,  by  volume,  i  cubic  foot  of  oxygen  to  3.76  cubic 
feet  of  nitrogen. 

For  every  pound  of  oxygen  employed  in  combustion,  4.35  Ibs.  of  air  are 
consumed;  or,  by  measure,  for  every  cubic  foot  of  oxygen  employed  in  com- 
bustion, 4.76  cubic  feet  of  air  are  consumed.  For  the  combustion  of  one 
pound  of  hydrogen,  of  carbon,  and  of  sulphur,  therefore,  the  quantities  of 
air  chemically  consumed  are  as  follows  :  — 

One  Pound. 

Hydrogen  consumes  .............  34.8  Ibs.,  or  457  cubic  feet,  of  air  at  62°. 

'6  lbs"  OT  '5»         do-  do- 


5-8  lbs.,  or    76        do.  do. 

Sulphur  consumes:  ...............  4.35  lbs.,  or    57         do.  do. 

The  process  of  their  combustion  is  indicated  in  the  following  tablets 


COMBUSTION  OF  HYDROGEN. 

Elements.  Process.  Products. 


' '  I  nitrogen,    26.8  pounds, ...  2  6. 8  pounds  nitrogen, 


35-8  35-8  35-8 


COMPLETE  COMBUSTION  OF  CARBON. 
i     pound  carbon,.. carbon,     i       pound,..  1      , 

x  i  6  pounds  air       {  ^    2'66  P»A  1 

lir''"  \  nitrogen,  8.94  pounds,  ...8.94  pounds  nitrogen. 


12.6  12.6  12.6 


COMBUSTION  OF  SULPHUR. 

i       pound  sulphur...  sulphur,  i       pound   )  -j 

.  oxygen,    i       pound   }  2       POUnds  sulPhurous  aci 

lir''"  \  nitrogen,  3.35  pounds.  ..3.  35  pounds  nitrogen. 


5-35  5-35  5-35 


4<DO  COMBUSTION. 

AIR  CONSUMED  IN  THE  COMBUSTION  OF  FUELS. 
Fuels  are,  for  the  most  part,  compounds  of  carbon,  hydrogen,  sulphur, 
and  oxygen,  in  various  proportions;  and  to  form  a  calculation  of  the 
quantity  of  air  chemically  consumed  in  the  combustion  of  a  fuel,  the  several 
quantities  required  for  each  combustible  element  of  the  fuel  are  to  be 
calculated.  When  oxygen  is  present  as  a  constituent  of  a  fuel,  it  exists  in 
combination  with  a  portion  of  hydrogen  as  water,  in  solid  fuels  at  least; 
and  in  any  fuel,  liquid  or  solid,  the  combined  oxygen  and  hydrogen  are 
driven  off,  in  the  process  of  combustion,  as  water  or  steam.  Such  portion 
of  the  constituent  hydrogen,  therefore,  as  is  thus  driven  off,  already  satur- 
ated with  oxygen,  is  to  be  excluded  from  the  calculation  for  the  quantity  of 
air  necessary  to  consume  the  remainder  of  the  hydrogen,  and  the  carbon 
and  sulphur.  Let  the  constituents  of  the  fuel  be  expressed  proportionally 
as  percentages  of  the  total  weight  by  their  initials  C,  H,  O,  S,  respectively ; 
then  the  volume  of  air  at  62°  F.  chemically  consumed  in  the  combustion 
of  one  pound  of  a  fuel,  is  expressed  for  each  combustible  as  follows : — 

Cubic  feet. 

For  the  carbon, 152  C  -•- 100. 

For  the  hydrogen, 457  (H  -  Q) -=- 100. 

8 

For  the  sulphur, 57  S-f- 100. 

The  quantity  Q  signifies  the  deduction  to  be  made  from  the  constituent 
hydrogen,  for  that  portion  which  forms  steam  with  the  constituent  oxygen, 
being  equal  in  weight  to  one-eighth  part  of  the  weight  of  the  oxygen.     The 
total  volume  of  air  is  the  sum  of  these  three  items  :— 
152  C  +  457  (H-0)  +  57  S 

100 

Putting  A  for  the  total  volume  of  air  at  62°,  and  reducing — 
i52(C  +  3(H-Q)  +  .4S) 

A    o  QV 

/i  — "5    v-Jl 

100 

A  =1.52(0  +  3  (H-Q)  +  .4S) (i) 

o 

RULE  i. — To  find  the  quantity  of  air  at  62°  F.,  under  one  atmosphere, 
chemically  consumed  in  the  complete  combustion  of  one  pound  of  a  given  fuel. 
Let  the  constituent  carbon,  hydrogen,  oxygen,  and  sulphur,  be  expressed 
as  percentages  of  the  whole  weight  of  the  fuel;  divide  the  oxygen  by  8, 
deduct  the  quotient  from  the  hydrogen,  and  multiply  the  remainder  by  3 ; 
multiply  the  sulphur  by  0.4;  add  these  two  products  to  the  carbon,  and  mul- 
tiply the  sum  by  1.52.  The  final  product  is  the  quantity  of  air  in  cubic  feet. 

To  find  the  weight  of  the  air  chemically  consumed,  divide  the  volume 
thus  found  by  13.14;  the  quotient  is  the  weight  of  the  air  in  pounds. 

Note. — In  making  ordinary  approximate  calculations,  the  sulphur  may  be 
omitted. 

QUANTITY  OF  THE  GASEOUS  PRODUCTS  OF  THE  COMPLETE  COMBUSTION  OF 
ONE  POUND  OF  FUEL.  It  By  Weight. 

Pound.  Pounds.  Pounds. 

i  carbon,  and     2.66  oxygen,  form  3.66  of  carbonic  acid. 

i  hydrogen,  and  8       oxygen,  form  9       of  steam. 

i  sulphur,  and     i        oxygen,  form  2        of  sulphurous  acid. 


GASEOUS   PRODUCTS   OF  COMBUSTION.  401 

Then  in  the  combustion  of  one  pound  of  fuel  the  weights  of  the  products 
are  as  follows : — 

3.66  C-f-  ioo  =  .c>366  C  =  the  weight  of  carbonic  acid (  a  ) 

9H  +  100=     .o9H  =  the  weight  of  steam (b  ) 

28+-  100  =      .02  S  =the  weight  of  sulphurous  acid (  c  ) 

To  this  is  to  be  added  the  weight  of  atmospheric  nitrogen  separated  from 
the  oxygen  chemically  consumed,  and  the  weight  of  the  constituent  nitrogen, 
N,  of  the  fuel.  The  quantity  of  atmospheric  nitrogen  is  3.35  times,  by 
weight,  that  of  the  oxygen  consumed;  and, 

Pound.  Pounds. 

For  i  carbon  there  are      2.66  x  3.35  =  8.93  nitrogen. 
For  i  hydrogen  there  are       8  x  3.35  =  26.8       do. 
For  i  sulphur  there  are          IX3-35— 3-35       do. 

Multiply  each  of  these  quantities  by  their  respective  percentages  of  combus- 
tible, and  divide  by  100;  the  sum  of  the  quotients  is  the  weight  of  nitrogen 
separated  from  the  atmospheric  oxygen  consumed.  To  this  is  to  be  added 
the  constituent  nitrogen  of  the  fuel: — 

8.93  C-T-IOO  =  . 0893  C. 
26.8  H-r-  loo-  .268  H. 
3.35  S -MOO  =  .0335  S. 

N-T-  1 00  =      .01  N. 

Thus,  the  total  weight  of  nitrogen  is  equal  to 

(.0893  C  +  .268  H  +  .0335  S  +  .oi  N) (d) 

Add  together  the  total  weights  of  carbonic  acid,  steam,  sulphurous  acid, 
and  nitrogen  above  noted,  and  put  w  for  the  total  weight  of  the  gaseous 
products  of  combustion,  then — 

o/  =  . 03660  +  .  09  H  +  .   02  $  +  (.0893  C  +  .268  H  +  .0335  S  +  .oi  N); 
orze/=    .126  C  +  .358  H  +  .053  S  +  .oi  N (  2  ) 

RULE  2. — To  find  the  total  weight  of  the  gaseous  products  of  the  complete 
combustion  of  one  pound  of  a  fuel.  Let  the  elements  be  expressed  as  per- 
centages of  the  fuel;  multiply  the  carbon  by  0.126,  the  hydrogen  by  0.358, 
the  sulphur  by  0.053,  and  the  nitrogen  by  .01,  and  add  together  those  four 
products.  The  sum  is  the  total  weight  of  the  gases  in  pounds. 

Note. — The  weight,  in  pounds,  of  the  carbonic  acid,  separately,  may  be 
found  from  the  quantity  (a),  above;  that  of  the  steam  from  (£),  that  of 
the  sulphurous  acid  from  (c)t  and  that  of  the  nitrogen  from  (d). 

2.  By  Volume. 

Multiply  the  weight  of  each  gaseous  product,  («),  (£),  (<r),  (*/),  by  the 
volume  of  one  pound  in  cubic  feet  at  62°  F.,  page  399.  Then 

Cubic  feet. 

.0366  C  x      8.59=    .315  0  =  Volume  of  the  carbonic  acid (e) 

.09  H     x  21.125  =  i. 9  H     =  Volume  of  the  steam (/) 

.02  S       x      5.85  =    .1178  =  Volume  of  the  sulphurous  acid...  (g) 
(.0893  C+   .268  H  +  .0335  S  +  .oi  N)    x  13.5  = 
(1.206  C   +3.6i8H  +  .45S     +.135  N)  =  Volume  of  the  nitrogen...  (h) 


402  COMBUSTION. 

Adding  together  and  reducing,  and  putting  V  =  the  total  volume  of  the 
gases, 

V=i.52  0  +  5.52  H  +  .567  S  +  .I35  N  ............  (3) 

RULE  3.  —  To  find  the  total  volume,  at  62°,  of  the  gaseous  products  of  the 
complete  combustion  of  one  pound  of  fuel.  Let  the  elements  be  expressed  as 
percentages;  multiply  the  carbon  by  1.52,  the  hydrogen  by  5.52,  the 
sulphur  by  .567,  and  the  nitrogen  by  .135,  and  add  together  the  four 
products.  The  sum  is  the  total  volume,  at  62°  R,  of  the  gases,  in  cubic 
feet. 

Note.  —  The  volume  of  the  several  gases  separately  may  be  found  from 
the  respective  quantities  (e\  (/),  (g),  (h). 

The  volume  of  the  gases  at  higher  temperatures  than  62°  F.  is  found  by 
the  formula  (2),  page  347;  namely, 

(4) 


As  t=  62°,  this  formula  becomes,  for  present  purposes, 


and  it  appears  that  the  initial  or  normal  volume,  as  at  62°,  under  one 
atmosphere,  is  doubled  when  the  temperature  is  raised  523°  higher;  and 
that  the  expanded  volume  at  any  other  temperature  /',  in  proportion  to  the 
normal  volume,  is  found  by  adding  461  to  the  temperature,  and  dividing 
the  sum  by  523. 

SURPLUS  AIR. 

If  the  quantity  of  surplus  air  that  enters  the  furnace  and  passes  away 
unconsumed,  be  expressed  as  a  percentage  of  the  air  chemically  consumed, 
it  is  found  directly  from  the  latter  when  this  is  known.  If  the  volume  is 
given,  the  weight  is  found  by  dividing  the  volume  at  62°  in  cubic  feet  by 
13.14. 

HEAT  EVOLVED  BY  THE  COMBUSTION  OF  FUEL. 

From  the  experimental  investigations  of  MM.  Favre  and  Silbermann,  the 
total  quantities  of  heat  evolved  by  the  combustion  of  one  pound  of  some 
combustibles  with  oxygen  were  determined  as  follows  :  — 

SIMPLE  BODIES.  Units  of  heat. 

Hydrogen  ..................................................  62,032 

Carbon  —  Wood  charcoal,  thoroughly  calcined  ......   1  4,  5  44 

Sugar  charcoal  .................................   1  4,470 

Gas-coke  .......................................   14,485 

Graphite,  from  blast-furnaces  ...............   13,972 

Natural  graphite  ..............................    14,033 

Diamond  (pure  carbon)  .....................   13,986 

Sulphur  .....................................................     4,03  2 


HEAT   EVOLVED   BY  COMBUSTION. 


403 


COMPOUND  BODIES.  Units  of  heat. 

Carbonic  oxide 4?325 

Light  carburetted  hydrogen 23,5 13 

Olefiant  gas 2 1 ,343 

Sulphuric  ether 1 6, 249 

Alcohol 12,929 

Turpentine 1 9,534 

Sulphuret  of  carbon 6,120 

Wax 18,893 

The  chemical  composition  of  the  compound  bodies  above  cited  is  as 
follows,  and  there  is  added  the  composition  of  a  few  other  combustibles : — 

Table  No.  132. — CHEMICAL  COMPOSITION  OF  COMPOUND  COMBUSTIBLES. 


Combustible. 

Combining  equivalents. 

In  loo  parts  by  weight. 

Carbon.     Hydrogen. 

Oxygen. 

Carbon. 

Hydrogen. 

Oxygen. 

Carbonic  oxide  . 

I 
2 

4 
4 
4 
20 

4 

4 

5 
6 
16 

;  <c  '   •      '  •  •  i  • 

I 

I 
2 

per  cent. 
42.9 

75-o 

85.7 
64.8 
52.2 
88.2 
81.6 
77.2 
79.0 

per  cent. 

25.0 

14.3 
13-5 
13.0 

n.8 

13-9 
13-4 
11.7 

per  cent. 
57-1 

21.7 

34-8 

4-5 
9.4 

9-3 

Light    carburetted  ) 
hydrogen  j 
Olefiant  gas 

Sulphuric  ether 

Alcohol  

Turpentine  

Wax  

Olive  oil  

Tallow 

The  heating  powers  of  the  compound  bodies  are  approximately  equal  to 
the  sum  of  the  heating  powers  of  their  elements.  Peclet  gives  a  number  of 
examples  in  proof  of  this.  Take  light  carburetted  hydrogen,  which  consists 
of  two  equivalents  of  carbon  and  four  of  hydrogen,  weighing  respectively 
2x6  =  12  and  i  x  4  =  4,  in  the  proportion  of  3  to  i,  or  ^  Ib.  of  carbon  and 
i^  Ib.  of  hydrogen  in  one  pound  of  the  gas.  The  elements  of  the  heat  of 
combustion  of  one  pound  are,  then, 


For  the  carbon  .......................  14,544  x 

For  the  hydrogen  .....................  62,032  x 


Units  of  heat. 

=    10,908 
=    15,508 


Total  heat  of  combustion,  calculated  ..............   26,416 

Total  heat,  by  direct  trial  ............................   23,5  13 

Excess  by  calculation  .....................     2,903 

Alcohol  has  4  of  carbon,  6  of  hydrogen,  and  2  of  oxygen.  Abstracting  the 
proportion  of  hydrogen  neutralized  by  the  oxygen,  there  are  to  be  dealt 
with,  4  of  carbon,  4  of  hydrogen,  and  2  of  water,  the  weights  of  which  are 


404  COMBUSTION. 

as  24,  4,  and  18;  total  46.  The  quantities  of  heat  evolved  in  the  combus- 
tion of  one  pound  of  alcohol  are,  therefore, 

Units  of  heat. 

For  the  carbon  14,544  x  24/46=      7588 

For  the  hydrogen 62,032  x  4/46=      5394 

Total  heat  evolved,  calculated 1 2,982 

Total  by  direct  trial 12,929 

Excess  as  calculated 53 

Olive  oil  consists,  in  100  parts,  of  77.2  carbon,  13.4  hydrogen,  and  9.4 
oxygen;  or  77.2  carbon,  12.2  hydrogen,  and  10.6  water.  The  heat  of 
combustion  is,  by  calculation, 

Units  of  heat. 

For  the  carbon 14,544  x  77-*/IOO  =    11,228 

For  the  hydrogen 62,032  x  «-*/IOO  =      7,568 

Total  heat 18,796 

Tallow  consists  of  79  carbon,  11.7  hydrogen,  and  9.3  oxygen,  in  100  parts; 
or  79  carbon,  10.54  hydrogen,  and  10.46  water.  The  heat  of  combustion 
is,  by  calculation, 

Units  of  heat. 

For  the  carbon 14,544  x  79/IOO      =    11,490 

For  the  hydrogen 62,032  x  IO-54/IOO  =      6,538 

Total  heat 18,028 

The  successive  evolvements  of  heat  in  burning  carbon, — when  carbonic 
oxide  is  formed,  and  when  it  is  converted  into  carbonic  acid, — are 
deduced  from  the  fact  that  one  pound  of  carbonic  oxide,  when  burned  with 
oxygen  to  form  carbonic  acid,  evolves  4325  units  of  heat.  As  the  oxide 
consists  of  6  of  oxygen  to  8  of  carbon,  a  pound  of  it  contains  6/i4tns  of  a 
pound  of  carbon,  the  combustion  of  which  has  produced  4325  units  of  heat. 
In  the  same  ratio,  one  pound  of  carbon,  as  carbonic  oxide,  would  evolve 

4325  x  14  -f  6  =  10,092  units  of  heat 

in  being  converted  from  oxide  into  acid.  Therefore,  the  heat  of  complete 
combustion,  14,544  units,  minus  10,092  =  4452  units,  is  the  heat  evolved  in 
the  conversion  of  the  carbon  into  the  oxide,  and  the  successive  develop- 
ments of  heat  by  the  combustion  of  one  pound  of  carbon  are  as  follows : — 

Units  of  heat. 

In  the  first  stage,  forming  carbonic  oxide 4A52>  or    3°  Per  cent. 

In  the  second  stage,  forming  carbonic  acid...   10,092,  or    70        „ 


Heat  evolved  by  complete  combustion...   14,544,  or  100        „ 

TABLE  OF  THE  HEATING  POWERS  OF  COMBUSTIBLES. 

The  experimental  results  of  MM.  Favre  and  Silbermann  are  adopted 
with  some  slight  revision,  recommended  by  M.  Peclet,  in  table  No.  133, 
column  5.  The  weight  of  oxygen,  column  2,  is  calculated  from  the  known 
equivalents  and  weights  of  the  elements,  as  given  in  table  No.  131,  page  398. 


HEATING   POWERS   OF  COMBUSTIBLES. 


405 


The  weight  of  air,  column  3,  is  4.35  times  the  weight  of  oxygen,  column  2, 
and  the  volume  of  air  at  62°,  column  4,  is  13.14  times  the  weight,  column  3. 
The  equivalent  evaporative  power,  columns  6  and  7,  is  expressed  by  the 
weight  of  water  evaporable  at  212°  by  one  pound  of  combustible — first,  if 
supplied  at  62°  F.,  by  dividing  the  total  heat  of  combustion,  column  5,  by 
1116°,  which  is  the  total  heat  of  atmospheric  steam  raised  from  water 
supplied  at  62°;  second,  if  supplied  at  212°  F.,  by  dividing  by  966°,  the 
total  heat  of  atmospheric  steam  raised  from  water  supplied  at  2 1 2°. 

Table  No.  133. — TOTAL  HEAT  EVOLVED  BY  COMBUSTIBLES  AND  THEIR 
EQUIVALENT  EVAPORATIVE  POWER,  WITH  THE  WEIGHT  OF  OXYGEN 
AND  VOLUME  OF  AIR  CHEMICALLY  CONSUMED. 


Combustibles. 

Weight  of 
oxygen 
consumed 
per  Ib.  of 
combustible. 

Quantity  of  air 
consumed  per 
pound  of 
combustible. 

Total  heat 
of  combustion 
of  i  pound  of 
combustible. 

Equivalent  evaporative 
power  of  i  pound  of 
combustible,  under  one 
atmosphere,  at  212°. 

cubic 

pounds 

pounds 

i  pound  weight. 

Ibs. 

Ibs. 

feet  at 

units. 

of  water 

of  water 

62°. 

at  62°. 

at  212°. 

Hvdrosen  .  . 

8.0 

34  8 

4.C7 

62  O"?2 

rr  6 

64..  2O 

Carbon,     making  ) 
carbonic  oxide...  ) 

i-33 

O^' 

5-8 

T-  J  / 
76 

^^j^O^ 

4,452 

JD'W 
4.0 

4.6l 

Carbon,     making  ) 
carbonic  acid....  J 

2.66 

n.6 

152 

14,500 

13.0 

15-0 

Graphite  .    . 

2.66 

ii.  6 

I  C2 

I  A.  OAO 

12.58 

1/4.  C  "2 

Carbonic  oxide  

o.57 

2.48 

A  J* 

33 

*4fo\tfJ,\t 

4,325 

3.88 

14-DO 
4.48 

Light  carburetted  ) 

hydrogen  J 

4.0 

17.4 

229 

23,513 

2I.O7 

24-34 

Bi-carburetted  hy-  j 

drogen,orolefiant  > 

3-43 

15.0 

196 

2i,343 

19.12 

22.O9 

eras  ...                      1 

Sulphuric  ether  .... 

2.60 

n-3 

149 

16,249 

14.56 

16.82 

Alcohol 

2.78 

12.  1 

J59 

12,929 

11.76 

I3-38 

Turpentine.    . 

3-29 

14.3 

188 

19^534 

17-5° 

2O.22 

Sulphur  

I.OO 

4-35 

57 

4,032 

3-6l 

4.17 

Wax  

3.24 

I4.I 

185 

18,893 

16.93 

19.56 

Olive  oil  

3.03 

I3.2 

173 

18,796 

16.84 

19.46 

Tallow 

2-95 

12.83 

169 

18,028 

16.15 

1  8.  66 

(  Supplementary.  ) 

Coal,   of  average  ) 
composition  J 

2.46 

10.7 

141 

14,133 

12.67 

14.62 

Coke,  desiccated... 

2.50 

IO.9 

J43 

!3,55° 

12.14 

14.02 

Wood,  desiccated.. 

1.40 

6.1 

80 

7,792 

6.98 

8.07 

Wood-charcoal,      ) 
desiccated             f 

2.25 

9.8 

129 

12,696 

11.38 

i3-i3 

Peat,  desiccated... 

1-75 

7.6 

IOO 

9,95  1 

8.91 

10.30 

Peat-charcoal,        ) 
desiccated  J 

2.28 

9.9 

129 

12,325 

11.04 

12.76 

406  COMBUSTION. 

From  the  table,  it  appears  that  when  carbon  is  not  completely  burned, 
and  becomes  carbonic  oxide,  it  produces  less  than  a  third  of  the  heat 
yielded  when  it  is  completely  burned.  For  the  heating  power  of  carbon 
an  average  of  14,500  units  will  be  adopted.  The  heating  power  of 
hydrogen  is  about  four  and  a  quarter  times  that  of  carbon. 

The  calculation  for  the  heating  power  of  a  combustible  may  be  reduced 
to  a  simple  formula.  Let  C,  H,  O,  and  S  represent,  as  before,  the  per- 
centages of  carbon,  hydrogen,  oxygen,  and  sulphur,  in  100  parts.  The 
elements  of  the  heat  evolvable  are  as  follows : — 

Of  the  carbon, 14,500  C  -f- 100. 

Of  the  hydrogen, 62,032  (  H  -  Q)-MOO. 

Of  the  sulphur, 4,032  S^ioo. 

The  quantity  Q  is  a  deduction  made  from  the  hydrogen  to  satisfy  the 

constituent  oxygen  of  the  fuel:   being  an  eighth  of  the  weight  of  the 
oxygen.     The  total  evolvable  heat  is 

14,500  C  +  62,032  ( H  -  Q)  +  4032  S 

—  ,or 

IOO 

14,500  C  +  (14,500  x  4.28  (H  -  Q) )  +  (14,500  x  .28  S) 

§ ; 

100 

or,  putting  h  for  the  total  heat, 

£=145  (C +  4.28  (H-Q) +  0.288) (6) 

8 

RULE  4. — To  find  the  total  heating  power  of  one  ponnd  of  a  combustible,  of 
which  the  percentages  of  the  constituent  carbon,  hydrogen,  oxygen,  and  sulphur 
are  given.  From  the  hydrogen  deduct  one-eighth  of  the  oxygen,  and 
multiply  the  remainder  by  4.28;  multiply  the  sulphur  by  0.28;  add  the 
two  products  to  the  carbon;  and  multiply  the  sum  by  145.  The  final 
product  is  the  total  heating  power  of  one  pound  of  the  combustible,  in 
units  of  heat. 

Note. — The  item  of  sulphur  as  a  combustible  may  be  ignored  in  cal- 
culations for  ordinary  purposes. 

Dividing  the  second  member  of  the  formula  (6)  by  1116°,  the  total  heat 
of  steam  at  212°  raised  from  water  at  62°;  or  by  966°  if  the  water  be 
supplied  at  212°;  the  quotients  express  the  equivalent  evaporative  power  of 
the  combustible.  Putting  e  for  the  evaporative  power,  in  pounds  of  water 
per  pound  of  combustible, — 

^  =  o.I3  (0  +  4.28  (H-Q) +  0.288), (7) 

8 

when  the  water  is  supplied  at  62°;  and 

<?=o.i5  (0  +  4-28  (H-Q)  +  o.28  S), (8) 

when  the  water  is  supplied  at  2 1 2°. 

RULE  5. — To  find  the  total  evaporative  power  of  one  pound  of  a  combustible, 
of  which  the  percentages  of  the  constituent  carbon,  hydrogen,  sulphur,  and 
oxygen  are  given.  From  the  hydrogen  deduct  one-eighth  of  the  oxygen, 
and  multiply  the  remainder  by  4.28;  multiply  the  sulphur  by  0.28;  add 
these  two  products  to  the  carbon,  and  multiply  the  sum  by  0.13  when  the 


TEMPERATURE  OF   COMBUSTION.  407 

water  is  supplied  at  62°,  or  by  0.15  when  the  water  is  supplied  at  212°. 
The  final  product  is  the  total  evaporative  power  of  one  pound  of  the 
combustible  in  pounds  of  water,  evaporated  at  2 1 2°. 

Note. — When  the  total  heating  power  is  known,  divide  it  by  1116,  when 
the  water  is  supplied  at  62°;  or  by  966  when  the  water  is  supplied  at  212°. 
The  quotient  is  the  equivalent  evaporative  power. 

The  equivalent  evaporative  power,  from  water  supplied  at  212°,  is  found 
roughly  by  dividing  the  total  heat  by  1000. 

TEMPERATURE  OF  COMBUSTION. 

The  temperature  of  combustion  is  settled  by  the  several  quantities  and 
specific  heats  of  the  products  of  combustion.  One  pound  of  carbon  when 
completely  burned  yields  3.66  Ibs.  of  carbonic  acid,  and  8.94  Ibs.  of 
nitrogen.  Multiply  these  by  the  respective  specific  heats  of  the  gases — 

FOR  CARBON. 

For,, b.  Carbon.  Specific      Units  of 

Carbonic  acid, 3.66  Ibs.  x  .2164=    .792  for  i°  F. 

Nitrogen, 8.94  Ibs.  x    .244  =  2.181        „ 

12.60  Ibs.  x    .236  =  2.973         „ 

showing  that  the  products  of  combustion  absorb  2.973  units  of  heat  in 
rising  i°  F.  of  temperature.  Divide  the  total  heat  of  combustion,  14,500 
units,  by  2.973,  and  the  quotient  is  4877°  F.  Add  the  initial  temperature, 
say  62°,  making  4939°  F.  the  temperature  of  combustion. 

FOR  HYDROGEN. 

For  x  Ib.  Hydrogen.  S^c        Umtsof 

Gaseous  steam, —  9     Ibs.  x.475  =   4-275  f°r  l0  F. 
Nitrogen, 26.8  Ibs.  x  .244=    6.539        „ 

35.8  Ibs.  x. 302  =  10.814       i, 

The  total  heat  of  combustion,  62,032  units  -4- 10.8  =  5744°.  Add  62°  for  the 
temperature  of  combustion  of  hydrogen,  making  5806°  F. 

FOR  SULPHUR. 

For  i  Ib.  Sulphur.  Sg^fic        U£(jjft°f 

Sulphurous  acid,.. 2       Ibs.  x  .1553 •  =    .311  for  i°  F. 
Nitrogen, 3.35  Ibs.  x    .244=   .817 


5.35  Ibs.  x    .211  =  1.128        „ 

The  total  heat  of  combustion,  4032  units  4-1.128  =  3575°.    Add  62°  for  the 
temperature  of  combustion  of  sulphur,  making  3637°  F. 

For  coal  of  average  composition  (the  calculation  for  which  will  be  given 
in  detail)  there  are  11.94  Ibs.  of  gaseous  products,  of  which  the  mean 
specific  heat  is  .246;  and 

11.94  x  .246  =  2.935  units  of  heat  for  i°  F. 

The  total  heat  of  combustion,  14,133  units -i- 2.935  =  4815°.     Add  62°  for 
the  temperature  of  combustion  of  average  coal,  making  4877°  F. 

If  surplus  air  be  mixed  with  the  products  of  combustion,  and  equal  in 


408 


COMBUSTION. 


quantity  to  the  air  chemically  consumed,  the  total  weight  of  gases  for  one 
pound  of  coal  is  increased  to  22.64  Ibs.,  having  a  mean  specific  heat, 
.242;  and 

22.64  x  -242  —  5'478  units  of  heat  for  i°  F. 


62°  for 
than 


The  total  heat  of  combustion,  14,133  units  -7-5. 478  =  2580°.     Add 
temperature  of  combustion,  making  2642°  F.;   which  is  little  more 
half  the  temperature  of  undiluted  products  of  combustion. 

From  the  annexed  table,  No.  134,  it  appears  that  the  specific  heat  of  the 
products  of  combustion  is  in  general  about  .250;  excepting  that  for 
hydrogen,  which  is  .302;  and  that  for  sulphur,  which  is  .211. 

Table  No.   134. — WEIGHT  AND  SPECIFIC  HEAT  OF  THE  PRODUCTS  OF 
COMBUSTION  AND  THE  TEMPERATURE  OF  COMBUSTION. 

(With  only  the  net  supply  of  air  chemically  necessary.) 


One  Pound  of  Combustible. 

Gaseous  Products  for  One  Pound  of  Combustible. 

Weight. 

Specific 
Heat. 

Heat  to 
Raise  the 
Temper- 
ature i°  F. 

Temperature  of  Com- 
bustion, measured  from 
Initial  Temperature. 

Hydrogen 

pounds. 

35-80 
11.97 
15.90 

15-35 
14.21 
I3.84 

8-45 
12.00 
12.  60 
11.77 
15.21 
10.09 
18.40 
12.94 

5-35 
17.22 
12.18 
22.57 

water  =  i. 
.302 
.256 
•257 
.256 
.258 
.256 

•253 
.246 

.236 
.236 

•257 
.270 
.268 

•245 
.211 

.244 

•257 
.242 

units. 
I0.8l4 
3-063 
4.089 

3-93° 
3.666 

3.540 
2.136 
2.924 

2-973 

2.778 

3-9*4 
2.680 

4-933 
3.189 
1.128 
4.196 
3.127 
5-467 

Fahr. 

5733° 
5305 
5219 
5*93 
5128 

5°93 
5i38 
5027 

4877 
4878 
4826 
4825 
4766 
4432 
3575 
3527 
3470 
2688 

ratio. 
100 
92 
91 
9° 
89.3 
89 
89 

87 
85 
85 
84 
84 
83 

77 
62 
61 
60 

47 

Sulphuric  ether         

Olefiant  gas  

Petroleum  

Olive  oil  

Tallow  

Wood  desiccated 

Coal  (average) 

Carbon  

Coke  

Wax  

Alcohol  

Light  carburetted  hydrogen... 
Coal,  with  TO  percent,  more  air 
Sulphur  

Coal,  with  50  percent,  more  air 
Turpentine  

Coal,  with  100  p.  cent,  more  air 

FUELS-COAL. 


The  fuels,  or  combustibles,  generally  used  are  coal,  coke,  wood,  wood- 
charcoal,  peat,  peat-charcoal,  and  refuse  tan-bark.  To  these  may  be 
added  petroleum  and  other  oils;  recently,  straw  has  been  used. 

Coal  may  be  arranged  in  five  classes : — '• 

i  st.  Anthracite,  or  blind  coal,  consisting  almost  entirely  of  free  carbon. 

2d.  Dry  bituminous  coal,  having  from  70  to  80  per  cent,  of  carbon. 

3d.  Bituminous  caking  coal,  having  from  50  to  60  per  cent,  of  carbon. 

4th.  Long  flaming  or  cannel  coal. 

5th.  Lignite,  or  brown  coal,  containing  from  56  to  76  per  cent,  of  carbon. 

The  anthracites  have  specific  gravities  varying  from  1.35  to  1.92.  They 
retain  their  form  when  exposed  to  a  temperature  of  ignition;  though,  if  too 
rapidly  heated,  they  fall  to  pieces.  The  flame  is  generally  short,  of  a  blue 
colour.  The  coal  is  ignited  with  difficulty;  it  yields  an  intense  local  or 
concentrated  heat;  and  combustion  generally  becomes  extinct  while  yet  a 
considerable  quantity  of  the  fuel  remains  on  the  grate. 

The  dry,  or  free-burning,  bituminous  coals,  are  rather  lighter  than  the 
anthracites;  varying  in  specific  gravity  from  1.28  to  1.44.  They  contain  a 
relatively  small  proportion  of  volatilizable  matter, — about  15  per  cent.;— and 
they  soon  arrive  at  the  temperature  of  full  ignition.  They  swell  consider- 
ably in  coking,  and  thus  is  facilitated  the  access  of  air,  and  the  rapid  and 
complete  combustion  of  their  fixed  carbon.  In  some  cases,  where  the 
combustion  is  slow,  the  masses  of  coke  scarcely  cohere,  and  the  original 
forms  of  the  pieces  of  the  coal  are  in  some  measure  preserved. 

The  bituminous  caking  coals  have  the  same  range  of  specific  gravity  as 
the  dry  bituminous  coals.  They  contain  the  maximum  proportion  of 
volatilizable  matter,  averaging  about  30  per  cent,  of  their  whole  weight. 
They  develop  much  of  the  hydrocarbon  gases,  and  burn  with  a  long  flame. 
They  swell  considerably,  and  give  a  coherent  coke,  which  preserves  nothing 
of  the  original  form  of  the  coal. 

SMALL  COAL. 

In  South  Wales,  where  the  recognized  system  of  working  is  by  "  pillar  and 
stall,"  upwards  of  40  per  cent,  of  the  actual  contents  of  the  vein  of  steam- 
coal  is  lost. 

According  as  the  small  coal  is  raised  to  the  surface  or  not,  the  pit  is  said 
to  be  worked  on  the  "  altogether  coal,"  or  on  the  "  separation  "  principle. 
In  the  former  case,  from  45  to  50  per  cent,  of  small  coal  passes  through  the 
screen;  in  the  latter  case,  where  only  hand-picked  coal  is  sent  to  the 
surface,  the  small  amounts  to  from  5  to  10  per  cent.  The  small  coal  is 
screened  into  three  sizes,  known  as  " nuts,"  " peas "  or  "beans," and  "duff" 


410  FUELS. — COAL. 

or  "  waste."  Small  coal  generally  consists  of  what  passes  through  screens 
with  spaces  between  the  bars  from  ify  inches,  as  in  South  Wales,  to  ^s 
inch,  as  in  the  Newcastle  district.  The  duff  consists  of  what  passes 
through  meshes  ^  inch  square;  the  peas  or  beans  consist  of  what  does  not 
pass  through  these  meshes,  but  falls  between  bars  7/l6  inch  apart.  The 
remainder  of  the  small  is  nuts. 

The  relative  proportions  of  large  and  small  coal,  on  the  "  altogether " 
system,  and  the  "  separation "  system,  brought  to  the  surface,  in  the 
Newcastle  district,  may  be  taken  as  follows : — 

ALTOGETHER.        SEPARATION. 

Round  coal 46.  i  per  cent.  80.29  per  cent. 

Small  coal: — 

Nuts 20.9       „  12.50       „ 

Beans 17.6       „  3.85 

Duff 15.4       „  3-36 


100.00  IOO.OO 

The  relative  market  values  of  the  different  sizes  of  coal  as  raised,  are 
illustrated  by  the  following  list  of  quotations  from  a  certain  colliery, 
delivered  free  on  board  at  Sunderland. 

Round  coals,  IDS.  to  us.,  average  105.  6d.  per  ton,  say  100 

Treble-screened  nuts 8s.  „  or  76 

Double-screened  nuts 75.  „  or  67 

Peanuts 6s.  „  or  57 

Single-screened  small 6s.  „  or  57 

Pea  nuts  and  duff  mixed 45.  6d.  ,,  or  43 

Duff 35.  6d.  „  or  33 

The  quantity  of  small  coal  separated  from  the  coal  brought  by  railway 
to  London  is  found,  at  the  end  of  the  journeys,  in  passing  through  screens 
at  the  staiths,  to  amount  to  from  5  to  n  percent.,  averaging,  probably, 
7  y^  per  cent.  This  represents  the  breakage  of  coal  between  the  loading 
at  the  pit's  mouth  and  the  discharging  in  London. 

The  breakage  of  coal  conveyed  by  sea  is  also  considerable.  Between 
the  colliery  and  the  ship,  it  has  been  estimated,  in  one  case,  at  5  per  cent.; 
and  when  double-screened  at  Cardiff,  at  from  8  to  8^  per  cent.  Again, 
the  quantity  of  small  coal  made  by  loading  into  and  unloading  from  the 
ship  is  stated  to  be  from  15  to  20  per  cent.1 

In  France,  at  St.  Etienne,  the  quotations  were  respectively  as  follows, 
for  round  coal,  medium  coal,  and  slack: — 

Round  coal,     2  francs  per  100  kilogrammes,  or  i6s.  per  ton,  say  100 
Medium  coal,  1.25  „  „  „  or  los.        „        or      62 

Slack,  0.25  to  0.50  „  „  „  or  25.  to  45.  „    12  to  24 

Utilization  of  Small  Coal. — It  is  a  matter  of  national  importance  to  utilize 
the  immense  accumulations  of  small  coal,  both  above  and  below  ground. 

1  See  Coal  Economy,  by  Mr.  F.  C.  Danvers,  1872  ;  from  which  the  above  particulars 
of  small  coal  are  derived. 


SMALL  COAL.  411 

The  best  known  system  is  that  of  Warlich's  patent  fuel, — a  mixture  of 
small  coal  and  tar  or  pitch  moulded  into  blocks.  Each  ton  of  small  coal 
is  mixed  with  22  gallons,  or  242  Ibs.  of  tar,  which  is  over  10  per  cent,  of 
the  weight  of  the  coal.  It  is  then  formed  into  blocks,  and  baked  at  a 
temperature  of  800°  F.  for  nine  or  ten  hours.  The  volatile  matter  of  the 
tar  is  driven  off,  leaving  the  pitch  as  a  cement  for  the  coal.  In  the  process 
of  baking,  the  blocks  lose  5  per  cent,  of  their  weight. 

Wylam's  fuel  is  prepared  by  mixing  with  slack  about  7  or  8  per  cent,  of 
its  weight  of  pitch,  in  a  dry  state,  ground  fine.  The  mixture  is  passed  by 
means  of  an  Archimedian  screw  through  a  retort  maintained  at  a  dull  red 
heat,  by  which  it  is  softened,  when  it  is  moulded  under  great  pressure  by 
a  species  of  brickmaking  machine. 

Mezaline's  fuel,  like  Wylam's,  is  a  mixture  of  slack  and  pitch  ground  fine, 
in  a  pug-mill,  where  it  is  at  the  same  time  softened  by  superheated  steam 
introduced  into  the  mass  at  different  points,  thus  to  increase  the  cohesion 
of  the  particles.  Fuel  thus  prepared,  when  exposed  to  a  high  temperature, 
loses  four  per  cent,  of  its  weight,  representing,  no  doubt,  the  moisture 
acquired  from  the  steam. 

In  Barker's  fuel,  the  binding  medium  consists  of  a  mucilage  formed  by 
the  mixture  of  potato-farina  with  water,  in  the  proportion  of  about  i  to  4, 
with  a  small  quantity  of  carbolic  acid.  Thirty  gallons  of  the  mucilage  was 
mixed  with  one  ton  of  coal,  and  the  mixture  baked  for  nine  hours  at  a 
temperature  of  300°  F.  This  mixture  was  not  hard  enough  to  stand  rough 
usage  or  exposure;  and  more  recently  a  certain  proportion  of  powdered 
pitch  has  been  added  with  good  results.  The  bulk  of  i  ton  is  33  cubic  feet. 

In  Holland's  fuel,  lime  and  cement  are  mixed  with  small  coal; — making, 
of  course,  a  large  quantity  of  ash  when  burned. 

Washing  of  Small  Coal. — Coal-washing  has  long  been  practised  on  the 
Continent  for  the  purpose  of  separating  from  the  small  coal  the  greater  part 
of  the  schists,  pyrites,  and  other  matters  mixed  with  it  when  it  is  extracted 
from  the  mine.  The  clean  coal  thus  obtained  is  useful  principally  for 
the  manufacture  of  coke  for  metallurgical  purposes  and  for  locomotives. 
The  advantages  derivable  from  the  washing  of  coal  are  beginning  to  be 
appreciated  in  England. 

Coal  is  washed  by  two  different  methods.  By  one  system,  a  wooden 
box  is  divided  into  two  compartments,  by  a  partition  which  descends  nearly 
to  the  bottom,  leaving  a  communication  between  the  two,  of  which  one  is 
smaller  than  the  other.  The  larger  of  the  two  is  fitted  with  two  grates,  one 
above  the  other,  of  which  the  upper  one  is  formed  of  bars  with  interspaces 
of  0.4  inch  in  width,  and  the  lower  is  a  plate,  usually  of  copper,  pierced 
with  numerous  small  holes.  The  smaller  compartment  contains  a  piston. 
Coal  being  filled  into  the  larger  compartment  upon  the  grate,  the  whole 
box  is  filled  with  water,  and  the  piston  set  in  motion.  By  the  action  of  the 
piston  the  water  is  caused  to  traverse  the  coal  upwards  and  downwards, 
when  the  heavier  particles  of  schist,  &c.,  fall,  and  are  collected  upon  the 
lower  grate.  When  the  space  is  filled  up  to  the  level  of  the  upper  grate 
with  deposit,  the  coal  is  removed. 

On  the  second  system,  the  operation  is  continuous.  Water  flows  in  a 
long  shallow  trough,  slightly  inclined,  with  cross  partitions  at  intervals  from 
top  to  bottom,  carrying  with  it  the  small  coal,  which  is  delivered  into  it  at 


412 


FUELS.— COAL. 


the  upper  end.     The  denser  particles  are  deposited  in  the  first  compart- 
ments, and  the  lighter  particles  in  the  last  ones. 

DETERIORATION  OF  COAL  BY  EXPOSURE. 

Coal  deteriorates  or  decays  to  a  greater  or  less  degree  by  exposure  to 
the  atmosphere,  by  disintegration  or  crumbling,  and  also  by  the  gradual 
combustion  of  the  volatilizable  elements.  Atmospheric  oxygen  is  absorbed, 
and  converts  the  hydrocarbons  into  water  and  carbonic  acid.  It  has  been 
proved  in  one  case,  in  Germany,  that  bituminous  coal,  after  having  been 
exposed  for  nine  months,  lost  half  its  value  as  fuel;  coal  exposed  for  three 
months  to  a  temperature  of  284°  R,  lost  all  its  hydrocarbons.  The  coke 
manufactured  from  coal  thus  deteriorated  is  inferior  to  what  is  made  from 
coal  freshly  mined. 

The  above  experimental  evidence  corroborates  the  fact  that  the  decay  of 
coal  proceeds  more  rapidly  in  the  hotter  climates.  Dryness  is  unfavourable 
to  the  change,  while  moisture  accelerates  it.  When  sulphur,  or  sulphuret 
of  iron  (iron-pyrites),  is  present  in  considerable  quantity  in  a  coal  still 
changing  under  the  action  of  the  air,  a  second  powerful  heating  cause  is 
introduced,  and  both  acting  together  may  produce  "  spontaneous  combus- 
tion." The  presence  of  sulphur,  or  iron-pyrites  alone,  if  in  considerable 
quantity,  is  sufficient  to  excite  combustion. 

BRITISH   COALS. 

COMPOSITION  OF  BITUMINOUS  COALS. — DR.  RICHARDSON'S  ANALYSES,  1838. 

The  first  accurate  analyses  of  bituminous  coals  were  made  by  the  late 
Dr.  Richardson,  of  Newcastle-on-Tyne.  The  coals  submitted  to  analysis 
were — ist,  Splint  coal;  2d,  cannel  coal;  3d,  cherry  coal;  4th,  caking  coal. 
The  following  table,  No.  135,  contains  the  results  of  his  analyses.  The 
total  average  composition  of  the  samples  analyzed  was  about  81  per  cent, 
of  carbon,  5^  per  cent,  of  hydrogen,  9  per  cent,  of  oxygen  and  nitrogen, 
and  5  per  cent,  of  ash. 

Table  No.  135. — COMPOSITION  OF  BITUMINOUS  COALS. 
BY  DR.  RICHARDSON,  1838. 


Coal,  and  Locality  of  Beds. 

Carbon. 

Hydrogen. 

Oxygen 
and 

Nitrogen. 

Ashes. 

Splint,  from  Wylam  

per  cent. 
74.82 

per  cent. 
6.l8 

per  cent. 
5OQ 

per  cent. 

I?    QI 

Do.,    from  Glasgow  

82.Q2 

C.4.Q 

•   y 

IO  4.6 

I.  17 

Average  . 

78  8? 

c  8^ 

7  78 

7   ^2 

O'^o 

/  •  j^ 

Cannel,  from  Wigan,  Lancashire  .... 
Do       from  Edinburgh 

83.75 
6?  60 

5.66 
54.O 

8.04 

12  A.1 

2-55 

I/L  ^  7 

•T-W 

**to 

'•'+•  j  / 

Average  

7^.68 

C.C-? 

IO  27. 

8.^6 

BRITISH   COALS. 
Table  No.  135  (continued). 


413 


Coal,  and  Locality  of  Beds. 

Carbon. 

Hydrogen. 

Oxygen 
and 

Nitrogen. 

Ashes. 

Cherry,  from  Jarrow,  Newcastle  

per  cent. 

84  8  c 

per  cent. 
SO1? 

per  cent. 
8  A.1 

per  cent. 
I  6? 

Do.,     from  Glasgow  

»*^«**3 

8l.2I 

>W3 
r   AC. 

"•tO 
1  1  02 

sr/ 

1.4.2 

O'T-J 

*-y* 

Average  

81.  o^ 

^.2^ 

IO.I7 

I.cc 

Caking,  from  Garesfield,  Newcastle.. 
Do.,     from  South  Hetton,  Durham 

87.95 
83.27 

5-24 
5-17 

5-42 
9.04 

i-39 

2.52 

Average 

85  61 

5  2O 

723 

i  06 

/••*o 

j-.yv/ 

Total  average 

80  80 

5    A  C 

8  8s 

4QO 

•40 

.yw 

WEIGHT  AND  COMPOSITION  OF  BRITISH  AND  FOREIGN  COALS. 
BY  MESSRS.  DELABECHE  AND  PLAYFAIR,  1847-50. 

An  extensive  series  of  analyses  and  of  trials  of  British  coals  were  con- 
ducted by  Sir  Henry  Delabeche  and  Dr.  Lyon  Playfair,  at  the  College  for 
Civil  Engineers,  Putney,  in  the  years  1847-50,  to  the  order  of  the  govern- 
ment. The  results  of  their  investigations  were  published  in  three  Reports 
on  Coals  suited  to  the  Royal  Navy,  in  the  years  1849,  1850,  1851. 

Samples  of  98  British  coals  were  analyzed  and  tried  for  their  evaporative 
performance,  namely: — 37  Welsh  coals,  18  Newcastle  coals  (Hartley  dis- 
trict), 7  Derbyshire  and  Yorkshire  coals,  28  Lancashire  coals,  8  Scotch 
coals — Total,  98  coals. 

In  addition  to  these  there  were  analyzed  and  tried,  one  sample  of 
anthracite  from  Ireland,  six  patent  fuels,  and  24  foreign  coals. 

The  chief  results  of  these  analyses  and  trials,  compiled  from  the  reports, 
are  averaged  and  embodied  in  table  No.  136,  together  with  deductions  as 
to  the  total  heat  of  combustion  of  the  fuels.  The  specific  gravity,  and  the 
weight  and  bulk,  of  the  coals,  are  given  in  columns  2,  3,  4,  5;  and  the 
chemical  composition  in  columns  6,  7,  8,  9,  10,  n.  The  quantity  of  coke 
produced  from  each  coal  is  given  in  column  12.  The  total  heat  of  com- 
bustion is  given  in  units  of  heat  in  column  13,  and  also  in  equivalent 
evaporative  efficiency  in  columns  14,  15,  when  the  water  is  supplied  at 
62°  and  at  212°  R,  and  evaporated  at  atmospheric  pressure.  These 
columns,  13,  14,  15,  have  been  calculated  by  means  of  formulas  (  6  ),  (  7  ), 
(  8  ),  page  406.  The  evaporative  efficiency  found  by  the  trials  is  given  in 
column  1 6. 


414 


FUELS.— COAL. 


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<  ° 

O  P^H 

U  M 


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|!l             !     ^ 

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O     ONOO    tx  tx,  tx. 

CO    ON 

at  of  Combustion 
pound  of  coal. 

Equivalent  evapo- 
rative power 
from  212°. 

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Averaged  groups. 

Welsh,  37  samples 
Newcastle,  18  „ 
Derbyshire  &  Yorkshire,  7  „ 
Lancashire,  28  „ 
Scotch,  8  „ 

Average  of  British  samples  
Anthracite,  Ireland  

ca 

CO 

c 

I-J            Oj 

"S     '5 

O         'J^ 

wT      S      H 

- 

BRITISH   AND   FOREIGN   COALS. 


415 


There  are  very  great  variations  in  the  chemical  composition  and  properties 
of  coals.     In  British  coals,  the  constituents  vary  in  quantity  as  follows : — 

Carbon,  from  about  70  to  91  per  cent,  of  the  gross  weight. 

Hydrogen,  from  3^  to  nearly  7  per  cent. 

Oxygen,  from  about  */>,  to  20  per  cent. 

Nitrogen,  from  a  mere  trace  to  2  J/s  per  cent. 

Sulphur,  from  nothing  to  5  per  cent. 

Ash,  from  1/s  to  15  per  cent. 

Coke,  from  49  to  93  per  cent. 

The  average  composition  of  British  coals  deduced  from  the  table,  is  as 
follows : — 

Carbon about  80  per  cent. 

Hydrogen „        5 

Nitrogen „        i  */5  „ 

Sulphur „        ii^   „ 

Oxygen „       8       „ 

Ash „       4 


Fixed  carbon,  or  coke  . 


„   100 
61 


The  foreign  coals,  from  Van  Diemen's  Land,  and  from  Chili,  had  only 
from  63  to  66  per  cent,  of  constituent  carbon,  with  28  per  cent,  of  oxygen 
and  ash. 

Welsh  Coals. — It  may  be  noted  here  that  Mr.  G.  J.  Snelus,  in  1871, 
made  an  analysis  of  Llangennech  coal,1  of  which  the  particulars  are  sub- 
joined, with  those  of  a  few  other  coals,  extracted  from  the  Reports  of  Dela- 
beche  and  Playfair,  for  comparison.  "  The  Ebbw  Vale  coal,"  it  is  said, 
"  may  be  taken  to  represent  the  Monmouthshire  steam  coals ;  and  Powell's 
Duffryn  represents  the  Merthyr  and  Aberdare  coals,  highly  esteemed  for 
locomotives  and  ocean  steamers."  There  is  a  close  correspondence  between 
the  analyses  of  Llangennech  coal  made  in  1848  and  in  1871. 


Class  of  Coal, 
and  Date  of  Analysis. 

Carbon. 

Hydrogen. 

Nitrogen. 

Sulphur. 

Oxygen. 

Ash. 

Coke. 

Ebbw  Vale,  1848  
Powell's  Duffryn,  1848... 
Llangennech,  1848  
Llangennech,  1871  
Graicrola  1848 

per  cent 
89.78 
88.26 
85-46 
84.97 
84  8? 

per  cent. 

tJI 

4.20 
4-26 

^  8d 

per  cent. 
2.16 

1-45 
1.07 

1-45 
4i 

per  cent. 
1.02 
1.77 
,29 
.42 

Af 

per  cent. 

% 

2.44 

3-50 
7  10 

p.  cent. 

*•£ 

3.26 

6.54 
5-40 

I.  ^O 

p.  cent. 

77-5 
84-3 
83.7 
86.7 
85.5 

1  See  Appendix  to  the  Report  of  the  Judges,  Mr.  F.  J.  Bramwell  and  Mr.  W. 
Menelaus,  on  the  Trials  of  Portable  Steam  Engines  at  Cardiff  in  1872 ;  for  a  full 
Report  on  the  Coal  used  in  the  Trials  of  Steam  Machinery  by  the  Royal  Agricultural 
Society. 


416  FUELS.— COAL. 

PATENT  FUELS. 

The  patent  fuels  tried  by  Delabeche  and  Playfair  consisted  of  mixtures 
of  bituminous  or  tarry  matter  with  small  bituminous  coal.  They  had  an 
average  of  83.4  per  cent,  of  constituent  carbon,  and  5  per  cent,  of  hydrogen, 
with  less  than  3  per  cent,  of  oxygen,  and  6  per  cent,  of  ash.  Three  patent 
fuels  produced  an  average  of  74  per  cent,  of  coke.  Warlich's  patent  fuel 
was  the  richest  in  carbon,  of  which  it  contained  90  per  cent.;  of  hydrogen, 
5.56  per  cent;  of  ash,  2.91  per  cent.  It  yielded  85  per  cent,  of  coke;  and 
it  evaporated,  by  trial,  10.36  Ibs.  of  water,  per  pound  of  fuel,  reckoned  from 
water  supplied  at  2 1 2°. 

WEIGHT  AND  BULK  OF  BRITISH  COALS. 

The  average  specific  gravity  of  coal,  as  by  the  table  No.  136,  is  1.279; 
it  varies  from  1.20  to  1.39. 

The  average  weight  of  coal  is  80  Ibs.  per  cubic  foot,  solid;  the  weight 
varies  from  78  to  86  Ibs. 

The  average  weight  is  50  Ibs.  per  cubic  loot,  heaped;  the  weight  varying 
from  45  to  58  Ibs. 

The  average  bulk  of  one  ton,  heaped,  of  coal,  is  44^  cubic  feet;  the 
bulk  varying  from  38  to  49  cubic  feet. 

The  average  specific  gravity  of  patent  fuels  is  1.167;  tne  average  weight 
is  73/^  Ibs.  per  cubic  foot,  solid,  and  65  Ibs.  per  cubic  foot,  heaped.  The 
bulk  of  one  ton,  heaped,  is  34^2  cubic  feet. 

These  averages  show  the  advantage  of  the  patent  fuels  in  point  of  com- 
pactness, over  coals;  for  though  they  are  the  lighter  fuel,  they  occupy  less 
space  per  ton  than  coals,  on  account  of  the  regular  forms  in  which  the 
blocks  are  manufactured,  and  the  facility  for  stowing  them  without  much 
interspace. 

HYGROSCOPIC  WATER  IN  BRITISH  COALS. 

The  hygroscopic  water  in  coal, — apart  from  what  is  chemically  combined 
with  it, — varies  considerably.  In  the  analyses  of  Delabeche  and  Playfair, 
in  which  the  specimens  were  dried  at  212°  F.,  it  varied  from  0.61  to  9.31 
per  cent,  of  the  weight  of  the  coal.  The  following  are  examples : — 


HYGROSCOPIC   WATER. 


Powell's  Duffryn  coal 1.13  per  cent. 

Mynydd  Newydd  0.61  ,, 

Pentrefelin  0.70  „ 

Park  End  coals,  Lydney 2.78  „ 

Ebbw  Vale 1.34  „ 

Resolven 1.55  „ 

Pontypool i. 60  „ 

Grangemouth  coal 6.42  „ 

Broomhill  coal 9.3 1  „ 

Wallsend  Elgin 2.49  „ 

Fordel  splint 8.40  „ 

Warlich's  patent  fuel  0.92  „ 

Bell's  patent  fuel 0.90  „ 

Wylam's  patent  fuel 1.38  „ 

Hartley  coals  6.19  to  10.17  „ 

Steamboat  Wallsend 1.14  „ 

Andrew's  House,  Tanfield 6.58  „ 


BRITISH   COALS. 


417 


HYGROSCOPIC  WATER. 

Cannel  coal,  Wigan i.oi  per  cent. 

Stavely 8.54 

Vancouver's  Island 7.21        „ 

Chirique , 9.11        „ 

Sydney,  New  South  Wales 3.25        „ 

Juan  Fernandez 6.00        „ 

It  appears  from  this  that  the  Welsh  coals  and  the  patent  fuels  contained 
the  least  proportion  of  hygroscopic  water. 

TORBANEHILL  OR  BOGHEAD  COAL. 

The  Boghead  coal  is  a  special  mineral  found  on  the  estate  of  Torbane- 
hill,  Linlithgowshire.  Its  colour  varies  from  dark  snuff-brown  to  brownish- 
black.  It  is  exceedingly  hard;  the  fracture  is  slaty  and  conchoidal.  When 
struck  with  a  hammer,  it  gives  a  woody  sound.  Its  specific  gravity  varies 
from  1.155  to  1.260,  the  average  being  1.189. 

In  composition,  Boghead  coal  occupies  the  opposite  end  of  the  scale  to 
anthracite, — having  a  comparatively  small  percentage  of  carbon,  and  a  large 
excess  of  hydrogen.  According  to  Dr.  Penny's  analysis  of  the  coal,  dried 
at  2 1 2°,  the  composition  is  as  follows : — 

per  cent. 

Carbon 63.94 

Hydrogen 8.86 

Nitrogen 0.96 

Sulphur o.  3  2 

Oxygen 4.70 

Ash  ..  21.22 


100.00 

As  the  oxygen  amounts  to  only  4.7  per  cent.,  it  leaves  free  a  surplus 
of  8^  per  cent,  of  hydrogen,  to  form  hydrocarbons  with  the  constituent 
carbon,  when  the  coal  is  distilled ;  and  it  is  found  that  coal  of  the  above 
composition  yields  67  per  cent,  of  volatile  matter,  and  31  per  cent,  of  ash. 
The  composition  varies  in  different  specimens,  as  may  be  observed  in  the 
following  analyses  of  four  specimens  taken  from  the  pit  at  different  dates, 
table  No.  137: — 

Table  No.  137. — COMPOSITION  OF  BOGHEAD  COAL. 


COMPOSITION. 

COAL. 

gravity. 

Fixed 
carbon. 

Volatile 
matter. 

Sulphur. 

Ash. 

Water. 

COKE. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

Brown,  1849.... 

I-I55 

II-3 

71.0 

o-3 

16.8 

0.6 

28.1 

Do.,    1851.... 

1.160 

7-1 

71.00 

0.2 

21.2 

o-5 

28.3 

Black,    1851.... 

i.  218 

9.25 

62.70 

0-35 

26.5 

i.  20 

35-75 

Do.,     1853.... 

1.188 

10.52 

67.11 

0.32 

21.0 

1.05 

3i-52 

Averages  

1.180 

9-54 

67.95 

0.29 

21.4 

0.84 

30-94 

27 

41 8  FUELS. — COALS. 

From  the  table  it  appears  that  the  fixed  carbon  averages  only  9%  per 
cent.;  and  that,  including  ash,  the  coke  averages  only  31  per  cent.,  the  vola- 
tile matter  exceeding  two-thirds  of  the  whole  weight  of  the  coal.  When 
distilled  at  comparatively  low  temperatures,  Boghead  coal  affords  large 
quantities  of  paraffin,  paraffin  oil,  &c. : — a  discovery  made  by  Mr.  Young. 


AMERICAN    AND    FOREIGN    COALS. 
BY  PROFESSOR  W.  R.  JOHNSON,  1843-44. 

The  results  of  an  investigation  of  the  qualities  of  American  coals,  at  the 
Navy  Yard,  Washington,  for  the  Navy  Department  of  the  United  States, 
conducted  by  Professor  W.  R.  Johnson,  were  published  in  "  A  Report  to 
the  Navy  Department  of  the  United  States,  on  American  Coals,"  in  1844. 

Thirty-nine  samples  of  coal,  and  three  samples  of  coke,  were  tried; 
and  the  general  results  are  given  in  table  No.  138. 

The  constituents,  so  far  as  the  analyses  extended,  were  in  the  following 
proportions : — 

Volatile  matter,  other  )       T  /  „.        T , 

than  moisture, }    2^  to  34#  Per  cent.,  average  16.17  per  cent. 

Hygrometric  moisture,.,  o      to    3^  „  „ 

Sulphur, o      to    2^  „  „ 

Fixed  carbon, 53      to  91  „  „ 

Earthy  matter, 4^  to  15  „  „ 

About 100.00 

Coke  (fixed  carbon  and  earthy  matter), 82.50  per  cent. 

The  proportions  of  volatile  matter,  fixed  carbon,  ash,  and  coke,  were  for 
the  three  classes  of  American  coal  as  follows : — 

Volatile  Fixed  .  ,  ~  , 

matter.  carbon.  Ash"  Coke' 

Anthracites, 3.97  ...  88.54  ...     6.28  ...  94.82 

Free  burning  bituminous  coals,. ...15.11   ...   73.21   ...   10.27   •••  83.48 
Bituminous  caking  coals, 29.43   ...  58.29  ...   10.90  ...  69.19 


Averages, 16.17   •••  73-35  •••     9-*5    •••  82.50 

WEIGHT  AND  BULK  OF  AMERICAN  COALS. 

The  specific  gravity  of  American  coal  varies  from  1.283  to  1-610,  and  it 
averages  1.400. 

The  weight  of  solid  coal  varies  from  80  to  100  pounds  per  cubic  foot, 
and  it  averages  87^  pounds. 

The  weight  of  heaped  coal  varies  from  45  to  56  pounds  per  cubic  foot, 
and  it  average?  51 24  pounds. 

The  bulk  of  one  ton  of  heaped  coal  varies  from  49^/3  to  40  cubic  feet, 
and  it  averages  43^  cubic  feet. 


AMERICAN   AND   FOREIGN   COALS. 


419 


Table  No.  138. — AMERICAN  COALS: — AVERAGE  WEIGHT,  BULK,  AND 
COMPOSITION,  1843. 

(Compiled  from  the  Report  of  Professor  W.  R.  Johnson.) 
WEIGHT,  BULK,  COKE,  AND  ASH. 


COAL. 

Specific 
gravity. 

WEIGHT  AND  BULK. 

Coke 
produced 
from  coal. 

Ash  and 
clinkers 
left  by 
combus- 
tion. 

One 

cubic  foot, 
solid. 

One 

cubic  foot, 
heaped. 

Bulk  of 
one  ton, 
heaped. 

Anthracites 

1.500 

1.358 
1.342 
I.3I8 

pounds. 
93-78 

84.93 
83.90 
82.39 

pounds. 

53.05 
32.13 
52.84 
49.28 
49.31 

cubic  feet. 

42.35 
69.76 
42.42 
45.71 

45-51 

per  cent. 
94.82 

83.68 
69.01 
65.27 

per  cent. 
8.60 

14-94 
11.27 
8.48 
7.98 

Cokes  

Free-burning  bituminous 
Bituminous  caking  

Foreign  and  Western  

Average  of  the  three  ) 
classes  of  American  > 
coals  1 

I.4OO 

87.54 

5L72 

43-49 

82.50 

9.42 

COMPOSITION. 


COAL. 

COMPOSITION,  IN  PERCENTAGES  OF  THE 
TOTAL  WEIGHT. 

Moisture. 

Volatile 
matter, 
other  than 
moisture. 

Sulphur. 

Fixed 
carbon. 

Earthy 
matter. 

Anthracites,  Pennsylvania  

per  cent. 
I.I9 

i-37 

1.56 
2.50 

per  cent. 

3-97 

15.11 

29-43 
32.68 

per  cent. 
O.04 

0.42 

I.OI 

O.24 

per  cent. 
88.54 

73.21 

58.29 
57-42 

per  cent. 
6.28 

14.94 

IO.27 
10.90 

7.85 

Coke,  two  samples  from   Mid-  } 
lothian   and   NeiFs  Cumber-  > 
land  coal,  Virginia  j 

Free-burning  bituminous,  Mary-  ) 
land  and  Pennsylvania  j 

Bituminous  caking,  Virginia  
Foreign  and  Western  bituminous. 

Average  of  the  three  classes  ) 
of  American  Coals  J 

i-37 

16.17 

0.49 

73-35 

9-15 

420  FUELS. — COAL. 


FRENCH   COALS. 

French  coals  are  divided  into  five  classes,  according  to  their  behaviour  in 
the  furnace : — 

i  st.  Bituminous  caking  coals  (houilles  grasses  marechales). 

2d.  Bituminous  hard  coals  (houilles  grasses  et  dures),  differing  from  the 
first  by  having  less  fusibility;  the  coke  is  more  dense  than  that  of  the 
first,  and  is  best  for  blast  furnaces. 

3d.  Bituminous  coals,  burning  with  a  long  flame  (houilles  grasses  a  longues 
flammes] ;  they  are  still  less  fusible  or  caking  than  the  preceding,  and  are 
best  for  boiler  and  other  furnaces.  They  are  known  by  the  designation 
flenu,  and  are  similar  to  Lancashire  cannel  coal. 

4th.  Dry  coals,  with  a  long  flame  (houilles  seches  a  longues  flammes].  The 
coke  has  not  much  coherence.  These  coals  are  burned  on  grates;  they 
are  less  durable  than  the  foregoing. 

5th.  Dry  coals,  with  a  short  flame  (houilles  seches  a  courtes  flammes\ 
These  coals  burn  with  some  difficulty,  and  are  used  chiefly  for  burning 
bricks,  and  in  lime-kilns;  in  breweries  for  drying  malt,  and  for  domestic 
fires. 

Anthracites  are  classed  by  themselves. 

The  coal,  as  it  comes  from  the  mine,  large  and  small  together,  is  known 
as  tout-venant — "as  it  comes."  In  the  market,  the  coal  from  a  mine  is  dis- 
tinguished, according  to  the  size  of  the  pieces,  into,  ist,  le gros,  round  coal; 
2d,  la  gaillette,  coal  of  medium  size,  in  pieces  5  or  6  inches  in  diameter, 
which  is  separated  by  screening  from  the  third  sort;  3d,  le  memt,  slack, 
which  is  subdivided  into  three  kinds : — gailletin,  the  size  of  nuts ;  tete  de 
moineau,  smaller  than  gailletin — literally  the  size  of  a  sparrow's  head;  and 
fine,  which  is  again  distinguished  mtofine  menue&n&fine  pottssier,  coal  dust. 

UTILIZATION  OF  THE  SMALL  COAL. 

The  menu,  or  small  coal,  is  made  into  briquettes,  or  rectangular  blocks; 
being  agglomerated  by  means  of  tar,  and  compressed  into  moulds,  as  has 
already  been  pointed  out  in  describing  English  patent  fuels,  with  some 
slight  differences  of  treatment,  ist.  The  small  coal  is  mixed  with  pitch, 
and  compressed  in  moulds  to  form  blocks.  These  blocks  have  great 
durability,  and  do  not  deteriorate  by  exposure  to  air.  2d.  When  the 
slack  is  derived  from  rich  bituminous  coals  it  is  filled  into  cast-iron  moulds, 
which  are  so  closed  that  nothing  but  gas  can  escape  from  them.  The 
moulds  are  heated  in  a  furnace  to  upwards  of  900°  F.,  where  they  remain 
from  half  an  hour  to  three  hours,  according  to  the  quality  of  the  coal.  By 
the  action  of  the  heat  the  coal  becomes  a  kind  of  paste,  and  tends  to 
swell;  but  it  is  on  the  contrary  powerfully  compressed  by  the  moulds.  3d. 
For  the  slack  of  dry  coals  a  certain  proportion  of  the  slack  of  bituminous 
coal  is  mixed  with  it,  to  give  cohesive  power. 

COMPOSITION  OF  FRENCH  COALS. 

The  table  No.  139  contains  the  specific  gravity  and  composition  of  a 
number  of  French  coals.  For  the  first  section,  comprising  Regnault's 
analyses,  compiled  from  a  table  by  M.  Peclet,  the  samples  were  dried  at  a 
temperature  of  120°  C,  or  about  250°  F.;  and  the  loss  of  weight,  repre- 


FRENCH   COALS. 


421 


senting  moisture,  varied  from  1.36  to  1.60  per  cent.  The  quantity  of 
nitrogen  was  in  general  very  small  in  the  anthracites;  and  in  the  other 
coals  it  was  from  1.50  to  2.0  per  cent.  The  united  weights  of  oxygen  and 
nitrogen  have  therefore  been  taken  by  M.  Peclet  to  represent  the  quantity 
of  oxygen,  in  calculating  the  heating  power,  according  to  the  principle 
already  explained,  page  403. 

Table  No.  139. — MEAN  DENSITY,  COMPOSITION,  AND  HEATING  POWER 

OF  FRENCH  COALS. 


COALS. 

Regnault,  22  samples. 
Marsilly,  79  samples. 

Specific 
gravity. 

Quan- 
tity of 
coke. 

Composition. 

Hydro- 
gen in 
excess. 

Heating 
Power. 

Carbon. 

Hydro- 
gen. 

Oxygen 
and  ni- 
trogen. 

Ash. 

(Regnault.  ) 
Anthracites. 

.498 
.319 
•293 
.303 
.362 

.265 

•293 
.197 
.289 
.280 

per  cent. 

88.83 
74.81 
67.54 
60.86 
54-72 

71-5° 
81.79 
86.58 
80.75 
74.66 

per  cent. 

86.17 
88.56 

87-73 
82.94 
76.48 

83.85 
86.38 
86.65 
86.50 
84.94 

per  cent. 

2.67 
4.88 
5.08 
5-35 
5.23 

5-19 
4.51 
4.l8 

4-52 
5.15 

per  cent. 

2.85 
4.38 

1§ 

16.01 

8.09 
5.46 
5-23 
5-39 
7.02 

per  cent. 

8.56 
2.19 

I.S4 
3-08 
2.28 

2.85 
3-66 

3-95 
3-52 
2-93 

per  cent. 

2-43 

4.27 

4-3° 
4-15 
3-09 

4.24 
3-82 
3.52 
3.85 
4.22 

units  of  heat 

14,038 

15,525 
I5>422 
14,622 
13,041 

14,884 
14,931 
14,787 
14,976 
15,003 

Bituminous  hard  coals.. 
Bituminous  caking  coals 
Bitum.  coals,  long  flame 
Dry  coals,  long  flame.  .  . 

(Marsilly.  ) 
Mons  Basin. 

Mons  Centre  Basin  
Charleroi  Basin  
Valenciennes  Basin  
Calais  Basin.. 

Average. 

I.3IO 

74-20 

85.02 

4.48 

6.87 

3-46 

3-79 

14,723 

Note. — The  averages  are  here  deduced  from  averages ;  being  averages  of  averages,  and 
are  to  be  accepted  as  approximate,  not  necessarily  exact  results. 

For  the  second  section,  the  samples  were  dried  by  exhaustion  in  the 
receiver  of  an  air-pump  during  from  twelve  to  twenty-four  hours. 

It  appears  from  the  table  that  the  average  composition  of  French  coals 
is  as  follows: — 

Carbon, 85      percent. 

Hydrogen, 4^        „ 

Oxygen  arid  nitrogen, 7  „ 

Ash, 3}^ 

Sulphur, ?          „ 


100 


The  average  specific  gravity  is  1.310,  giving  a  weight  of  81.68  Ibs.  per 
cubic  foot  solid. 

According  to  Peclet  the  weight  of  heaped  coal  from  different  mines  is  as 
follows,  in  table  No.  140;  to  which  are  added  the  weight  of  one  cubic 
foot  heaped,  and  the  volume  of  one  ton  heaped. 


422  FUELS.  —  COAL. 

Table  No.  140.  —  WEIGHT  AND  VOLUME  OF  FRENCH  COALS, 


HEAPED. 


MINE. 

Weight  of 
one  hectolitre, 
heaped. 

Weight  of 
one  cubic  foot, 
heaped. 

Volume 
of  one  ton, 
heaped. 

Labarthe..                     

kilogrammes. 

88 

pounds, 
tc  o 

cubic  feet. 
4O  7^ 

Auvergne  and  Blanzy  

87 

00>v 

CA    ? 

•H-w-  /  0 
41  22 

Combelle     

86 

c^.7 

41.  7O 

Latauoe  . 

8«5 

C5.I 

42.  IQ 

Saint-  Etienne  

84 

C2X 

42.6Q 

Decise 

8? 

qi  8 

43  21 

Mons  

"O 

80 

D  A- 
^O.O 

•ro**  * 

44.83 

Creusot 

70 

40  3 

4^.30 

Averages  of  bituminous  coals 

84 

52.5 

42.75 

Anthracite           .       ...        

oo 

^6.2 

4O.OO 

An  abstract  of  a  resume  of  analyses  of  French  and  other  coals  and 
lignites,  by  MM.  Scheurer-Kestner  and  Charles  Meunier-Dollfus,  is 
given  in  table  No.  141,  together  with  the  observed  heat  of  combustion. 
The  figures  have  reference  to  pure  fuel,  from  which  the  ash  has  been 
separated,  in  terms  of  the  gaseous  constituents  only. 

Table  No.  141. — FRENCH  AND  OTHER  COALS  AND  LIGNITES.     ANALYSIS 
OF  GASEOUS  CONSTITUENTS  AND  OBSERVED  HEAT  OF  COMBUSTION. 

(Scheurer-Kestner  and  Meunier-Dollfus.} 
The  fuel  is  assumed  to  be  dry  and  pure — without  any  ash. 


Designation  of  Combustible. 

Gaseous  Elements. 

Heat  of 
combustion  oi 
i  pound  pure 
(observed). 

Carbon. 

Hydro- 
gen. 

Oxygen 
and 
nitrogen. 

COAL. 
Ronchamp  3  samples 

per  cent. 
88.59 
Sl.IO 
90.60 
78.58 
87.02 

84.45 
83.94 
91.08 
92.49 
96.66 

9T-45 
82.67 

per  cent. 
4.69 

4-75 
4.IO 

5-23 

4.72 
4.21 

4.43 
3^3 
4.04 

1-35 
4-5° 

5-o7 

per  cent. 
6.72 
14.15 
5-30 
16.19 
8.26 
11.32 
11.63 

5-°9 
3-47 
1.99 

4-05 
12.26 

units. 
16,416 
15.320 
16,994 

14,985 
l6,40O 
16,663 
16,290 

15,804 

1  6,  1  08 

14,866 

15*651 

14,438 

Sarrebruck   7      do 

Creusot        4      do. 

Blanzy  •  —  Montceau  

Do         Anthracitic  

Angin      

Denain      .        

English  :  —  Bwlf      

Do          Powell-Duffryn  

Russian  :  —  Grouchefski  anthracite.  .  .  . 
Do.         Miouchi,  bituminous  
Do.         Goloubofski,  flaming  

INDIAN   COALS. 
Table  No.  141  (continued}. 


423 


Gas 

eous  Elem 

snts. 

Heat  of 

Designation  of  Combustible 

Carbon. 

Hydro- 
gen. 

Oxygen 
and 
Nitrogen. 

Combustion 
of  i  Ib.  pure 
(observed). 

LIGNITES. 
Rocher  bleu  

per  cent. 
72.Q8 

per  cent. 
4.  04. 

per  cent. 
22  08 

units. 
1  1  67O 

Manosque  bituminous 

7O  ^7 

544 

23  QQ 

T  -2    2  C  3 

Do         dry 

/w'  j  / 

66  ii 

•*Hh 

4  8"; 

^o-yy 
28  84 

io>^oo 

12   ?84 

Bohemia  bituminous 

w.  jj. 

76  s8 

8  27 

I  C    T  C 

••"^O^T- 
14  26"? 

Russian  Xoula 

/  "•  0" 

7-2  72 

V»^J 

6  OQ 

AD*  *  3 
2O  IQ 

T-2  8"?7 

Lignite,  passing  to  fossil  wood  

/  O'/  ^ 

66.1;  I 

•vy 

4  72 

*\j.  j.y 

28  77 

•••OJ^O/ 
II  444 

Fossil  wood,  passing  to  lignite  

'"•3  ->• 
67.6O 

4.  CC 

27  85 

I  I."?6o 

T-'OO 

•*  /<v») 

INDIAN   COALS. 

In  July,  1860,  Mr.  R.  Haines,  acting  chemical  analyst  to  the  Bombay 
government,  reported  on  samples  of  coal  from  Australia,  the  Nerbudda 
Valley,  and  Nagpore.  The  following  are  the  principal  results  contained  in 
the  report : — 

The  Australian  coal  is  jet-black  and  brilliant,  very  brittle,  and  breaks 
with  a  cubical  fracture  like  Newcastle  coal. 

The  Nerbudda  coal  is  dull  black,  heavy,  very  hard,  being  pulverized 
with  difficulty;  it  has  a  laminated  structure  and  slaty  cleavage;  it  has,  here 
and  there,  interspersed  in  its  substance,  small  lumps  of  half-formed  coal 
like  charcoal. 

The  Nagpore  coal  is  very  similar  in  appearance  to  the  Nerbudda  coal, 
and  has  the  same  texture,  except  that  the  laminae  are  alternately  dull 
and  glossy. 

The  Australian  coal  is  bituminous,  and  it  cokes  like  Newcastle  coal. 
The  Nerbudda  and  Nagpore  coals  do  not  even  cohere  in  coking.  The 
ash  of  the  Australian  coal  is  of  a  dirty  white  colour,  and  that  of  the  other 
coals  is  similar  in  appearance. 

The  results  of  analysis  of  these  coals  are  given  in  table  No.  142,  to- 
gether with  similar  results  from  English  coals.  The  products  are  divided 
into  solid,  or  "coke,"  and  volatile;  and  in  the  last  two  columns  are  given 
separately  the  sulphur  and  the  ash,  the  first  of  which  is  included  in  the 
volatile  matter  and  the  second  in  the  coke. 

The  proportions  of  ash,  or  incombustible  matter,  are  respectively  as 
follows : — 

COAL. 

Australian, 8.38  per  cent. 

Nerbudda  Valley, 18.09       „ 

Nagpore, 18.73        » 

English, 3.66       „ 


424 


FUELS. — COAL. 


Table  No.  142. — COMPARATIVE  COMPOSITION  OF  AUSTRALIAN,  NERBUDDA, 
NAGPORE,  AND  ENGLISH  COALS.     BY  MR.  R.  HAINES,  1860. 


Locality  or  Description. 

Specific 
gravity. 

Coke. 

Volatile 
matter. 

Sulphur. 

Ash. 

Australia 

I.  312 

per  cent. 
68  27 

per  cent. 
11  73 

per  cent. 
O  C.O 

per  cent. 

8  38 

Nerbudda  Valley 

I.4.4O 

\j\j.^  f 

66  6? 

Ox'  I  O 

3337 

o  60 

1  8  OQ 

Nagpore.                               .  . 

I./J.I7 

'•'-'O 

76.00 

JO'O  / 

24  OO 

O  34 

l8  73 

W<OT- 

i<J'  1  O 

Welsh  steam  coal  :  — 
From  

.27^ 

62.C 

37.  c. 

O  33 

I  2< 

To 

•7  CQ 

88  i 

1  1  Q 

V'OO 

5O7 

A-^0 
6  Q4 

Average 

•o  jw 

•2  JO 

80  o 

A  ±.y 

20  o 

.<w>/ 

I  2  C 

w.y^j. 
3OO 

Scotland  :  — 
From  

.200 

40  3O 

CO.7O 

l*3 

O  33 

I   13 

To  

.316 

Co  i  c 

3V^/W 

40  8  c. 

w<oo 

1^7 

•'*O 

14  ^7 

Average  .  .  . 

.260 

C.4.OO 

H-^-'-'O 

46  oo 

**Jl 

I.IO 

XifO  / 
4  OO 

Newcastle  :  — 
From  

1.23 

62.7O 

37.30 

o  06 

0.  2O 

To 

1.  31 

72  3O 

27  7O 

i  Sc; 

I  3  QI 

Average  . 

••••o  x 

1  28 

/  ^'O^ 

66  oo 

^  /•  /  w 
34  OO 

*«wo 

I  OO 

^O'V-1 

4  oo 

Oit"v-' 

Average  of  English  coals.  . 

1.28 

66  66 

•?•?   -J-? 

112 

3  66 

OO'O  O 

An  official  memorandum  was  addressed  to  the  Indian  government,  in 
January,  1867,  by  Dr.  Oldham,  superintendent  of  the  geological  survey  of 
India,  containing  the  results  of  analysis  of  eighty-one  samples  of  Indian 
coal:  showing  the  volatile  matter,  the  fixed  carbon,  and  the  ash.  These 
results  are  given  in  table  No.  143,  and  a  column  is  prefixed  showing  the 
percentage  of  coke,  which  is  arrived  at  by  adding  that  of  the  ash  to  that 
of  the  fixed  carbon.  For  comparison,  the  results  of  a  similar  analysis  of 
English  coals  saleable  at  Calcutta,  are  added. 

The  distinguishing  characteristic  of  the  Indian  coal  is  the  great  propor- 
tion of  ash  it  contains,  varying  from  i^  per  cent,  though  in  only  one 
instance,  to  59  per  cent.,  and  averaging,  for  81  samples,  23  per  cent.  The 
English  coal  saleable  at  Calcutta  has  only  an  average  of  2.7  per  cent.  The 
following  are  the  average  compositions  of  Indian  and  of  English  coal  at 
Calcutta,  deduced  from  table  No.  143: — 

Indian  coals.        English  coals, 
per  cent.  per  cent. 

Coke, 70.2  ...  70.8 

Fixed  carbon, 47.3  ...  68.1 

Volatile  matter, 29.6  ...  29.2 

Ash, 22.9  ...  2.7 

showing,  notwithstanding  the  great  excess  of  ash  in  the  composition  of  the 
Indian  coals,  that  the  quantity  of  volatile  matter  is  about  the  same  as  in 
the  English  coals,  about  29  per  cent.  In  the  absence  of  a  full  chemical 
analysis,  it  is  impossible  to  say  how  much  of  this  consists  of  carbon, 


INDIAN   COALS. 


425 


oxygen,  hydrogen,  and  nitrogen  individually;   and  therefore  the  heating 
power  of  the  volatile  matter  cannot  be  estimated. 

Table  No.  143. — COMPOSITION  OF  INDIAN  COALS,  1867. 

Compiled  from  a  Report  by  Dr.  Oldham. 


Locality. 

Coke 
(sum  of  fixed 
carbon  and 
ash). 

Fixed 
carbon. 

Volatile 
matter. 

Ash. 

Kurhurbali  Field  :  — 
From                    

per  cent. 
7^.2 

per  cent. 
^O.Q 

per  cent. 
12.6 

per  cent. 
4.8 

To                         

87.4 

7^.1 

24.8 

^0.2 

Average  .  .  , 

7Q.Q 

62.8 

2O.2 

I7.I 

Rajmahal  Hills:  — 
From 

C4.4 

2S.2 

28.8 

i*^ 

To 

y*r*f 

71.2 

1:7.6 

44-8 

37-6 

Average  . 

60.8 

44.2 

-20.  -2 

16.6 

Ranigunj  Field:  — 
From     

CQ.O 

3Q.2 

2EJ.6 

1.75 

To 

7^O 

6?  8 

38.7 

-2C.2 

Average 

/  J'w 

6c  o 

**o*** 

co.o 

3C.O 

I^.O 

Sherria  Field:  — 
From 

\O-w 

CC.4 

ow< 
30.8 

18.0 

1.7 

To  

OJ'T- 

86.0 

68.4 

44.6 

28.8 

Average  

60.0 

<;6.3 

3I.O 

12.7 

Central  India  (Pench  River)  :— 
From  

^2.  0 

30.3 

I4.O 

2.2 

To 

86  o 

61  6 

CA  o 

48.7 

Average 

6*  6 

47.4 

32.8 

18.2 

Madras  (Godavery  River) 

"0-" 

81.0 

23  2 

IQ.O 

^7.8 

Total  averages  of  Indian  coals. 

70.2 

47-3 

29.6 

22.9 

English  coal,  saleable  at  Calcutta  : 
Averages  

70  8 

68.1 

2Q.2 

2.7 

It  may  be  added  that  Dr.  Oldham,  in  1859,  analyzed  two  specimens  of 
anthracitic  coal  from  Kotlee,  in  the  Punjab,  and  found  their  composition 
as  follows : — 

Carbon. 


per  cent. 

No.  i 90.5 

No.  2 90.0 


Volatile  matter. 

per  cent. 
...       4.0       .. 

6.0 


Ash. 
per  cent. 

5-5 
4.0 


Much  of  the  Indian  coal  is  peculiarly  liable  to  disintegration  from  ex- 
posure to  the  atmosphere,  particularly  in  the  hot  seasons.  Coal  from  the 
new  Chanda  coalfields  is  reported  to  have  fallen  to  so  small  pieces,  after  a 
short  period  of  exposure,  as  to  have  become  unfit  as  fuel  for  locomotives. 


426  FUELS.— COAL. 

COMBUSTION   OF   COAL. 

When  coal  is  exposed  to  heat  in  a  furnace,  the  carbon  and  hydrogen, 
associated  in  various  chemical  unions,  as  hydrocarbons,  are  volatilized  and 
pass  off.  At  the  lowest  temperature,  naphthaline,  resins,  and  fluids  with  high 
boiling  points  are  disengaged;  next,  at  a  higher  temperature,  volatile  fluids 
are  disengaged;  and  still  higher,  olefiant  gas,  followed  by  common  gas, 
light  carburetted  hydrogen,  which  continues  to  be  given  off  after  the  coal 
has  reached  a  low  red  heat.  As  the  temperature  rises,  pure  hydrogen  is 
also  given  off,  until  finally,  in  the  fifth  or  highest  stage  of  temperature  for 
distillation,  hydrogen  alone  is  discharged.  What  remains  after  the  distil- 
latory process  is  over,  is  coke,  which  is  the  fixed  or  solid  carbon  of  coal, 
with  earthy  matter,  the  ash  of  the  coal. 

The  hydrocarbons,  especially  those  which  are  given  off  at  the  lowest 
temperatures,  being  richest  in  carbon,  constitute  the  flame-making  and 
smoke-making  part  of  the  coal.  When  subjected  to  degrees  of  heat  much 
above  the  temperatures  required  to  vaporize  them,  they  become  decom- 
posed, and  pass  successively  into  more  and  more  permanent  forms  by 
precipitating  portions  of  their  carbon.  At  the  temperature  of  low  redness, 
none  of  them  are  to  be  found,  and  the  olefiant  gas  is  the  densest  type  that 
remains,  mixed  with  carburetted  hydrogen  and  free  hydrogen.  It  is  during 
these  transformations  that  the  great  body  of  smoke  is  made,  consisting  of 
precipitated  carbon  passing  off  uncombined.  Even  olefiant  gas,  at  a 
bright  red  heat,  deposits  half  its  carbon,  changing  into  carburetted  hydro- 
gen; and  this  gas,  in  its  turn,  may  deposit  the  last  remaining  equivalent  of 
carbon  at  the  highest  furnace  heats,  and  be  converted  into  pure  hydrogen. 

Throughout  all  this  distillation  and  transformation,  the  element  of  hydro- 
gen maintains  a  prior  claim  to  the  oxygen  present  above  the  fuel ;  and  until 
it  is  satisfied,  the  liberated  carbon  remains  unburned. 

SUMMARY  OF  THE  PRODUCTS  OF  DECOMPOSITION  IN  THE  FURNACE. 

Reverting  to  the  statement  of  the  average  composition  of  coal,  page  415, 
it  was  found  that  the  fixed  carbon  or  coke  remaining  in  the  furnace  after 
the  volatile  portions  of  the  coal  are  driven  off,  averages  6 1  per  cent,  of  the 
gross  weight  of  the  coal.  Taking  it,  for  round  numbers,  at  60  per  cent., 
the  proportion  of  carbon  volatilized  in  combination  with  hydrogen,  will  be 
20  per  cent. — making  up  the  total  of  80  per  cent,  of  constituent  carbon  in 
average  coal. 

Of  the  5  per  cent,  of  constituent  hydrogen,  i  part  is  united  to  the  8  per 
cent,  of  oxygen,  in  the  combining  proportions  to  form  water,  and  the 
remaining  4  parts  of  hydrogen  are  found  partly  united  to  the  volatilized 
carbon,  and  partly  free. 

These  particulars  are  embodied  in  the  following  summary  of  the  condi- 
tion of  the  elements  of  100  pounds  of  average  coal,  after  having  been 
decomposed,  and  prior  to  entering  into  combustion : — 


COMBUSTION  OF  COAL.  427 

100  Pounds  of  Average  Coal  in  the  Furnace. 

Composition.  Ibs.  Ibs.  Decomposition. 

Carbon  j  v^arilized 20     1  f6°      fi^ed  carbon. 

Hydroeen  5  I  2*     hydrocarbons  and  free  hydrogen. 


Nitrogen IT/S 

Ash 4 

About loo  J 


nitrogen. 


4      ash. 


IOO 


showing  a  total  useful  combustible  of  86^  per  cent.,  of  which  26^  per 
cent,  is  volatilized.  Whilst  the  decomposition  proceeds,  combustion  pro- 
ceeds, and  the  25^  per  cent,  of  volatilized  portions,  and  the  60  per  cent, 
of  fixed  carbon,  successively,  are  burned. 

It  may  be  added  that  the  sulphur  and  a  portion  of  the  nitrogen  are  dis- 
engaged in  combination  with  hydrogen,  as  sulphuretted  hydrogen  and 
ammonia.  But  these  compounds  are  small  in  quantity,  and,  for  the  sake 
of  simplicity,  they  have  not  been  indicated  in  the  above  synopsis. 

QUANTITY  OF  AIR  CHEMICALLY  CONSUMED  IN  THE  COMPLETE 
COMBUSTION  OF  COAL. 

Take  coal  of  average  composition.  Then,  applying  the  rule  i,  page  400,. 
the  carbon  C  =  80,  the  available  hydrogen  (H  —  Q)  =  4,  and  the  sulphur 
S=  1.25,  and 

C  +  3  (H  -  O)  +  .4  S  =  80  +  1 2  +  .5  -  92.5, 

o 

and  92.5  x  1.52  =  140.6  cubic  feet  of  air  at  62°,  the  quantity  chemically 
consumed  by  one  pound  of  average  coal. 

To  find  the  proportions  in  which  this  quantity  of  air  is  appropriated  for 
the  volatilized  and  the  fixed  portions  of  the  coal,  as  above  divided,  for 
loo  Ibs.  of  the  fuel: — 

FOR  THE  VOLATILIZED  PORTION — 

Hydrogen 4      Ibs.  x  457  =  1828  cubic  feet. 

Carbon 20      Ibs.  x  152  =  3040          „ 

Sulphur i%  Ibs.  x    57  =      71          „ 

4939  cubic  feet. 

FOR  THE  FIXED  PORTION — 

Carbon 60      Ibs.  x  152  = 9120        „ 

Total  useful  combustible,  85^  Ibs.  *4>°59         » 

showing  that  14,059  cubic  feet  of  air  at  62°  are  required  for  the  complete 
combustion  of  100  Ibs.  of  coal  of  average  composition.  It  is  equivalent 


428 


FUELS. — COAL. 


to  140.6  cubic  feet  of  air  for  one  pound  of  coal,  as  already  found,  or  in 
round  numbers,  140  cubic  feet,  of  which  there  are  required, 

For  the  volatilized  portions 50  cubic  feet,  or    36  per  cent. 

For  the  fixed  portion 90          „          or    64        „ 


140 


100 


The  weight  of  this  quantity  of  air,  dividing  the  volume  by  13.14,  is 
10.7  Ibs. 

The  following  table,  No.  144,  gives  the  composition  of,  and  the  quanti- 
ties of  air  chemically  consumed  in  the  complete  combustion  of,  British 
coals  of  the  highest,  the  lowest,  and  average  heating  powers,  placed 
together  for  comparison.  It  appears  from  the  table  that  the  quantity  of  air 
chemically  consumed  in  the  combustion  of  one  pound  of  British  coal  varies, 
according  to  the  composition  of  the  coal,  from  116  to  163  cubic  feet  at 
62°:— 

Table  No.  144. — COMPARATIVE  STATEMENT  OF  COMPOSITION,  HEAT  OF 
COMBUSTION,  AND  AIR  CHEMICALLY  CONSUMED  BY  BRITISH  COALS 
-  OF  THE  HIGHEST,  LOWEST,  AND  AVERAGE  QUALITY. 


Air  chemically 

/"•       1 

Total  heat  of 

consumed  in 

Coal 
(Selected  from  Delabeche  and 

Carbon. 

Hydrogen. 

Oxygen. 

Sulphur. 

combustion 
of  one  pound 

the  complete 
combustion  of 

Playfair's  Report). 

of  coal. 

one  pound 

of  coal. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

units. 

cubic  feet  at 
62°. 

Warlich's  patent  fuel. 

90.02 

5-56 



1.62 

16,495 

I63 

Ebbw  Vale 

80.78 

e  1C 

O.  7Q 

I.  O2 

l6,22I 

161 

Haswell  Wallsend... 

vy*  1  w 

83.47 

3'  *J 

6.68 

O  .7 
8.17 

O.O6 

15,502 

153 

Coal  of  average  com-  ) 
position                 j 

8O.OO 

5.00 

8.00 

*'*S 

!4>i33 

140 

Ince  Hall,  Pember-  \ 

ton  five  feet  (low-  > 

68.72 

4.76 

18.63 

I-35 

JI>525 

116 

est  British)  J 

Chirique,  Chili  (low-  ) 
est  foreign)  J 

38.98 

4.01 

13-38 

6.14 

7,349 

74 

GASEOUS  PRODUCTS  OF  THE  COMPLETE  COMBUSTION  OF  COAL. 

The  quantity  of  the  gaseous  products  is  found  by  rules  2  and  3,  pages 
401,  402.  Take,  for  example,  the  case  of  coal  of  average  composition. 

i.  By  weight. — The  percentages  of  carbon,  hydrogen,  sulphur,  and 
nitrogen  are  respectively  80,  5,  1.25,  1.20.  Then,  by  rule  2,  page  402,  the 
weight  of  the  gaseous  products,  taken  collectively,  of  the  combustion  of 
one  pound  of  coal,  is 

(.126  x  80) +  (.358  x  5) +  (.053  x  i. 25)  + (.01  x  1.20)=  11.94  pounds. 


COMBUSTION   OF  COAL.  429 

The  weights  of  the   gases   individually  are  given  by  the   expressions 
a,  b,  c,  d,  page  401,  as  follows:  — 

Pounds.     Per  cent. 

Carbonic  acid  ..............................................  0366x80      =2.93    or  24.5 

Steam.  .......................................................  09  x  5        =    .45    or    3.8 

Sulphurous  acid  ...........................................  02x1.25  =   .025  or    0.2 

Nitrogen  =  (.0893  x  80)  +  (.268  x  5)  +  (.0335  x  1.25)  +  (.01  x  1.20)  =  8.536  or  71.5 

11.94     100.0 

2.  By  volume.  —  The  total  volume  is  found  by  rule  3,  page  402;  thus:  — 
(1.52  x  80)  +  (5.  52  x  5)  +  (.567  x  i.  25)  +  (.135  x  i.  20)  -150.07  cubic  feet. 

The  volumes  in  detail  are,  by  the  expressions  e,  f,  g,  h,  page  401,  as 
follows  :  — 


Carbonic  acid  ......................................  .  .........  315x80      =   25.2    or  17 

Steam  .........................................................  1.9      x5        =     9.5    or    6 

Sulphurous  acid  .............................................  117x1.25  =     0.15     trace 

Nitrogen  =  (i.2o6x  80)  +  (3.618  x  5)  +  (.45  x  1.25)  +  (.135  x  1.20)  =  115.29  or  77 

150.14      100 

showing  that  the  12  pounds  of  gaseous  products  have  a  volume  of  150 
cubic  feet  at  62°,  equal  to  12^  cubic  feet  per  pound.  The  element  of 
nitrogen  is  nearly  three-fourths  by  weight,  and  fully  three-fourths  by  volume, 
of  the  total  quantity  of  gaseous  products. 

The  relatively  larger  volume  of  the  gaseous  products  at  the  higher  tem- 
perature at  which  they  enter  the  chimney,  is  found  by  the  formula  (  2  ), 
page  347,  repeated  at  page  402.  If  the  final  temperature  be  500°  R,  the 
final  volume  of  the  gaseous  products  for  one  pound  of  average  coal  is, 


150  5°°  +  4l  =  276  cubic  feet; 
62  +  461 

or  nearly  double  the  volume  at  62°.  At  585°  temperature,  the  volume 
would  be  exactly  double,  or  300  cubic  feet;  and  at  1108°  F.  it  would  be 
just  three  times  the  normal  volume  at  62°. 

SURPLUS  Am. 

The  quantity  of  surplus  air  which  passes  off  with  the  producik  of  cbfrit  ' 
bustion  into  the  chimney,  is  to  be  added  to  that  of  these  products  to  find 
the  total  weight  or  volume  of  the  gases  in  the  chimney,  as  already  stated,   A  ,  * 
page  402. 

Taking  the  case  of  coal  of  average  composition,  suppose  that  the 
quantity  of  surplus  air  is  equal  to  that  which  is  chemically  consumed 
by  the  fuel;  then  it  amounts  to  140  cubic  feet  by  volume,  or  10.7  pounds 
by  weight,  for  one  pound  of  coal  consumed;  adding  these  to  the  weight 
and  the  volume  of  the  products  of  combustion  above  found,  there  is 

Cubic  feet  at  62°.   Weight  in  Ibs. 

Gaseous  products  of  combustion  per  Ib.  of  coal,  ...150  ......   12 

Surplus  air  „  „      ...140  ......   10.7 

Total  escaping  gases,  ..................  290  ......  22.7 


430  FUELS.— COKE. 

When  the  quantity  of  surplus  air  is  less  than  that  which  is  chemically 
consumed,  the  volume  and  weight  to  be  added  to  those  of  the  products  of 
combustion,  are  less  than  140  cubic  feet  and  10.7  pounds  respectively,  in 
the  same  proportion. 

The  total  quantity  of  escaping  gases,  therefore,  produced  by  the  com- 
bustion of  one  pound  of  average  coal,  varies  according  to  the  proportion  of 
surplus  air — 

From  150  cubic  feet  to  290  cubic  feet,  at  62°; 
From  276         „          to  533         „          at  500°; 
From  12  pounds  to  22.7  pounds  in  weight. 

It  is  here  assumed  that  the  maximum  quantity  of  surplus  air  does  not 
exceed  the  quantity  of  air  chemically  consumed. 

TOTAL  HEAT  OF  COMBUSTION  OF  BRITISH  COALS. 

The  total  heat  of  combustion  of  coal  of  average  composition,  having 
80  per  cent,  of  carbon,  5  per  cent,  of  hydrogen,  8  per  cent,  of  oxygen, 
and  1.25  per  cent,  of  sulphur,  is,  by  rule  4,  page  406, 

145  (80  +  4.28  (5  -  8/s  )  +  (o.28  x  1.25)  )  =  14,133  units. 

The  heating  power,  expressed  in  pounds  of  water  evaporable  under  one 
atmosphere  by  one  pound  of  the  fuel,  is,  by  rule  5,  p.  406,  as  follows: — 
When  the  water  is  supplied  at  62°  the  total  evaporative  power  is 

0.13  (80  +  4.28  (5  -  8/8  )  +  (o.28  x  1.25))  =  12.67  pounds  of  water. 
When  the  water  is  supplied  at  212°  the  evaporative  power  is 
0.15  (80  +  4.28  (5  -  8/8  )  +  (o.28  x  1.25)  )  =  14.62  pounds  of  water. 

The  total  heat  of  combustion  of  British  coals  is  given  in  table  No.  136. 
page  414;  and  for  contrast  in  table  No.  144,  above. 


COKE. 

The  quantity  of  residuary  coke  in  various  coals,  was  found  by  laboratory 
analysis  as  follows  (see  previous  tables) : — 


(Excluding  anthracites.) 


COKE.  COKE. 


per  cent.  per  cent. 

English  coals 50  to  72  Average  61.4 

American  coals 64  to  86  „        76.4 

French  coals 53  to  76  „       64.5 

Indian  coals 521084  „        70.2 

Anthracite  coke  scarcely  deserves  the  name;  it  is  without  cohesion,  and 
pulverulent.  The  best  coke, — from  bituminous  coal, — is  clean,  crystalline, 
and  porous;  and  it  is  formed  in  columnar  masses.  It  has  a  steel-gray 
colour,  possesses  a  metallic  lustre,  with  a  metallic  ring  when  struck,  and  is 
so  hard  as  to  be  capable  of  cutting  glass. 


QUALITY   OF   COKE. 


431 


The  quality  of  coke  obviously  depends,  in  a  great  measure,  on  the  propor- 
tions of  the  constituent  hydrogen  and  oxygen  of  the  coal  from  which  it  is 
made,  which  regulate  the  degree  of  fusibility  of  the  coal  when  exposed  to 
heat.  Taking,  for  example,  the  particulars  of  the  coke  produced  from  the 
French  coals,  table  No.  139,  and  arranging  the  averages  for  each  kind  of 
coal  in  the  order  of  the  quantity  of  hydrogen  in  excess,  the  nature  of  the 
coke  produced,  as  described  by  M.  Peclet,  was  as  follows : — 


AVERAGES. 

Hydrogen. 

Oxygen 

and 
Nitrogen. 

Hydrogen 
in  excess. 

Nature  of  the 
Coke. 

Anthracites               

per  cent. 
2.67 

per  cent. 
2  8q 

per  cent. 
2.4.1? 

pulverulent 

Dry  coals,  long  flame  

C.23 

••"a 

16.01 

*"T-O 
3OQ 

in  fragments 

Bituminous  coals  long  flame 

51  e 

8  6^ 

A     T  C 

porous 

Bituminous  hard  coals 

'OJ 

488 

A  1& 

4-  A0 
427 

porous 

Bituminous  caking  coals 

«;  08 

e  6; 

••  / 

A    1Q 

very  porous 

D'w 

D'WD 

*t'O^J 

Showing  a  series  of  five  coals,  with  an  ascending  series  of  hydrogen  in 
excess,  from  2.43  to  4.30  per  cent.  The  nature  of  the  cokes  advances 
correspondingly  from  pulverulent  or  powdery,  to  very  porous  or  excessively 
fused  and  raised.  The  first  is,  in  fact,  a  failure  as  a  coke,  and  the  second, 
with  3.09  per  cent,  of  hydrogen  in  excess,  barely  coheres,  being  in  frag- 
ments; the  third  and  fourth,  with  about  4.20  per  cent,  of  hydrogen  in 
excess,  produce  a  porous  and  cohesive  coke,  and  the  fifth  an  excessively 
porous  coke, — bright,  but  comparatively  light  for  metallurgical  operations. 

From  this  it  appears  that  a  coal  having  less  than  3  per  cent,  of  hydrogen 
in  excess,  is  unfit  for  coke-making;  and  that,  for  the  manufacture  of  good 
coke,  coal  containing  at  least  4  per  cent  of  free  hydrogen  is  required. 

It  is  not  clear  in  what  manner  the  presence  of  free  hydrogen  operates  in 
fusing  the  substance  of  the  coal;  unless,  probably,  that  the  hydrogen 
being  in  combination  with  carbon  in  various  proportions  to  form  tar 
and  oils,  softens  the  fixed  carbon,  and  forms  a  pasty  mass,  which  is  raised 
like  bread  by  the  expansion  of  the  confined  gases  and  vapours  seeking  to 
escape.  The  increasing  proportions  of  volatilized  matter  which  is  raised  by 
heat,  successively,  from  anthracites,  bituminous,  and  caking  coals,  are  clearly 
exemplified  by  the  analyses  of  American  coals  in  table  No.  138,  page  419; 
and  they  evidently  have  relation  to  an  increase  of  the  hydrogen  in  excess 
above  that  required  to  form  water  with  the  constituent  oxygen.  They  are 
as  follows: — 


AMERICAN  COALS. 


VOLATILIZED 

MATTER. 

per  cent. 


Anthracite 5.16 

Free  burning  bituminous  coals 16.48 

Bituminous  caking  coals 3°-99 


COKE 

PRODUCED. 
per  cent. 
94.82 
83.68 
69.01 


The  increasing  volatilized  matter  explains,  as  above  suggested,  the  increas- 
ing porosity  and  bulk  of  the  coke  yielded  by  the  respective  coals. 


432  FUELS.— COKE. 

ANTHRACITIC   COKE. 

By  a  process  recently  introduced  at  Swansea  by  Messrs.  Penrose  & 
Richards,  and  described  by  Mr.  W.  Hackney,1  anthracite  has  been  success- 
fully used  as  the  basis  of  a  coke,  manufactured  from  the  following  mixture: — 
Anthracite,  60  per  cent;  Bituminous  coal,  35  per  cent;  Pitch,  5  per  cent. 
The  materials  are  crushed  and  mixed  together  through  a  disintegrator.  The 
yield  of  coke  is  80  per  cent  of  the  weight  of  the  charge.  The  coke  is 
steel-gray  in  colour,  and  so  hard  as  to  scratch  glass  easily;  and  it  is  about 
23  per  cent,  heavier  than  the  best  Welsh  coke.  It  burns  in  a  common  fire, 
or  in  a  blast  furnace,  without  any  sign  of  crumbling. 

QUANTITY  OF  COKE  YIELDED  BY  COAL. 

The  quantity  of  coke  produced,  on  the  large  scale,  from  coal  varies  from 
60  to  80  per  cent,  in  weight  The  following  are  examples  of  the  yield : — 

COALS.  COKE  PRODUCED. 

Andrew's  House,  Tanfield 65  per  cent  of  the  coal. 

Bristol 60  to  63.5       „  „ 

Kilsyth 60  „  „ 

Mons 77  to  80          „  „ 

Seraing 67  „  „ 

In  general,  the  yield  of  good  coke  is  about  two-thirds,  or  66  per  cent,  of 
the  coal. 

The  whole  of  the  coke  matter  in  coal  cannot  be  extracted  from  it,  on  the 
large  scale;  a  portion  of  it  is  burned  off.  Thus,  Seraing  coal,  from  which 
67  per  cent,  of  coke  was  made,  on  the  large  scale,  yielded  80  per  cent  of 
coke,  by  laboratory  analysis. 

WEIGHT  AND  BULK  OF  COKE. 

Coal  expands  in  volume  in  the  coking  process,  insomuch  that  the 
volume  of  the  resulting  coke  is  greater  by  from  10  to  30  per  cent,  than  that 
of  the  coal  from  which  it  is  made.  Tanfield  coke  has  1 1  per  cent,  more 
volume  than  the  coal  from  which  it  is  made;  and  as  the  specific  gravity  of 
the  coal  is  1.26,  that  of  the  coke  is  0.74,  calculated  as  follows: — 

1.26  x 5 =  0.74. 

I.I  I   X   IOO 

The  weight  and  volume  of  Tanfield  coal,  and  of  coke  made  from  it,  are 
as  follows: — 

Specific          Weight  of  i  cubit         Weight  of  i  cubic        Volume  of  i 
gravity.  foot,  solid.  foot,  heaped.  ton,  heaped. 

Tanfield  coal...   1.26  78.57103.  52.19105.  42.92  cubic  feet 

Do.      coke      0.74          46.14  „  30.00   „  74.66         „ 

Mickley  coke  weighs  28  Ibs.  per  cubic  foot,  heaped,  and  measures  in  bulk 
80  cub.  ft.  per  ton.  Gas  coke  weighs  from  12)^  cwt.  to  1 5  cwt.  per  chaldron. 

1  In  a  paper  read  at  the  meeting  of  the  Iron  and  Steel  Institute,  1875,  published  in 
Engineering,  November  12,  1875. 


COMPOSITION   OF   COKE.  433 

The  American  cokes,  from  Midlothian,  Va.,  and  Cumberland,  Md., 
averaged'  a  weight  of  32.13  Ibs.  per  cubic  foot,  heaped,  and  a  volume  of 
69.8  cubic  feet  per  ton. 

The  coke  used  for  smelting  furnaces  in  France  weighs,  ordinarily,  25  Ibs. 
per  cubic  foot,  heaped,  and  measures,  in  bulk,  about  90  cubic  feet  per  ton. 
Of  the  Seraing  coking  coal,  and  the  coke  produced  from  it,  the  weight  and 
bulk  are  as  follows, — assuming  that  the  coal  is  the  same  as  "average  New- 
castle coal,  with  which  it  is  almost  identical  in  chemical  composition : — 

Weight  of  i  cubic  Volume  of  i  ton, 

foot,  heaped.  heaped. 

Seraing  coal 50  Ibs.  45  cubic  feet. 

Do.     coke  31   „  72 

From  the  foregoing  particulars  it  may  be  gathered  that  coke  of  good  quality 
weighs  from  40  to  50  Ibs.  per  cubic  foot  solid,  and  about  30  Ibs.  per  cubic 
foot,  heaped;  and  that  the  average  volume  of  one  ton  is  75  cubic  feet, 
varying  from  70  to  80  cubic  feet  per  ton. 

COMPOSITION  OF  COKE. 

For  all  purposes,  the  less  ash  there  is  in  coke  the  more  valuable  it  is. 
Pure  coke,  if  such  there  be,  consists  entirely  of  carbon.  But  in  practice, 
coke  consists  of  carbon,  sulphur,  and  ash.  The  purest  coke  known  is 
Ramsay's  Garesfield  coke.  The  composition,  as  ascertained  by  analysis 
by  Dr.  Richardson,  is  as  follows : — 

Carbon 97.6    percent. 

Sulphur 0.85       „ 

Ash 1.55       „ 

IOO.OO 

The  composition  of  Durham  coke  varies  within  the  following  limits:  — 

Carbon 85  to  92  per  cent. 

Sulphur  i|  to    2       „ 

Ash 4toi2       „ 

Dr.  Muspratt  gives  the  results  of  nineteen  analyses  of  coke  of  the  usual 
qualities  supplied  to  manufacturers;  they  are  here  given  in  table  No.  145, 
arranged  in  the  order  of  the  percentages  of  carbon,  the  first  in  the  list  being 
Ramsay's  coke  above-mentioned. 

For  the  service  of  locomotives  on  railways,  coke,  besides  being  dense 
and  hard,  should  not  contain  more  than  6  per  cent,  of  ash  to  insure  its 
passing  as  coke  of  good  quality;  with  9  per  cent,  of  ash,  it  is  of  mediocre 
quality;  with  12  per  cent,  of  ash,  it  is  decidedly  bad  coke. 

The  washing  of  coal  destined  for  the  formation  of  coke  has  already  been 
described.  Its  effect  in  removing  the  earthy  matter  and  in  improving  the 
quality  of  the  coke  has  already  been  referred  to.  Suppose  a  coal  which,  in 
its  ordinary  condition,  yields  a  coke  containing  from  10  to  15  per  cent,  of 
ash,  the  effect  of  previously  washing  the  coal  would  be  to  reduce  the 
quantity  of  ash  in  the  coke  to  from  4  to  6  per  cent. 


434 


FUELS. — COKE. 


Table  No.  145. — COMPOSITION  OF  COKES. 

Arranged  from  data  given  by  Dr.  Muspratt. 


No.  OF  COKE. 

COMPOSITION. 

No.  OF  COKE. 

COMPOSITION. 

Carbon. 

Sulphur. 

Ash. 

Carbon. 

Sulphur. 

Ash. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

I 

97.60 

0.85 

i-55 

II 

92.70 

.60 

5-70 

2 

96.42 

0.83 

2-75 

12 

92.44 

-56 

6.00 

3 

95-51 

1.64 

2.85 

J3 

91.49 

.46 

7-05 

4 

94.67 

1.07 

4.26 

14 

9I.l6 

•19 

7-65 

5 

94-31 

0.72 

4-97 

15 

90-53 

.01 

8.46 

6 

94.21 

0.69 

5.10 

z.6 

89.87 

.78 

8.35 

7 

94.08 

0.88 

5-°4 

17 

89.69 

.96 

8-35 

8 

93-54 

0.76 

5-7o 

18 

85.85 

2.08 

12.07 

9 

93-4i 

0.79 

5.80 

19 

84.82 

0.78 

14.40 

10 

93-05 

1.58 

5-37 

Average  of  ) 
19  cokes  j 

93-44 

1.22 

5-34 

MOISTURE  IN  COKE. 

Coke  is  capable  of  absorbing  from  15  to  20  per  cent,  of  its  weight  of 
water.  It  has  been  found  to  absorb  as  much  as  8  per  cent,  of  water  on  its 
way  from  the  ovens  to  its  destination  in  uncovered  waggons.  Directly 
exposed  to  rain,  it  may  absorb  as  much  as  50  per  cent,  of  its  weight  of 
water;  the  most  part  of  which  is  afterwards  quickly  evaporated,  leaving 
from  5  to  10  per  cent,  in  the  coke. 

Loss  OF  COMBUSTIBLE  MATTER  IN  THE  CONVERSION  OF  COAL  INTO  COKE. 

Peclet  quotes  the  experience  with  Alais  coal,  a  bituminous  hard  coal, 
having  the  average  composition  of  the  coals  used  for  the  manufacture  of 
coke  at  Seraing : — 

Per  cent. 

Carbon, 89.27 

Hydrogen...... 4.85 

Oxygen  and  nitrogen, 4.47 

Ash, 1.41 


100.00 


The  yield  of  coke  is  67  per  cent,  and  deducting  the  ash,  1.41  per  cent, 
there  remains  65.59  per  cent,  as  carbon  in  the  coke.  The  total  loss  of 
combustible  matter  in  parts  of  the  coal  is  then — 

Per  cent. 

Carbon, 89.27-65.59  =  23.68 

Hydrogen, 4.85 

The  heat  of  combustion  of  the  lost  carbon  and  hydrogen  is,  by  rule  4, 
page  406, 


COMBUSTION   OF  COKE.  435 


145  (23.68  +  4.28  (4.85  -)  )  =  6o96  units, 

o 

showing  a  loss  of  40  per  cent,  of  the  total  heating  power  of  the  coal,  which 
is  15,606  units. 

AlR   CONSUMED    IN    THE    COMPLETE    COMBUSTION   OF   COKE. 

The  quantity  of  air  chemically  consumed  in  the  complete  combustion  of 
coke  is  found  by  means  of  rule  i,  page  400.  Take,  for  example,  coke 
of  average  composition,  having  93.44  per  cent,  of  carbon,  and  1.22 
per  cent,  of  sulphur.  By  the  formula,  the  volume  of  air  at  62°  chemically 
consumed  is 

1.52  (93.44  +  (1.22  x  0.4)  )  =  i-52  x  93.93  =  142.8  cubic  feet. 

To  find  the  weight  of  this  quantity  of  air,  divide  the  volume  by  13.14, 
and  the  quotient  is  the  weight,  10.87  Ibs. 

Similarly,  the  air  chemically  consumed  by  the  best  and  worst  cokes,  in 
table  No.  145,  is  found,  and  the  quantities  for  the  three  cokes  are  here 
placed  together  for  comparison  :  — 

Carbon.       Sulphur.          Ash.        QUANTITY  OF  AIR  CHEMICALLY  CONSUMED. 
per  cent,      per  cent.       per  cent.  Volume  at  62°.  Weight, 

No.  i  ............  97.60      0.85         1.55       148.9  cubic  feet.      11.33  Ibs. 

No.  19  ..........  84.82       0.78       14.40       128.9         )>  9-81    ?? 

Average  coke...  93.44       1.22         5.34       142.8         „  10.87   „ 

GASEOUS  PRODUCTS  OF  THE  COMBUSTION  OF  COKE. 

The  combustible  elements  of  coke  —  carbon  and  sulphur  —  produce  car- 
bonic acid  and  sulphurous  acid.  These,  together  with  the  nitrogen  of  the 
air  chemically  consumed,  constitute  the  products  of  combustion.  By 
rule  2,  page  401,  the  weight  of  these  products  is  as  follows,  for  coke  of 
average  composition  —  with  93.44  per  cent,  of  carbon  and  1.22  per  cent. 
of  sulphur  :  — 

PRODUCTS.  Pounds.        Per  cent. 

Carbonic  acid,  ..........................  93.44  x  .0366  =3.  42  or    28.4 

Sulphurous  acid,  ........................   1.22  x  .02       =0.24  or      2.0 

Nitrogen,  ............  (93-44  x  .0893)  +  (1.22  x  .0335)^8.38  or    69.6 


Total  weight, 1 2.04  or  100.0 

showing  a  total  weight  of  12  Ibs.  of  gaseous  products  for  one  pound  of 
average  coke — the  same  weight  as  was  found  for  average  coal  (page  428). 

The  volume  of  the  gaseous  products,  at  62°  F.,  is  found  from  the  per- 
centages of  the  combustibles  by  the  data  (e)  (g)  (h\  page  401,  respec- 
tively as  follows : — 

PRODUCTS.  Cubic  feet  at  62°.    Per  cent. 

Carbonic  acid, 93-44  x -3  J 5  =    29-43  or    20-6 

Sulphurous  acid, 1.22  x  .117  —     0.14  or      o.i 

Nitrogen, (93.44  x  1.206) +  (1.22  x    .45)  =  113.25  or    79.3 

Total  volume, 142.82  or  100.0 

Showing  a  total  volume,  as  at  62°,  of  about  143  cubic  feet  of  gaseous 
products  for  one  pound  of  average  coke. 


436 


FUELS. — LIGNITE   AND   ASPHALTE. 


HEATING  POWER  OF  COKE. 

The  heating  power  of  coke  is  calculated  directly  from  the  quantity  of 
constituent  carbon,  if  the  sulphur  be  neglected.  Taking,  for  example, 
the  first  and  the  last  samples  of  coke,  of  which  the  analyses  are  given 
in  table  No.  145,  with  coke  of  average  composition,  the  percentages  of 
constituent  carbon  are  as  follows,  to  which  are  added  the  heating  powers  of 
one  pound  of  the  fuels,  calculated  by  rules  (  4  )  and  (  5  ),  page  406 :— 


COKE. 


Constituent    Total  Heating 
Carbon.  Power. 

per  cent.         units  of  heat. 


No.     i 97-6c 

No.  19 84.82 

Average  coke 93-44 


12,300 


Total  Evaporative 

Power  from 
water  at  212°. 

14.64  pounds. 
12.72   „ 
14.02   „ 


TEMPERATURE  OF  COMBUSTION  OF  COKE. 

The  temperature  of  combustion  of  carbon,  which  is  the  combustible 
matter  of  coke,  was  found,  page  407,  to  be  4877°  F.  when  completely 
burned.  It  may  therefore  be  assumed  that  the  temperature  of  combustion 
of  coke  is  under  5000°  F. 

LIGNITE   AND   ASPHALTE. 

Lignite,  or  as  it  is  occasionally  called,  brown  coal,  though  it  is  often  found 
of  a  black  colour,  belongs  to  a  more  recent  formation — the  tertiary — than 
coal.  It  is  in  fact  an  imperfect  coal.  Brown  lignite  is  sometimes  of  a 
woody  texture,  sometimes  earthy.  Black  lignite  is  either  of  a  woody 
texture,  or  it  is  homogeneous,  with  a  resinous  fracture.  Some  lignites,  more 
fully  developed,  are  of  a  schistose  character,  with  pyrites  in  their  composi- 
tion. The  coke  produced  from  various  lignites  is  either  pulverulent,  like 
that  of  anthracite,  or  it  retains  the  forms  of  the  original  fibres.  Lignite  is 
less  dense  than  coal. 

The  table  No.  146  contains  the  composition  of  lignites  of  various  quali- 
ties, including  the  hygrometric  moisture. 

Table  No.  147  contains  the  results  of  analyses  and  other  particulars  of 
lignites  and  of  asphalte,  according  to  Regnault.  See  also  table  No.  141, 
page  423. 

Table  No.  146. — DENSITY  AND  COMPOSITION  OF  VARIOUS  LIGNITES, 
INCLUDING  HYGROMETRIC  MOISTURE. 


Locality  and  description. 

Specific 
gravity. 

Carbon. 

Hydro- 
gen. 

Oxygen. 

Nitro- 
gen. 

Ash. 

Mois- 
ture. 

Meissner—  red  brown,  woody 
Rheinhardswalde  —  gray   or  "J 
black,  with  abundance  of  > 
resin                                     ) 

1.  12 
I-I3 

p.  cent. 
51.24 

58.78 

p.  cent. 
4.17 

4.04 

p.  cent. 
52.33 

20.80 

p.  cent. 
0.17 

0.15 

p.  cent. 
0.80 

5-94 

p.  cent. 
10.30 

10.28 

Meissner  —  brilliant  black,  ~) 
fracture     fibrous,     lustre  > 
vitreous  J 

1.32 

70.0 

3-19 

17-59 

0.12 

547 

3.63 

Hirschberg-brownish  black,  ) 
in  tree-like  masses     ) 

i-35 

60.30 

4.86 

20.17 

0.12 

3-17 

n-39 

• 

COMPOSITION   OF   LIGNITE   AND   ASPHALTE. 


437 


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Nitroge 


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438  FUELS.— WOOD. 

ASPHALTE. 

Asphalte,  like  lignite,  has  a  large  proportion  of  hydrogen.  It  has  less 
than  9  per  cent,  of  oxygen  and  nitrogen,  and  thus  leaves  8^  per  cent,  of 
free  hydrogen,  and  it  accordingly  yields  a  porous  coke. 

The  average  composition  of  perfect  lignite  and  of  asphalte  may  be  taken 
in  whole  numbers  as  follows : — 

Lignite.  Asphalte. 

Carbon 69  per  cent.  79  per  cent. 

Hydrogen 5  9 

Oxygen  and  nitrogen 20        „  9        „ 

Ash 6  3 

IOO  IOO 

Coke,  by  laboratory  analysis 47        „  9        „ 

The  lignites  are  distinguished  from  coal  by  the  large  proportion  of 
oxygen  in  their  composition — from  13  to  29  per  cent.,  which  goes  far  to 
neutralize  the  hydrogen,  so  that  for  the  first  and  second  lignites  the  free 
hydrogen  is  less  than  3  per  cent.  For  the  third — bituminous  lignite — the 
free  hydrogen  amounts  to  nearly  6  per  cent.,  and  the  varied  effect  of  the 
proportion  of  free  hydrogen  is  visible  on  the  nature  of  the  coke  of  lignite, 
as  was  found  in  the  case  of  the  coke  of  coal.  Thus, 

With  2.64  per  cent,  of  free  hydrogen,  the  coke  is  pulverulent. 
„     1.75  „  „  „  like  wood  charcoal. 

„     5.76  „  „  ,,  raised  and  porous. 

The  small  yield  of  coke  from  asphalte — only  9  per  cent. — though  the 
constituent  carbon  amounts  to  79  per  cent.,  is  evidently  caused  by  the  great 
amount  of  free  hydrogen  volatilizing  a  large  proportion  of  the  carbon. 

TOTAL  HEATING  POWER  OF  LIGNITE  AND  ASPHALTE. 

The  total  heating  power  of  lignite  and  asphalte,  in  units  of  heat,  and  their 
equivalent  evaporative  powers  in  water  from  212°,  under  one  atmosphere, 
are  as  follows : — 


Fuel. 

Heating  power. 

Total  evaporative  power 
in  water  from  212°  per 
pound  of  fuel. 

Perfect  lignite  

units  of  heat. 
11,678 

pounds. 
12   IO 

Imperfect  lignite  
Bituminous  lignite 

...        9,834     ... 
I  A  AA.Q 

IO.I8          
IA  06 

Asphalte  

16  6c  c 

I  7  2A 

It  may  be  observed,  with  reference  to  the  lignites  noted  in  table  No.  141, 
that  the  more  perfectly  converted  lignites  possess  the  greatest  heating 
power.  There  is  a  fine  distinction  between  lignite  passing  to  fossil  wood, 
and  fossil  wood  passing  to  lignite;  their  heating  powers  are  nearly  equal 
to  each  other,  and  both  are  less  than  the  heating  powers  of  the  perfect 
lignites. 


HYGROMETRIC   MOISTURE   IN  WOOD. 


439 


WOOD. 

Wood,  as  a  combustible,  is  divisible  into  two  classes: — ist.  The  hard, 
compact,  and  comparatively  heavy  woods,  as  oak,  beech,  elm,  ash;  2d.  The 
light-coloured,  soft,  and  comparatively  light  woods,  as  pine,  birch,  poplar. 
In  France,  firewood  is  classed  as  fresh  wood  (bois  neuf),  carried  by  land  or 
water  to  its  destination;  raft  wood  (bois  flotte],  floated  to  its  destination; 
and  peeled  wood  (bois  pelard),  or  oak  stripped  of  its  bark. 

According  to  M.  Leplay,  green  wood,  when  cut  down,  contains  about 
45  per  cent,  of  its  weight  of  moisture.  In  the  forests  of  Central  Europe, 
wood  cut  down  in  winter  holds,  at  the  end  of  the  following  summer,  more 
than  40  per  cent,  of  water.  Wood  kept  for  several  years  in  a  dry  place 
retains  from  15  to  20  per  cent,  of  water. 

Wood  which  has  been  thoroughly  desiccated,  will,  when  exposed  to  air 
under  ordinary  circumstances,  absorb  5  per  cent,  of  water  in  the  first  three 
days;  and  will  continue  to  absorb  it,  until  it  reaches  from  14  to  16  per 
cent.,  as  a  normal  standard.  The  amount  fluctuates  above  and  below  this 
standard,  according  to  the  state  of  the  atmosphere. 

M.  Violette  found  that,  by  exposing  green  wood  to  a  temperature  of 
212°  F.,  it  lost  45  per  cent,  of  its  weight,  which  accords  with  the  observa- 
tion of  M.  Leplay.  He  further  found  that  by  exposing  small  prisms  of  wood 
half  an  inch  square  and  eight  inches  long,  cut  out  of  billets  that  had  been 
stored  for  two  years,  to  the  action  of  superheated  steam,  for  two  hours,  they 
lost  from  1 5  to  45  per  cent,  of  their  weight,  according  to  the  temperature  of 
the  steam,  which  varied  from  257°  F.  to  437°  F.  (125°  C.  to  225°  C).  The 
following  are  the  particulars  for  four  woods : — 


Loss  OF  WEIGHT. 

TEMPERATURE  OF 

DESICCATION. 

Oak. 

Ash. 

Elm. 

Walnut. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

125°  C.  or  257°  F. 

15.26 

14.78 

I5.32 

!5.55 

150         „    302 

17.93 

16.19 

I7.O2 

17.43 

175         »   347 

32.13 

21.22 

36.94 

21.79 

200         „   392 

35.80 

27-51 

33.38  (?) 

41-77  (?) 

225         „   437 

44.31 

33.38 

40.56 

36-56 

The  hardest  wood,  oak,  lost,  according  to  this  statement,  more  weight 
than  the  softer  woods.  The  observations  queried  appear  to  have  been 
errors  of  observation.  At  a  temperature  of  200°  C.,  or  392°  F.,  wood 
becomes  visibly  altered,  and  the  alteration,  or  decomposition,  may  likely 
commence  at  a  lower  temperature;  and  it  may  be  that  the  losses  of  weight 
are  not  entirely  due  to  a  reduction  of  hygrometric  water.  A  higher  tempera- 
ture than  2 1 2°  F.  appears  to  be  necessary  to  disengage  all  the  water. 

Ordinary  firewood  contains,  by  analysis,  from  27  to  80  per  cent,  of 
hygrometric  moisture. 


440 


FUELS. — WOOD. 
COMPOSITION  OF  WOOD. 


M.  Chevandier,  in  1844,  published  the  results  of  analysis  of  five  woods, — 
beech,  oak,  birch,  poplar,  and  willow.  The  woods  were  reduced  to  powder, 
and  desiccated  at  a  temperature  of  140°  C.,  or  284°  R,  before  being  sub- 
mitted to  analysis.  The  results  of  analysis  are  given  in  table  No.  148. 

Table  No.  148. — COMPOSITION  OF  WOODS. 

(From  Analysis  by  M.   Eugene  Chevandier,   1844.) 


WOODS. 

COMPOSITION. 

Carbon. 

Hydrogen. 

Oxygen. 

Nitrogen. 

Ash. 

Beech 

per  cent. 

49-36 
49.64 

50.20 

49-37 
49.96 

iposing  tl 

S0-1? 
49.96 
51.24 

49-5° 
5i-54 

per  cent. 

6.01 

5-92 
6.  20 

6.21 

5-96 

le  Fagots 

6.12 

6.  02 

6.22 

6.09 
6.26 

per  cent. 
42.69 
4I.l6 
41.62 
41.60 
39-56 

40.38 
4I.IO 
40.17 

40.43 
36.21 

per  cent. 
0.91 
1.29 

I-I5 

0.96 
0.96 

•°5 
.OO 

•05 
.OO 
.41 

per  cent. 
I.OO 
1.97 

0.81 
1.86 
3-37 

1.77 
1.90 
1.32 
2.98 
4-57 

Oak 

Birch         

Poplar  

Willow  

Twigs  and  Branches  con 
Beech 

Oak 

Birch  . 

Poplar.            .        .        .... 

Willow  

Average  of  woods  

49.70 
50.46 

6.06 

6.14 

41.30 
39-65 

•05 
.11 

i.  So 

2.50 

Average  of  fagots  

There  is  a  remarkable  nearness  to  identity  in  the  composition  of  these 
woods,  and  also  in  the  composition  of  the  trunk  and  the  branches. 

The  results  show  that  the  composition  of  woods  is  practically  as  follows : — 

Carbon  50  per  cent. 

Hydrogen 6       „ 

Oxygen 41       „ 

Nitrogen i       „ 

Ash...  2 


100 


Showing  that  there  is  only  56  per  cent,  of  combustible  matter,  that  there 
is  a  large  quantity  of  oxygen,  nearly  sufficient  to  neutralize  the  whole  of  the 
hydrogen,  and  that  there  is  only  2  per  cent,  of  ash. 

The  above-mentioned  analysis  is  corroborated  by  the  analysis  of  M. 
Violette,  who  ascertained  the  composition  of  different  parts  of  the  same 
tree,  desiccated  at  a  temperature  of  80°  C.  or  176°  F.,  with  the  results 
given  in  the  following  table  No.  149: — 


COMPOSITION   OF  WOOD. 


441 


Table  No.   149. — COMPOSITION    OF   THE   VARIOUS    MEMBERS 
OF   ONE   TREE. 

(From  Analysis  by  M.  Violette.) 


MEMBERS  OF  THE  TREE. 

COMPOSITION. 

Carbon. 

Hydrogen. 

Oxygen  and 
Nitrogen. 

Ash. 

Leaves      

per  cent. 
45.01 

52.50 
48.36 
48.85 
49.90 
46.87 
48.00 
46.27 
48.92 
49.08 
49-32 

50-37 
47-39 

45.06 

per  cent. 
6.97 

7.31 
6.60 

6-34 

6.61 

5-57 
6.47 

5-93 
6.46 

6.O2 
6.29 
6.07 

6.26 

5-°4 

per  cent. 
40.91 
36.74 
44-73 
41.12 

43-36 
44.66 

45-I7 
44-75 
44.32 
48.76 
44.11 
41.92 
46.13 
43-5° 

per  cent. 
7.12 

3-45 
0.30 

3-68 
0.13 
2.90 

0-35 
2.66 
0.30 

i.i3 

0.23 
1.64 

O.22 

5-oi 

0111         i            f  bark  .  . 

Small  branches....  |wood 

,     ,  .                ,       f  bark 

Medium  branches  j  ^  

f  bark  . 

Large  branches.  ..|wood  ; 

rr,      i                       (  bark  .  . 

Trunk                      < 

"  {  wood  

fbark.. 

Large  root..        .   <          i 

(  wood..     . 

Medium  root          i  bark;  

"  \  wood  

Small  roots  with  bark  

Averages  :  — 
Leaves  

45.01 
49.00 
48.66 
45.06 

6.97 

6.21 

6.45 

5-°4 

40.91 
43.00 
44.64 
43-50 

7.12 
2.58 
0.25 

5-oi 

Bark  . 

Wood..  .. 

Small  roots  with  bark 

Here  it  appears  that  the  composition  of  the  wood  is  about  the  same 
throughout  the  tree,  and  that  of  the  bark  also;  that  the  wood  and  the  bark 
have  about  the  same  proportion  of  carbon,  49  per  cent.,  but  that  the  bark 
has  more  ash  than  the  wood.  The  leaves  and  the  small  roots  have 
less  carbon  than  the  wood, — only  45  per  cent.;  and  more  ash, — 7  and  5 
per  cent. 

The  leaves  when  dried  at  100°  C.  lost  60  per  cent,  of  water,  and  the 
branches  45  per  cent. 

COMPOSITION  OF  ORDINARY  FIREWOOD. 


The  respective  percentages  of  the  constituent  elements  of  stacked  wood 
in  its  ordinary  state  are,  of  course,  reduced  in  amount  when  the  water  is 
taken  into  account.  Thus,  in  the  following  analysis  of  ordinary  firewood, 
containing  25  per  cent,  of  moisture,  the  carbon  constitutes  only  37.5  per 
cent,  of  the  fuel : — 


44: 


FUELS.— WOOD. 


per  cent. 

Hygrometric  water 25 

Carbon 37.5 

Hydrogen 4.5 

Oxygen 30.75 

Nitrogen 0.75 

Ash 1.5 


IOO.OO 


WEIGHT  AND  BULK  OF  WOOD. 


The  density  of  a  large  number  of  woods  has  already  been  given  in  table 
No.  65,  page  208.  These  values  can,  in  most  instances,  only  be  given 
as  approximate,  for  the  density  changes  with  the  hygrometric  condition  of 
the  wood.  The  specific  gravity  varies  from  1.35,  that  of  pomegranate, 
to  0.24,  that  of  cork  wood. 

The  density  of  the  ligneous  fibre  of  which  wood  is  formed,  has  been 
ascertained  by  M.  Violette,  from  a  great  number  of  observations.  The 
samples  of  wood  were  reduced  to  powder  in  a  mortar,  and  dried  at  a 
temperature  of  100°  C.  He  found  that  the  fibre  of  all  woods  had  the  same 
density,  and  that  its  specific  gravity  was  1.50. 

It  is  said  that  the  quantity  of  intersticial  space  in  a  closely-packed  pile  of 

Table  No.  150. — OF  THE  WEIGHT  AND  BULK  OF  WOODS  IN  FRANCE. 


Woods  in  ordinary  state  of  dryness. 

Weight  of  one 
cubic  foot,  heaped. 

Bulk  of 
one  ton,  heaped. 

Firewood  

pounds. 
21.  Q  tO  23  4 

cubic  feet. 
IO2  3  tO  Q^  7 

Wood  for  charring,  hard  and  soft,  cut  up 
Do.         do.       hard  wood,  cut  up... 
Oak,  cut  up 

18.8 
23-4 

22  A.  tO  23  7 

IIQ.I 

95-7 

TOO  to  04.  4 

Do.    branches. 

*.<}  \.\j  ^^.  i 

17  -2 

132  A 

Do.    small  branches  

A  /  'O 

IQ.8 

1  13.  1 

Beech,  cut  up  

237 

Q4  A 

Do.     branches 

*3*  / 

IQ  O 

V4-H- 
Il8  0 

Do.     small  branches 

j-y.^ 

IQ  6 

1  14.  2 

Yoke-elm,  cut  up  

j.y.v-» 
23    I 

07  o 

Do.       branches 

186 

I2O.3 

Do        small  branches 

IQ.  C 

1  14.6 

Birch,  cut  up  

21.  1 

106.  i 

Do.    branches  f  

16.8 

133.3 

Fir  

16.0 

1  40.  1 

Alder,  cut  up. 

18  3 

122.4 

Willow,  cut  up 

18.0 

124.7 

Aspen,  cut  up              ...    . 

17.  0 

I3I.4 

Pine  in  the  United  States  

21.  0 

106.0 

Averages 

20.  o 

1  14.0 

COMBUSTION   OF   WOOD.  443 

wood,  consisting  of  uncloven  stems,  is  30  per  cent,  of  the  gross  bulk ;  for 
cloven  stems,  the  intersticial  space  amounts  to  from  40  to  50  per  cent. 

A  cord  of  pine  wood, — that  is,  of  pine  wood  cut  up  and  piled, — in  the 
United  States,  measures  4  feet  by  4  feet  by  8  feet,  and  has  a  volume  of 
128  cubic  feet.  Its  weight,  in  ordinary  condition,  averages  2700  Ibs.;  or 
21  Ibs.  per  cubic  foot. 

A  "corde"  of  wood,  in  France,  has  a  volume  of  4  cubic  metres,  or  141 
cubic  feet. 

Firewood  is  measured,  in  France,  by  the  vote,  of  which  the  volume  is 
2  cubic  metres,  or  2  s feres.  As  the  length  of  the  billets  is  1.14  metres,  or 
3.74  feet,  the  half-zwV,  or  sfere,  measures  1.14  metres  x  0.88  metre  x  i  metre, 
equal  to  i  cubic  metre,  or  35.3  cubic  feet;  and  the  vote  is  equal  to  70.6 
cubic  feet  in  bulk.  The  weight  of  the  vote  of  firewood,  in  Paris,  is  from 
700  to  750  kilogrammes,  or  from  1544  to  1653  Ibs.,  averaging  1600  Ibs. 

The  vote  of  wood  for  making  charcoal,  in  the  forests  of  the  Ardennes, 
weighs  1324  Ibs. ;  it  consists  of  one-fourth  oak  and  beech,  one-fourth  poplar 
and  willow,  and  one-half  elm.  The  hard  wood  for  charring,  of  the  forests 
of  the  Meuse,  weighs  1653  Ibs.  per  vote. 

The  above  and  other  particulars  given  by  M.  Chevandier  are  collected 
and  arranged  in  table  No.  150,  showing  the  weight  and  bulk  of  ordinarily 
dry  wood. 

QUANTITY  OF  AIR  CHEMICALLY  CONSUMED  IN  THE  COMPLETE 
COMBUSTION  OF  WOOD. 

In  terms  of  the  average  percentages  of  carbon,  hydrogen,  and  oxygen,  in 
wood,  page  440,  the  quantity  of  air  consumed  is,  by  the  rule  i,  page  400, 

1.52  (50  +  3  (6  -— )  )=  1.52  x  52.625  =  80  cubic  feet, 
8 

or  80  4- 13. 14  =  6.09  Ibs. 

GASEOUS  PRODUCTS  OF  THE  COMBUSTION  OF  WOOD. 

For  one  pound  of  dry  wood  the  products  are,  by  the  expressions  ( a ), 
(b\(c\  page  401, 

PRODUCTS.  Pounds.       Per  cent. 

Carbonic  acid, 50  x  .0366  =  1.83  or    21.7 

Steam, 6  x  .09       =0.54  or      6.4 

Nitrogen,....  (50  x  .0893) +  (6  x  .268)  + (i  x.oi)      =6.08  or    71.9 

Total  weight  of  products, 8.45       100.0 

says  8*4  Ibs.  weight  of  products. 

The  volumes  of  the  products  at  62°  are,  by  the  expressions  (e),  (/), 
(/&),  page  401, 

PRODUCTS.  Cubic  feet.       Per  cent. 

Carbonic  acid, 50  x    .315  =  15.75  or    14.4 

Steam, 6x1.9      =11.40  or    10.4 

Nitrogen, (50  x  1.206) +  (6  x  3.618)  =  8a'.bi  or     75.2 

Total  volume  of  products  for  i  Ib.  of  wood,  109.16       100.0 
being  about  13  cubic  feet  per  Ib.  weight  of  gaseous  products. 


444  FUELS.  —  WOOD-CHARCOAL. 

TOTAL  HEAT  OF  COMBUSTION  OF  WOOD. 

The  total  heat  of  combustion  of  dry  wood  is,  by  rule  4,  page  406. 
145  (50  +  4.28  (6  -  II)  )  -  145  x  53.745  =  7792  units, 

which  is  a  little  more  than  half,  or  54^  per  cent.,  of  that  of  coal,  and  is 
equivalent,  by  rule  8,  page  406,  to  the  evaporation  of  0.15  x  53.745  =  8.07 
Ibs.  of  water  at  212°. 

When  the  wood  holds  25  per  cent,  of  water,  there  is  only  75  per  cent,  or 
three-quarter  pound  of  wood-substance  in  one  pound;  and  the  total  heat 
of  combustion  is  75  per  cent,  of  7792  units,  or  5844  units,  which  is  only 
41  per  cent,  of  that  of  average  coal.  Similarly,  the  equivalent  evaporative 
power  is  reduced  to  6.05  Ibs.  of  water  at  212°,  of  which  the  equivalent 
of  a  quarter  of  a  pound  is  appropriated  to  the  vaporizing  of  the  contained 
moisture. 

TEMPERATURE  OF  THE  COMBUSTION  OF  WOOD. 

It  is  found,  in  the  manner  already  shown,  page  407,  that  2.136  units  raise 
the  temperature  of  the  products  i°  F.  The  total  heat  of  combustion, 
7792  units  -^2.136  =  3648°  F.;  and  3648  +  62  =  3710°  F.,  is  the  temperature 
of  combustion. 

When  the  wood  holds  25  per  cent,  of  water,  the  weight  of  the  direct 
products  is  75  per  cent,  of  8.45  Ibs.,  or  6.34  Ibs.;  and  the  total  heat  of 
combustion  is  5844  units,  of  which  1116°  (total  heat  of  steam)  ^4  =  279 
units,  are  appropriated  to  evaporate  a  quarter  of  a  pound  of  water  from  62°, 
leaving  5844  -  279  =  5565  units  of  heat  available  for  raising  the  temperature 
of  the  gases.  To  raise  the  direct  products  one  degree  of  temperature, 
there  are  required  — 

Units. 

2.136  x  24  =  .............................................    1.  602 

The    evaporated    water,    as    gaseous    steam,  ) 


Total  for  i°  F  .......................    1.721 

Then,  5565  -^1.721  =3234°  F.,  the  temperature  of  combustion.     It  is  only 
88.6  per  cent,  of  the  temperature  for  perfectly  dry  wood. 

In  order  to  obtain  the  maximum  heating  power  from  wood  as  fuel,  it  is 
the  practice,  in  some  works  on  the  Continent,  —  as  glass  works  and  porcelain 
works,  —  where  intensity  of  heat  is  required,  to  dry  the  wood-fuel  thoroughly, 
even  using  stoves  for  the  purpose,  before  using  it. 

WOOD-CHARCOAL. 

When  wood  is  exposed  to  heat  it  is  at  first  desiccated  and  afterwards 
carbonized.  Under  temperatures  up  to  300°  F.,  the  wood  is  simply  desic- 
cated. Under  temperatures  over  300°  the  gaseous  elements  are  driven  off, 
until  at  650°  the  wood  yields  a  charcoal  which  is  black,  solid,  and  brittle. 
The  gases  are  not  completely  driven  off  except  under  much  higher  temper- 
atures. 

Wood  charcoal,  completely  converted,  is  black,  solid,  brittle,  and  friable; 
it  preserves  the  form  and  structure  of  the  wood  from  which  it  is  made. 
Though  easily  pulverized,  it  makes  a  very  hard  powder. 


THE   CARBONIZATION    OF   WOOD. 


445 


The  following  are  the  results  of  the  experiments  of  M.  Violette  on  the 
carbonization  of  wood.  He  experimented  on  black  elder-wood,  formed 
into  prisms  2.4  inches  long  and  0.4  inch  in  diameter,  made  up  in  sets 
of  twenty  prisms.  Each  set  was  dried  separately  at  a  temperature  of  300°, 
in  a  current  "of  superheated  steam,  to  which  it  was  subjected  during 
two  hours.  The  carbonization  was  effected  by  the  same  medium,  at  least 
up  to  662°  F.;  and  in  crucibles  placed  in  a  furnace,  at  higher  temperatures. 
The  temperatures  arrived  at  when  in  the  furnace  were  checked  by  the 
melting  of  small  pieces  of  various  metals  placed  in  the  crucible  with  the 
samples. 

The  table  No.  151  gives  the  weight  of  the  products  obtained  as  the  result 
of  carbonization  at  the  given  temperatures : — 

Table  No.    151. — YIELD    OF    CHARCOAL    FROM    BLACK    ELDER   WOOD, 
CARBONIZED  AT  DIFFERENT  TEMPERATURES. 

(By  M.   Violette.) 


Temperature  of 
Carbonization. 

Weight  of 
gross  product 
from 
dry  wood. 

Observations. 

Cent. 

Fahr. 

per  cent. 

150° 

302° 

IOO 

\ 

160 

320 

98 

170 

338 

94-55 

1  80 

356 

88.59 

190 

374 

81.99 

These  products  are  only  wood  more  and 

200 

392 

77.10 

)       more  altered,  but  they  are  not  charcoal. 

2IO 

410 

73-14 

They  are  called,  in  France,  brulots. 

22O 

428 

67.50 

230 

446 

55-37 

240 

464 

50.79 

250 

482 

49.67 

t 

260 

500 

40.23 

270 

518 

37-14 

280 

536 

36.16 

(  Brown  charcoal  (charbon  roux).   Commences 
(      to  be  friable. 

290 

554 

34-09 

300 

572 

33.61 

3IO 

590 

32.87 

320 

608 

32.23 

330 

626 

3z-77 

340 

644 

3L53 

Very  black  charcoal. 

35° 

662 

29.26 

432 

810 

18.87 

Melting  point  of  antimony.   Charcoal  very  hard. 

1023 

1873 

18.75 

Do.            silver.              Do.          do. 

1  100 

2OI2 

18.40 

Do.            copper.            Do.          do. 

1250 

2282 

17.94 

Do.           gold.                Do.          do. 

1300 

2372 

17.16 

Do.            steel.                Do.          do. 

1500 

2732 

17.31 

Do.            iron.                 Do.          do. 

? 

15.00 

Do.           platinum.         Do.          do. 

446 


FUELS. — WOOD-CHARCOAL. 


From  this  table,  it  appears  that  charcoal,  properly  so-called,  is  not 
formed  until  a  temperature  of  536°  F.  is  reached.  From  536°  to  644°  F., 
brown  charcoal  (charbon  ronx),  from  36  to  31^  per  cent.,  is  formed. 
Beyond  644°  F.  the  charcoal  is  black,  and  the  yield  diminishes  with  the 
increase  of  temperature,  until,  at  the  unknown  temperature  of  melting 
platinum,  it  becomes  just  15  per  cent,  of  the  weight  of  the  dried  wood 
from  which  it  is  produced. 

Brown  charcoal  is  flexible,  unctuous,  and  soft  to  the  touch;  black  char- 
coal is  rigid,  brittle,  and  harsh  to  the  touch. 

COMPOSITION  OF  CHARCOAL. 

The  composition  of  these  charcoals  varies  with  the  temperatures  at  which 
they  are  produced,  as  may  be  seen  by  the  annexed  table,  No.  152,  showing 
the  results  of  analysis  of  some  of  the  charcoals  obtained : — 

Table  No.  152. — COMPOSITION  OF  CHARCOAL  PRODUCED  AT  VARIOUS 

TEMPERATURES. 

(By  M.  Violette.) 


Composition  of  the  Solid  Product. 

1 

Carbon 

Temperature 

for  a  given 

of  Carbonization. 

Oxygen, 

weight  of 

Carbon. 

Hydrogen. 

Nitrogen, 

Ash. 

wood. 

and  Loss. 

Centigrade. 

Fahrenheit. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

15°° 

302° 

47-51 

6.12 

46.29 

0.08 

47-51 

2OO 

392 

51.82 

3-99 

43-98 

0.23 

39.88 

250 

482 

65-59 

4.81 

28.97 

0.63 

32.98 

300 

572 

73-24 

4-25 

21.96 

0-57 

24.61 

35° 

662 

76.64 

4.14 

18.44 

0.61 

22.42 

432 

810 

81.64 

4.96 

I5-24 

1.61 

15.40 

1023 

1873 

81.97 

2.30 

14.15 

i.  60 

!5-3° 

1  1  00 

2012 

83.29 

1.70 

*3'79 

1.22 

!5-32 

1250 

2282                88.14 

1.42 

9.26 

1.  2O 

15.80 

1300 

2372                90.81 

1.58 

6.49 

I.I5 

15-85 

1500 

2732           94-57 

0.74 

3.84 

0.66 

16.36 

Melting  point 
of  platinum 

}- 

96.52 

0.62 

0.94 

1-95 

14.47 

From  this  table,  it  is  evident  how  materially  a  higher  temperature  operates 
in  driving  off  the  injurious  gases,  oxygen  and  nitrogen — injurious,  that  is, 
in  reducing  the  heating  power;  though  the  useful  gas,  hydrogen,  is  likewise 
driven  off — which  is  a  loss  for  heating  power.  The  carbon  and  the  ash, — 
the  solid  constituents, — on  the  contrary,  are  proportionally  increased.  At  the 
same  time,  there  is  an  absolute  loss  of  carbon,  though  less  in  degree  than  the 
diminution  of  the  gases,  as  the  temperature  rises.  The  rate  of  diminution 
of  the  absolute  quantity  of  carbon  for  a  given  weight  of  wood,  is  arrived  at 


COMPOSITION   AND  YIELD  OF  WOOD-CHARCOAL.  447 

by  multiplying  the  percentages  of  constituent  carbon  in  the  second  table 
(No.  152)  by  the  relative  percentages  of  gross  products  in  the  first  table 
(No.  151),  for  a  given  temperature,  as  given  in  the  last  column  of  the 
second  table.  Here,  the  absolute  quantity  of  carbon,  which  is  47.5  per  cent, 
in  the  dry  wood  in  its  natural  state,  at  150°  C,  is  reduced  to  15.4  per  cent., 
or  one-third,  at  432°  C.  or  810°  F.;  and  beyond  this  temperature,  however 
great  the  heat  may  be,  there  is  practically  no  further  diminution  of  the  carbon; 
that  is  to  say,  that  no  more  carbon  is  driven  off  by  raising  the  temperature, 
the  gaseous  elements  alone  being  driven  off. 

It  is  remarkable  that  the  proportion  of  ash  found  by  M.  Violette  is 
only  from  i  to  2  per  cent.,  whilst  the  ash  of  the  original  wood  averages 
i^  per  cent.;  for,  it  is  naturally  supposed  that  the  whole  of  the  ash  should 
be  concentrated  in  the  charcoal,  and  should  average  7^  per  cent,  suppos- 
ing that  the  yield  of  charcoal  is  one-fifth  the  weight  of  wood.  The  ash- 
element  must  have  been  withdrawn  with  the  gases  that  escaped  during  the 
process  of  carbonization.  It  is  found  that,  practically,  the  ash  of  the  char- 
coal of  commerce  amounts  to  from  7  to  8  per  cent. 

According  to  M.  Sauvage,  the  charcoal  manufactured  in  the  forests  is 
composed  as  follows : — 

Carbon, 79  per  cent. 

Hydrogen,  free 2        „ 

Hydrogen,  oxygen,  and  nitrogen, 1 1        „ 

Ash, 8        „ 

100 

YIELD  OF  CHARCOAL  BY  LABORATORY  ANALYSIS. 

M.  Violette  ascertained  that  the  greater  or  less  rapidity  with  which 
carbonization  is  effected  influence  materially  the  quantity  of  the  yield. 
He  obtained  by  slow  carbonization  twice  as  much  charcoal  as  by  rapid 
carbonization,  at  the  same  temperature;  but  he  does  not  give  the  details  of 
the  experiments  by  which  he  arrived  at  this  conclusion. 

He  further  found  that  when  wood  was  carbonized  in  close  vessels 
hermetically  sealed,  the  yield  was  decidedly  greater  than  in  open  vessels, 
thus : — 

Temperature  of  Carbonization.  In  Open  Vessels.  In  Closed  Vessels. 

yield.  yield. 

1 60°  C.  or  320°  F 97  per  cent 97.4  per  cent 

340°  C.  or  644°  F 29        „          79.1 

The  charcoal  obtained  at  180°  C.,  in  a  close  vessel,  was  brown,  friable, 
and  similar  to  that  produced  at  280°  C.  in  an  open  vessel  (table  No.  151); 
though  differently  constituted,  as  the  former  held  a  greater  proportion  of 
gaseous  matter,  and  also  more  ash  than  the  latter. 

Finally,  M.  Violette  ascertained  by  experiments,  similarly  conducted  in 
open  vessels  with  superheated  steam,  the  quantity  of  charcoal  for  various 
woods  and  other  ligneous  substances,  dried,  in  the  first  place,  at  150°  C.  or 
302°  F.,  and  then  exposed  to  a  temperature  of  300°  C.  or  572°  F.  These 
are  arranged  in  table  No.  153,  in  the  order  of  the  quantities  yielded:— 


448 


FUELS. — WOOD-CHARCOAL. 


Table  No.   153. — YIELD   OF  CHARCOAL   FROM  VARIOUS  WOODS,   DRIED 
AT  150°  C.,  OR  302°  R,  AND  CARBONIZED  AT  300°  C,  OR  572°  F. 

(By  M.  Violette. ) 


WOOD. 

Weight 
of  Charcoal. 

WOOD. 

Weight 
of  Charcoal. 

Cork  

per  cent. 
62.80 

Apple  tree  

per  cent. 

M.6o 

Decayed  Willow 

C2.I  7 

Elm 

-24.  en 

Wheat  straw 

46.00 

Hornbeam  . 

34.44 

Oak 

46.00 

Alder  

34.4O 

Yew  .              

46.06 

Birch  

34.17 

Beech  

44.  2  c; 

Plum  tree  

34.O6 

Pine  

41.48 

Maple  

33.  7^ 

Poplar  (leaves) 

4O.QCJ 

Willow 

33  74 

Do      (roots) 

4O.QO 

Black  elder 

33  6l 

Wild  pine 

4O.7^ 

Ash  . 

33.28 

Larch     .... 

4O.3I 

Pear  tree. 

31.88 

Chestnut  tree  . 

36.06 

Lime  tree  

31.8s 

Cherry  tree  

•2C.C-2 

Poplar  (stem)  

31.12 

Aspen  

34.87 

Sweet  chestnut  tree... 

30.86 

It  appears  from  this  table  that  cork,  the  lightest  of  woods,  yields  the 
largest  percentage  of  charcoal,  about  63  per  cent.;  and  that  poplar  and 
sweet  chestnut  tree  yield  the  lowest,  about  31  per  cent.  But  there  does 
not  appear  to  be  any  definite  relation  between  the  density  of  the  wood  and 
the  quantity  of  the  yield. 

CARBONIZATION  OF  WOOD  IN  STACKS — YIELD  OF  CHARCOAL. 

Wood  has  been  carbonized,  from  the  remotest  times,  in  heaps  on  the 
ground;  and  this  process  is  still  generally  followed  on  the  Continent.  The 
stack  or  pile  is  covered  with  a  mixture  of  earth  and  powdered  charcoal,  or 
with  turf.  A  few  openings  are  left  in  the  covering  to  admit  air  to  the 
interior,  as  well  as  a  larger  opening  at  the  summit.  When  the  stack  is  lit 
it  burns  rapidly,  and  so  soon  as  flame  appears  at  the  chimney  it  is  partially 
damped  down  by  a  turf.  The  progress  of  carbonization  is  indicated  by  the 
colour  of  the  smoke,  and  when,  finally,  the  mass  becomes  incandescent,  it 
is  covered  up  with  earth  and  allowed  to  cool.  By  this  process,  the  charcoal 
obtained  usually  amounts  in  weight  to  from  17  to  20  per  cent,  of  the  wood, 
and  to  more  than  this  in  the  larger  heaps.  From  25  to  30  per  cent.,  in 
volume  is  obtained  in  the  small  heaps,  and  from  30  to  34  per  cent,  in  the 
larger  heaps.  The  charring  requires  from  sixty  to  eighty  hours  to  produce 
a  good  quality  of  charcoal. 

In  the  departments  of  the  Ardennes  and  the  Meuse,  in  France,  according 
to  M.  Sauvage,  the  following  are  the  particulars  of  the  yield  of  charcoal 
from  wood.  In  the  case  of  the  Ardennes,  the  wood  prepared  for  carbon- 
ization is  a  mixture  of  one-fourth  oak  and  beech,  one-fourth  poplar  and 


MANUFACTURE  OF  WOOD-CHARCOAL.  449 

willow,  and  one-half  elm.  In  the  example  from  the  Meuse,  hard  wood  is 
used.  The  billets  are  about  30  inches  in  length;  they  are  piled  on  end,  in 
three  tiers.  The  stack  contains  from  60  to  90  cubic  metres,  or  from  80  to 
120  cubic  yards:  — 

Ardennes.  Meuse. 

Mixed  wood.  Hard  wood. 

Weight  of  a  cubic  metre  of  wood,.  .  .  662  Ibs.  827  Ibs. 

'3»  to  145  lbs-  '  ?6    " 


YibdvotL')ubic..me.tr.e.of..wood:  }  io^  to  "*  cub-  ft-  <*  to  '4  **.  ft- 

Percentage  of  yield,  in  weight,  .......    20  to  22  per  cent.  21  per  cent. 

Weight  of  a  cubic  metre  of  char-  )  „  ., 

coal  (heaped),  .....................  }  44°  lbs'  53°  lbs. 

It  is  obvious,  from  the  small  percentages  of  yield,  averaging  21  per  cent. 
for  the  mixed  woods  and  the  hard  wood,  that  much  of  the  substance  of 
the  wood  is  lost,  which  would  by  a  better  system  of  carbonization  be 
yielded  as  charcoal.  According  to  the  table  No.  153,  the  maximum  yield 
obtainable  from  the  mixed  woods  is  38  per  cent.;  and  from  the  hard  woods 
upwards  of  40  per  cent. 

MANUFACTURE  OF  BROWN  CHARCOAL. 

The  best  method  of  making  brown  charcoal  consists  in  heating  the  wood 
to  be  charred  in  a  close  vessel,  by  means  of  superheated  steam  introduced 
into  the  vessel.  The  required  temperature  is  thus  readily  obtained,  and  a 
homogeneous  product  is  yielded.  This  process  was  introduced  by  Messrs. 
Thomas  &  Laurens,  and  is  successfully  employed  in  France  and  Belgium 
in  the  production  of  brown  charcoal  for  the  manufacture  of  gunpowder, 
principally  for  fowling-pieces. 

DISTILLATION  OF  WOOD. 

The  distillation  of  wood  in  close  vessels  affords  evidence  of  the  increased 
yield  of  charcoal  obtainable  by  more  careful  treatment  than  in  the  open- 
air  stack.  In  France,  the  wood  is  distilled  in  large  iron  cylinders  or  retorts 
capable  of  holding  about  180  cubic  feet  of  wood,  as  piled;  and  the  opera- 
tion is  completed  in  from  seven  to  eight  hours.  By  this  process,  28  per 
cent,  of  charcoal  is  obtained,  with  the  products  of  distillation  in  addition. 
But  12*4  per  cent,  of  wood  is  consumed  as  fuel,  making  a  total  of  112^4 
parts  of  wood  for  a  yield  of  28  parts  of  charcoal;  which  reduces  the  avail- 
able yield  to  25  per  cent,  of  the  whole  quantity  of  wood  consumed,  as 
against  2  1  per  cent,  in  the  open-air  stacks  of  hard  wood.  There  is  a  gain, 
in  addition,  in  reduced  cost  of  labour,  and  in  the  value  of  the  yield  of 
pyroligneous  acid.  The  gases  are  directed  into  the  furnace  to  aid  as  fuel 
in  heating  the  retorts. 

CHARBON  DE  PARIS  (ARTIFICIAL  FUEL). 

Charbon  de  Paris,  or  Paris  charcoal,  is  a  mixture  of  two  parts  of  powdered 
charcoal  with  one  part  of  gas-tar,  formed  by  powerful  compression  into 


450 


FUELS.— WOOD-CHARCOAL. 


round  pieces  4  inches  long  and  1%  mcn  in  diameter,  and  submitted  to 
a  high  temperature.  It  takes  fire  easily,  and  burns  slowly  until  it  is  entirely 
consumed,  without  making  flame  or  smoke;  it  makes  from  20  to  22  per 
cent,  of  ash. 

WEIGHT  AND  BULK  OF  WOOD-CHARCOAL. 

It  does  not  appear  that  the  density  of  wood-charcoal,  as  manufactured, 
has  been  accurately  determined.  M.  Violette  determined  the  density  of 
the  matter  of  the  charcoal  of  black  alder,  reduced  to  impalpable  powder, 
so  as  to  extinguish  the  pores.  It  varied  according  to  the  temperature  of 
carbonization,  as  shown  in  table  No.  154: — 

Table  No.  154. — ABSOLUTE  DENSITY  OF  THE  CHARCOAL  OF  BLACK  ALDER, 
DRIED  AT  212°  R,  AS  POWDER. 

(By  M.  Violette.) 


Temperature  of  Carbonization. 

Specific 
gravity. 

Temperature  of  Carbonization. 

Specific 
gravity. 

Centigrade. 

Fahrenheit. 

Centigrade. 

Fahrenheit. 

150° 

302° 

1.507 

330° 

626° 

1.428 

170 

338 

1.490 

350 

662 

1.500 

IQO 

374 

1.470 

432 

810 

1.709 

2IO 

410 

1-457 

IO23 

1873 

1.841 

230 

446 

.416 

1250 

2282 

1.862 

250 

482 

.413 

I5OO 

2732 

1.869 

270 

5i8 

.402 

(  Fusing  point  } 

290 

554 

.406 

? 

1   of  the  plati-   > 

2.0O2 

3IO 

590 

.422 

(  num  retort,   j 

The  table  shows  that  the  density  at  302°  R,  and  of  course  at  inferior 
temperatures,  is  that  of  the  natural  wood,  dried  at  100°;  that  the  density  of 
the  charcoals  produced  at  from  302°  R  to  518°  R  was  reduced  by  the 
increasing  temperature  from  1.507  to  1.402. 

The  table  shows  briefly  as  follows : — 

1.  That  the  density  of  the  charcoal  at  302°  R,  is  that  of  the  natural 
wood  dried  at  212°,    namely  1.507. 

2.  The  density  of  the  charcoals  produced  at  from  302°  F.  to  518°  R  was 
reduced  from  1.507  to  1.402. 

3.  The  density  at  temperatures  above  518°  R  increases  with  the  temper- 
ature until  it  reaches  2.000,  or  double  that  of  water,  at  the  melting  point 
of  platinum. 

The  specific  gravity,  weight,  and  bulk  of  various  charcoals  are  given  in 
table  No.  65,  page  211,  and  they  are  here  abstracted  for  reference — supple- 
mented by  the  weight  and  bulk  of  the  charcoal  of  the  Ardennes  and  the 
Meuse,  derived  from  the  data,  page  449 : — 


MOISTURE   IN   WOOD-CHARCOAL. 


451 


WOOD  CHARCOAL. 

Specific  gravity. 

Weight  of 
a  cubic  foot. 

Bulk  of  one  ton. 

As  powder  

IXOO 

pounds. 

Q-3    CT 

cubic  feet. 
2/1  O 

In  small  pieces,  heaped  

O.AOZ 

yj'j 

2C.7 

8S  c 

As  manufactured,  heaped  

O.22Z 

•"O'O 
I4..O 

«->V.J 

160.0 

Ardennes  heaped 

O.2OI 

12.? 

180  o 

Meuse,  heaped  

O.2AI 

•"'0 
I  tJ.O 

1  4.0  O 

*J"? 

m.y.\s 

MOISTURE  IN  WOOD-CHARCOAL. 

Charcoal  absorbs  moisture  with  avidity.  The  charcoal  of  commerce  is 
usually  exposed  to  the  atmosphere,  and  open  to  rain-  and  it  contains 
generally  from  10  to  12  per  cent,  of  moisture. 

Charcoal  fresh  made,  from  different  woods,  was  exposed  by  M.  Nau,  for 
twenty-four  hours,  to  an  atmosphere  loaded  with  moisture,  and  the  weights 
of  water  they  absorbed  during  that  time  are  given  in  the  following  table, 
No.  155: — 

Table  No.  155. — MOISTURE  ABSORBED  BY  CHARCOALS  DURING 
TWENTY-FOUR  HOURS. 

(ByM.  Nau.) 


Wood  from  which  the  Charcoal 
was  made. 

Moisture 
Absorbed. 

Wood  from  which  the  Charcoal 
was  made. 

Moisture 
Absorbed. 

White  beech 

per  cent. 

o  8 

Horse  chestnut 

per  cent. 

6  06 

Ash             .     . 

4  06 

Elm 

6  60 

Oak     

4  28 

Alder 

7  Q'? 

Birch  

4AQ 

Scotch  fir 

/  -yo 

8  20 

Larch  

4  to 

Willow 

8  20 

Maple 

•  J^ 

4  80 

Italian  poplar 

8  50 

Pine  

Z.IA 

Fir 

8.  QO 

Red  beech  

J-  *-*+ 
5  -20 

Black  poplar 

r6  "?o 

•ow 

Showing  a  capacity  for  absorption  varying  from  0.8  to  16  per  cent. 

It  is  certain  that  the  period  of  exposure  was  not  sufficiently  long  to 
saturate  the  charcoals.  For  charcoals  have  been  known  to  absorb  increas- 
ing quantities  of  moisture  during  three  months. 

M.  Violette  made  some  observations  on  the  capacity  for  moisture  of 
charcoals  which  had  been  prepared  from  black  alder  at  various  temper- 
atures. The  samples  were  exposed  in  a  room  the  air  of  which  was  satur- 
ated with  moisture.  Observations  were  made  every  eight  days,  and  they 
lasted  three  months — until  the  charcoals  ceased  to  absorb  more  moisture. 
The  results  show  that  charcoal  is  less  absorbent  the  higher  the  temperature 
at  which  it  is  produced.  The  ordinary  black  charcoals,  produced  at  tern- 


452  FUELS.— PEAT. 

peratures  of  from  480°  to  750°  F.,  are  capable  of  absorbing  from  5  to  7  per 
cent,  of  water;  and  taking  the  extreme  observations,  the  absorption  ranges 
from  21  to  2.2  per  cent,  between  the  extreme  temperatures.  At  the  lower 
temperatures,  of  course,  the  charcoal  was  only  partially  converted. 

Am  CONSUMED  IN  THE  COMPLETE  COMBUSTION  OF  DRY  WOOD-CHARCOAL. 

According  to  the  analysis  of  M.  Sauvage,  page  447,  there  is  79  per  cent. 
of  carbon,  and  2  per  cent,  of  free  hydrogen,  in  forest-charcoal.  By  formula 
(i),  page  400,  the  volume  of  air  at  62°  F.  chemically  consumed  in  the 
complete  combustion  of  one  pound  of  charcoal,  is 

I-52  (79 +  (3  x  2)  )  =  I29  cubic  feet  of  air  at  62°. 
The  weight  of  the  air  is  129  -=- 13.14  =  9.8  pounds. 

GASEOUS  PRODUCTS  OF  THE  COMPLETE  COMBUSTION  OF  DRY 
WOOD-CHARCOAL. 

The  gaseous  products  of  combustion  consist  of  carbonic  acid,  steam, 
and  nitrogen,  and  the  total  weight  of  them  is  found  by  formula  ( 2 ),  page 
401,  as  follows: — 

(79  x  o.i  26) +  (2  x  0.356)  -  10.66  pounds. 

The  total  volume  of  the  gases,  as  at  62°,  by  formula  (3),  page  402,  is, 
(79  x  z-52)  +  (2  x  5-52)  =  I3I  cubic  feet  at  62°. 

HEAT  EVOLVED  BY  THE  COMPLETE  COMBUSTION  OF  WOOD-CHARCOAL. 

The  total  heating  power  of  dry  wood-charcoal,  having  79  per  cent,  of 
carbon  and  2  per  cent,  of  free  hydrogen,  is  by  formula  (6),  page  406: — 

145  (79 +  (4. 28  x  2)  )=  12,696  units  of  heat. 
The  total  evaporative  efficiency  is,  by  formula  (8),  page  406, 

°-I5  (79  +  (4-28  x  2)  )  =  I3-I3  pounds  of  water, 
evaporated  from  212°,  under  one  atmosphere. 

For  charcoal  containing  moisture  the  heating  power  is  less,  and  may  be 
estimated  in  the  manner  already  adopted  in  the  case  of  coke. 

PEAT. 

Peat  is.  the  organic  matter,  or  vegetable  soil,  of  bogs,  swamps,  and 
marshes, — decayed  mosses  or  sphagnums,  sedges,  coarse  grasses,  &c., — in 
beds  varying  from  i  or  2  feet  to  20,  30,  or  40  feet  deep.  The  peat 
near  the  surface,  less  advanced  in  decomposition,  is  light,  spongy,  and 
fibrous,  of  a  yellow  or  light  reddish -brown  colour;  lower  down,  it  is  more 
compact,  of  a  darker  brown  colour;  and,  in  the  lowest  strata,  it  is  of  a 
blackish  brown,  or  almost  a  black  colour,  of  a  pitchy  or  unctuous  feel, 
having  the  fibrous  texture  nearly  or  altogether  obliterated. 

Peat,  in  its  natural  condition,  generally  contains  from  75  to  80  per  cent, 
of  its  entire  weight,  of  water.  The  constituent  water  occasionally  amounts 


COMPOSITION   OF   PEAT. 


453 


to  85  or  even  to  90  per  cent.,  in  which  case  the  peat  is  of  the  consistency 
of  mire.  It  shrinks  very  much  in  drying;  and  its  specific  gravity  varies 
from  .22  or  .34  to  i. 06,  the  surface  peat  being  the  lightest,  and  the  lowest 
peat  the  densest.  Detailed  particulars  of  the  weight  and  specific  gravity  of 
peat  are  given  at  page  207. 

Table  No.  156. — CHEMICAL  COMPOSITION  OF  IRISH  PEAT, 
TAKEN  AS  PERFECTLY  DRY. 

(Sir  Robert  Kane.) 


DESCRIPTION  AND  LOCALITY 
OF  PEAT. 

Specific 
Gravity. 

Carbon. 

Hydrogen. 

Oxygen. 

Nitrogen. 

Ash. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

p.  cent. 

I.  Light  surface,  Philipstown.  .  .  . 

.405 

57.52 

6.83 

32.23 

1.42 

1.99 

2.  Rather  dense,         do. 

.669 

S8.S6 

5-91 

31.40 

•85 

3.30 

3.  Light  surface,  Wood  of  Allen 

•335 

58.30 

6-43 

3L36 

1.22 

2.74 

4.  Compact  and  dense,    do. 

•655 

56.34 

4.81 

30.20 

•74 

7-90 

5.  Light  fibrous,  Ticknevin  ,  
6.  Light  fibrous,  Upper  Shannon 

.500 
.280 

58.60 
58.53 

6.55 

5-73 

30.50 
32.32 

1.84 

•93 

2.63 

2.47 

7.  Very  dense,  compact,  do. 

•*S3 

59.42 

5-49 

30.50 

1.64 

2.97 

Averages 

S28 

58  18 

?  06 

•jj  21 

I  23 

3A<2 

Table  No.  157. — COMPOSITION  OF  SUNDRY  PEATS,  INCLUDING  MOISTURE. 
FIRST,  EXCLUSIVE  OF   MOISTURE. 


.DESCRIPTION  OF  PEAT. 

Moisture. 

Carbon. 

Hydrog. 

Oxygen. 

Nitrog. 

Sulphur. 

Ash. 

Coke. 

Good  air-dried  
Poor  air-dried  
Dense,  from  Gal  way 

per  cent. 

per  cent. 

59-7 
59-6 
59-5 

per  cent. 

6.0 

4-3 

7-2 

per  cent. 

31 

29 
24.8 

percent. 
.8 

2.3 

per  cent. 
.8 

p.  cent. 
2-4 

6-3 

5-4 

p.  cent. 
44-3 

Averages  

— 

59-6 

5-8 

29.6 

•3 

4.7 

— 

Good  air-dried  
Poor  air-dried  
Dense,  from  Galway 

SECON 
24.2 
29.4 
29-3 

D,  INCL 

45-3 
42.1 
42.0 

USIVE   0 

4.6 
3-1 

F  Mois 

24 

21 
17.5 

TURE. 

.1 
.0 

.6 

.2 

1.8 
4.4 
3-8 

31-3 

Averages  

27.8 

43-i 

4.3 

21.4 

3-3 

— 

When  wet,  peat  is  masticated,  macerated,  or  milled,  so  that  the  fibre  is 
broken,  crushed,  or  cut.  The  contraction  in  drying  is  much  increased  by 
this  treatment;  and  the  peat  becomes  denser,  and  is  better  consolidated 
than  when  it  is  dried  as  cut  from  the  bog.  Peat  so  prepared  is  known  as 
condensed  peat;  and  the  degree  of  condensation  varies  according  to  the 
natural  heaviness  of  the  peat.  Peat  from  the  lowest  beds  is  but  little 
condensed ;  but  peat  from  the  middle  and  upper  beds  is  condensed,  when 
dry,  to  from  two  to  three  times  its  natural  density.  So  effectively  is  peat 
consolidated  and  condensed  by  the  simple  process  of  breaking  the  fibres 


454 


FUELS. — PEAT. 


whilst  wet,  that  no  merely  mechanical  force  of  compression  is  equal  in 
efficiency  to  mastication. 

The  table  No.  156  contains  the  results  of  chemical  analysis  of  Irish  peat 
of  various  qualities  by  Dr.  Kane;  the  samples  were  desiccated  before  being 
submitted  to  analysis. 

Mr.  A.  M'Donnell  gives  the  composition  of  average  "  good  air-dried " 
peat  and  "poor  air-dried"  peat,  analyzed  by  Dr.  Reynolds,  as  in  table 
No.  157;  to  which  are  added  an  analysis  of  dense  peat  from  Galway,  made 
by  Dr.  Cameron : — 

From  the  above  tables,  it  appears  that  sulphur  is  rarely  found  in  Irish 
peat,  and  that  the  average  composition  of  the  peat  is  as  follows : — 


Perfectly  Dry. 
per  cent. 

Carbon 59 

Hydrogen 6 

Oxygen 30 

Nitrogen i  % 

Sulphur ? 

Ash  . .  4 


Including  25  per  cent, 
of  moisture. 
per  cent. 


22.; 

I 


Moisture 


100 


75 


IOO 


Ordinary  air-dried  peat  contains  from  20  to  30  per  cent,  of  its  gross 
weight  of  moisture.  If  dried  in  air  in  the  most  effective  manner,  it  contains 
at  least  15  per  cent,  of  moisture;  and  even  when  dried  in  a  stove,  it  seldom 
holds  less  than  7  or  8  per  cent. 

The  peats  named  in  table  No.  156  were  subjected  to  distillation,  when 
they  yielded  water,  tar,  charcoal,  and  gas,  in  the  proportions  shown  in 
table  No.  158:— 


Table  No.  158. — PRODUCTS  OF  DISTILLATION  OF  IRISH  PEAT. 


Description  and  Locality  of  Peat. 

Water. 

Crude  Tar. 

Charcoal. 

Gas. 

Nos.  i  and  2,  Philipstown  

per  cent. 

23  6 

per  cent. 
2.O 

per  cent. 
7.7.  S 

per  cent. 
36  Q 

»               3)  Wood  of  Allen  
4,            do. 

32'3 
^8.1 

3-6 

2.8 

o  /•  o 
39-i 

32.6 

O^'y 
25.0 
26.  Z 

n                5,  Ticknevin  

3^.6 

2.Q 

71.  1 

72.7 

J?               6,  Upper  Shannon  

38.1 

4.4 

21.8 

3^-7 

))                7)              5?             

21.8 

i-5 

19.0 

57-7 

Averages  . 

71.4. 

2.8 

2Q.2 

36.6 

O^'T- 

The  tar,  when  re-distilled,  yielded  water,  paraffine,  oils,  charcoal,  and  gas, 
The  water  yielded  chloride  of  ammonium,  acetic  acid,  and  wood-spirit. 


PEAT-CHARCOAL.  —  TAN.  455 

Heating  Power  of  Irish  Peat.  —  In  peat  of  average  composition,  as  given 
above,  the  heating  power  is  by  rule  4,  page  406, 

perfectly  dry,  ....................  145  (59  +  4.28  (6~y)  )  =  995  J  units  of  heat; 

'«  (44  +  4-8  (4.5  -1)  )  =  743S  units  of  heat. 


Deduct  for   evaporating  the   moisture,    %   lb., 

supplied  at  62°;  ni6°-^4  .....................  =    279    do.       do. 

Effective  heating  power  ...............  7156    do.       do. 

The  total  evaporative  power  of  i  lb.  of  fuel,  evaporating  at  212°,   is 
as  follows  :  — 

Perfectly  dry  Containing  25  per 

cent,  of  moisture. 

When  water  is  supplied  at  62°,  divisor  in6°...  8.91  Ibs.  6.41  Ibs. 

Do.  do.          212°,     do.        966°...  10.30  Ibs.  7.41  Ibs. 

British  and  foreign  peats  are  very  much  like  Irish  peat  in  composition  ; 
the  principal  variation  takes  place  in  the  proportion  of  ash. 


PEAT-CHARCOAL.  fc 

The  charcoal  of  ordinary  dried  peat  is  very  porous,  and,  in  general,  light 
and  fragile;  but  the  charcoal  of  condensed  peat  is  dense  and  solid.  It 
burns  easily  but  slowly :  small  incandescent  pieces  separated  from  the  fire 
continue  to  burn  until  the  whole  of  the  carbon  disappears.  Good  peat 
yields  from  30  to  40  per  cent,  of  charcoal;  and  the  charcoal  when  perfectly 
dry  consists  generally  of  from  85  to  90  per  cent,  of  carbon,  and  10  to  15 
per  cent,  of  ash. 

The  heating  power  of  one  pound  of  peat -charcoal,  containing  85  per 
cent,  of  carbon,  by  rule  4,  page  406,  is,  145  x  85  =  12,325  units  of  heat; 
equivalent  to  the  evaporation,  at  212°,  of  11.04  Ibs.  of  water  supplied  at  62°, 
or  of  12.76  Ibs.  supplied  at  212°.  The  temperature  of  combustion  is  that 
of  carbon,  4877°  F. 

In  France,  the  peat-charcoal  of  Essonne  contains  18.2  per  cent,  of  ash. 
In  the  Ardennes,  Bar  peat  carbonized  in  ovens  yields  44  per  cent,  of  char- 
coal; but  this  contains  one-third  volatile  matter,  one-fourth  ash,  and  only 
43  per  cent,  of  carbon. 

TAN. 

Tan,  or  oak-bark,  after  having  been  used  in  the  processes  of  tanning,  is 
burned  as  fuel.  The  spent  tan  consists  of  the  fibrous  portion  of  the  bark. 
According  to  M.  Peclet,  five  parts  of  oak-bark  produce  four  parts  of  dry 
tan;  and  the  heating  power  of  perfectly  dry  tan,  containing  15  per  cent,  of 
ash,  is  6100  English  units;  whilst  that  of  tan  in  an  ordinary  state  of  dryness, 
containing  30  per  cent,  of  water,  is  only  4284  English  units.  The  weight 
of  water  evaporated  at  212°  by  one  pound  of  tan,  equivalent  to  these  heat- 
ing powers,  is  as  follows : — 


FUELS. — STRAW,   LIQUID  FUELS. 


Perfectly  dry. 

Water  supplied  at  62° 5.46  Ibs.  3.84  Ibs. 

212° 6.31  „  4.44  „ 

STRAW. 

The  average  composition  of  wheat-straw  is  as  follows: — 

Water 14.23  per  cent. 

Organic  or  combustible  matter;  consisting  of  )    g 

carbon,  hydrogen,  oxygen,  and  nitrogen.,  j  7  '3°       » 
Ash 7.47 

100.00 

Chemists  have  not,  so  far  as  the  author  has  learned,  thought  it  worth 
while  to  record  the  proportions  of  the  organic  elements.  But  it  may  be 
supposed  that  the  composition  of  straw  is  similarly  proportioned  to  that  of 
peat.  The  weight  of  pressed  straw  is  from  6  to  8  Ibs.  per  cubic  foot. 

LIQUID  FUELS. 

Petroleum  is  a  hydro-carbon  liquid  which  is  found  in  abundance  in 
America  and  Europe.  According  to  the  analysis  of  M.  Sainte-Claire 
Deville,  the  composition  of  fifteen  petroleums  from  different  sources  was 
found  to  be  practically  constant.  The  average  specific  gravity  was  .870. 
The  extreme  and  the  average  elementary  composition  were  as  follows: — 

Carbon 82.0  to  87.1  per  cent.     Average  84.7  per  cent. 

Hydrogen 11.2  to  14.8       „  „         13.1        „ 

Oxygen 0.510    5.7        „  „          2.2        „ 


IOO.O 


The  total  heating  and  evaporative  powers  of  one  pound  of  petroleum 
having  this  average  composition  are,  by  rules  4  and  5,  page  406,  as 
follows  :  — 

Total  heating  power  ..........  =  145  (84.7  +  4.28  (13.1  -  ^)  )  =  20,  240  units. 

8 

Evaporative  power  :  evaporating  at  2  1  2°,  water  supplied  at  62°  =  =    18.13  Ibs. 
Do.  do.  do.         212°=    20.33  Ibs. 

Petroleum-Oils  are  obtained  in  great  variety  by  distillation  from  petro- 
leum. They  are  compounds  of  carbon  and  hydrogen,  ranging  from  CIO  H24 
to  C32  H64;  or,  in  weight, 

from  I  7I'42  carbon      1  to  I  73'77  carbon 
n  \  28.58  hydrogen  j  tO  (  26.23  hydrogen 

IOO.OO  IOO.OO 


COAL-GAS. 


457 


The  specific  gravity  ranges  from  .628  to  .792.  The  boiling  point  ranges 
from  86°  to  495°  F.  The  total  heating  power  ranges  from  28,087  to 
26,975  units  of  heat:  equivalent  to  the  evaporation,  at  212°,  of  from 
25.17  Ibs.  to  24.17  Ibs.  of  water  supplied  at  62°,  or  from  29.08  Ibs.  to 
27.92  Ibs.  of  water  supplied  at  212°. 

Schist-Oil,  like  petroleum,  consists  of  carbon,  hydrogen,  and  oxygen;  but 
there  is  less  hydrogen  and  more  oxygen,  as  may  be  seen  from  the  following 
analysis  by  St.-Claire  Deville: — 

From  From 

Vagnas  Schist.     Autun  Schist. 

Carbon 80.3  79.7 

Hydrogen 11.5  zi.8 

Oxygen 8.2  8.5 


100.0 


100.0 


Pine-wood  Oil,  analyzed  by  the  same  chemist,  contains  87.1  per  cent,  of 
carbon,  10.4  per  cent,  of  hydrogen,  and  2.5  per  cent,  of  oxygen. 


COAL-GAS. 

Mr.  Vernon  Harcourt  made  an  analysis  of  coal-gas,1  one  pound  of  which 


COAL-GAS. 
(Mr.  V.  Harcourt.) 

Carbon. 

Hydrogen. 

Oxygen. 

TotaL 

Olefiant  gas. 

per  cent. 

io-5 
39-7 
5-9 
1.9 

per  cent. 

1-7 

13.2 

8.1 

per  cent. 
7-9 

5-o 

per  cent 
12.2 

52-9 

13.8 
6.9 

8.1 
5-8 
°-3 

Marsh  gas. 

Carbonic  oxide  

Carbonic  acid  

Hydrogen  

Nitrogen  

Oxveen  .  .  . 

58.0 

23.0 

— 

IOO.O 

had  a  volume  of  30  cubic  feet  at  62°  F.     The  heating  power,  calculated  for 
the  three  elements,  is  as  follows: — 

Units. 

Carbonic  oxide 13. 8  per  cent  x    4,3254-100=       597 

Carbon 50.2       „        x  14,500-^  100  =    7,279 

Hydrogen 23.0       „        x  62,000-7-100=  14,260 


Total  heat  of  combustion 22,136 

This  is  equivalent  to  the  evaporation  of  19.84  Ibs.  of  water  from  62°  at 

1  See  a  paper  on  "Petroleum  and  other  Mineral  Oils,  applied  to  the  Manufacture  of 
Gas,"  by  Mr.  Owen  C.  D.  Ross,  in  the  Proceedings  of  the  Institution  of  Civil  Engineers, 
vol.  xl.  page  150. 


45  8  FUELS.— COAL-GAS. 

212°  F.,  or  to  22.92  Ibs.  from  and  at  212°.  The  heating  power  of  i  cubic 
foot  is  738  units,  equivalent  to  the  evaporation  of  .66  Ib.  or  .76  Ib.  of  water. 

Mr.  F.  W.  Hartley1  tested  the  heating  power  of  gas  manufactured  by 
the  South  Metropolitan  Gas  Company,  London.  By  means  of  his  gas 
calorimeter  he  determined  the  heating  power  of  one  cubic  foot  of  gas  at 
60°  F.,  under  30  inches  of  mercury,  to  be  622.15  units,  equivalent  to  the 
evaporation  of  .56  pound  of  water  from  60°  at  212°,  or  .64  pound  from  and 
at  212°. 

Taking  the  means  for  these  two  gases,  the  heating  power  of  i  cubic  foot 
at  62°  F.  is  equivalent  to  the  evaporation  of  .70  pound  of  water  from  and 


1  Report  on  the  Gas  Section  of  the  International  Electric  and  Gas  Exhibition  at  the  Crystal 
Palace,  1882-83,  Page  23-  It  ma7  ^ere  he  stated  that  according  to  the  results  of  tests  of 
several  "Instantaneous  Gas  Waterheaters,"  made  by  Mr.  D.  K.  Clark,  in  the  same  con- 
nection, much  more  heat  was  generated  than  was  deduced  from  Mr.  Hartley's  deter- 
minations. 

It  is  due  to  Mr.  Hartley  to  state  that  he  was  cognizant  of  the  usual  presumption  that 
the  potentiality  of  gas  as  a  heating  power  is  greater  than  what  is  deduced  from  such 
experiments.  "The  problem,"  he  says,  "  of  determining  with  ease  and  certainty  when  a 
quantity  of  two  to  three  cubic  feet  can  be  had,  the  absolute  calorific  value  of  a  combustible 
gas,  within  about  ]/2  per  cent,  of  the  truth,  is,  I  think,  completely  solved;  and  the  fact 
that  the  calorific  power  indicated  for  the  gas  in  question  is  much  below  that  which  ordi- 
nary coal-gas  is  presumed  to  possess,  in  no  degree  disturbs  my  belief  in  the  accuracy  of 
the  results  which  I  have  now  the  honour  to  submit." — Report,  page  23. 


APPLICATIONS   OF    HEAT. 


This  section  on  the  applications  of  heat  comprises  the  principles  of  the 
transmission  of  heat  through  solid  bodies.  The  consideration  of  the 
application  of  the  heat  of  furnaces  for  the  generation  of  steam  in  boilers 
will  be  taken  up  in  the  section  on  steam-boilers.  In  the  present  section, 
the  subjects  dealt  with  relate  to  the  heating  and  evaporation  of  water  by 
steam,  the  condensing  of  steam  by  water,  the  heating  of  air  by  hot  water 
and  by  steam,  the  warming  and  ventilation  of  buildings,  distillation,  cooling, 
drying,  blast-furnaces,  and  cognate  subjects. 


TRANSMISSION  OF  HEAT  THROUGH  SOLID  BODIES— 
FROM  WATER  TO  WATER  THROUGH  SOLID  PLATES 
OR  BEDS. 

With  a  view  to  educe  the  general  principles  of  the  transmission  of  heat 
through  solid  bodies,  M.  Peclet  made  a  series  of  experiments  on  the  trans- 
mission of  heat  through  plates  of  metal,  heated  on  one  side  by  heated  water, 
and  cooled  on  the  other  side  by  water  at  a  low  temperature.  He  found 
from  experiments  made  with  wrought  iron,  cast  iron,  copper,  lead,  zinc,  and 
tin,  that  when  the  fluid  in  contact  with  the  surface  of  the  plate  was  not 
changed  by  artificial  means,  the  rate  of  conduction  of  metals  was  not  only 
the  same  for  different  metals,  but  also  for  different  thicknesses  of  the  same 
metal.  Correctly  ascribing  this  uniformity  of  performance  to  the  presence 
of  a  stagnant  film  of  water  adhering  to  the  surfaces  of  the  plates,  which,  by 
its  inferior  conductivity,  negatived  in  a  greater  or  less  degree  the  conduc- 
tivity of  the  plates  themselves,  he  made  a  new  series  of  experiments  with 
lead  plates,  in  which  the  water  was  thoroughly  circulated  over  the 
surface;  and  he  found  that  the  quantity  of  heat  transmitted  through  the 
plates  was  inversely  proportional  to  the  thickness.  Having  by  this  means 
settled  the  constant  for  lead,  he  adopted  the  results  of  Depretz's  experi- 
ments on  the  conducting  power  of  metals  (see  page  331),  and  calculated 
the  constants  for  other  metals  from  these  data.  He,  further,  made  a  series 
of  experiments  on  the  conducting  or  transmissive  power  of  "  bad  conduc- 
tors" of  heat,  between  two  surfaces  of  water: — stone,  and  wood  and  other 
vegetable  substances,  which  were  incased  in  two  thin  coats  of  copper,  to 
prevent  the  absorption  of  water  by  them. 

M.  Peclet  lays  down  the  elementary  law  of  the  transmission  of  heat  as 
follows : — The  flow  of  heat  which  traverses  an  element  of  a  body  in  a  unit  of 
time  is  proportional  to  its  surface,  and  to  the  difference  of  temperature  of  the 
two  faces  perpendicular  to  the  direction  of  the  flow;  and  is  in  the  inverse 


460 


APPLICATIONS   OF   HEAT. 


ratio  of  the  thickness  of  the  element.  This  law,  he  maintains,  is  rigorously 
deduced  from  the  nature  of  the  motion  of  heat,  and  he  embodies  it  in  the 
following  formula: — 

M  =  (/-/-)-J.j  (i) 

in  which  /  and  f  are  the  temperatures  of  the  surfaces,  C  the  quantity  of 
heat  transmitted  per  hour  for  one  degree  of  difference  of  temperature 
through  one  unit  of  thickness,  and  E  the  thickness.  That  is  to  say,  using 
English  measures: — if  the  difference  of  temperatures  in  degrees  Fahren- 
heit be  multiplied  by  the  constant  C  for  the  given  material,  one  inch  thick, 
and  divided  by  the  thickness  in  inches,  the  quotient  is  the  quantity  of  heat 
in  English  units  passed  through  the  plate  per  square  foot  per  hour. 

The  quantities  of  heat  transmitted  through  plates  or  beds  of  metals  and 
other  solid  bodies,  one  inch  in  thickness,  for  i°  F.  difference  of  tempera- 
ture per  hour,  as  determined  by  M.  Peclet,  are  given  in  table  No.  159, 
being  the  values  of  the  constant  C  in  formula  (i).  The  conditions  are, 
that  the  surfaces  of  the  conducting  material  must  be  perfectly  clean,  that 
they  be  in  contact  with  water  at  both  faces  of  different  temperatures, 
and  that  the  water  in  contact  with  the  surfaces  be  thoroughly  and  con- 


Table  No.  159. — QUANTITIES  OF  HEAT  TRANSMITTED  FROM  WATER  TO 
WATER  THROUGH  PLATES  OR  BEDS  OF  METALS  AND  OTHER  SOLID 
BODIES  i  INCH  THICK,  PER  SQUARE  FOOT,  FOR  i°  F.  DIFFERENCE 
OF  TEMPERATURE  BETWEEN  THE  Two  FACES  PER  HOUR. 

Selected  from  M.  Peclet's  tables,  and  converted  for  English  measures. 


Substance. 

Quantity 
of 
heat. 

Substance. 

Quantity 
of 
heat. 

Substance. 

Quantity 
of 
heat. 

Gold 

units. 
62O 

Fir  across  fibre 

units. 
7  A 

Coke  powd 

units. 
80 

Platinum  
Silver  . 

604 

co6 

Fir,    along   the 
fibre 

•  /'T 
I    l6 

Iron  filings  — 
Cotton  wool 

1.26 

•22 

Copper  .  . 

OV" 

crc 

Caoutchouc 

-L'O^ 
I      l6 

Calico 

•6Z 

AO 

Iron  

000 

22^ 

Gutta-p  ercha 

'••y 

I  37 

Carded  wool 

,q.v 
•2C 

Zinc  

"^0 
22^ 

Glass 

*-'O  I 

6  q6 

Eider-down 

•oo 

-7  I 

Tin  

Sand 

*voy 

2   l6 

Canvas 

•ox 

A2 

Lead 

112 

Brick  powder'd 

112 

\Vhite  wntincf- 

Marble  
Plaster 

24 
2  6 

Chalk,      do. 
Ashes  of  wood 

.69 

r  -2 

paper  
Gray  paper  un- 

•34 

Terra  cotta  

4.8 

Wood-charcoal, 

•oo 

sized  

.10 

Oak,  across  fibre 

1.69 

powdered.... 

.63 

stantly  changed.  M.  Peclet  found  that  when  metallic  surfaces  became  dull, 
the  rate  of  transmission  of  heat  through  all  metals  became  very  nearly  the 
same. 

Mr.  James  R.  Napier  made  experiments  with  experimental  boilers  of  iron 
and  copper  of  various  thicknesses,  over  a  gas  flame,  and  he  found  only  a 


HEATING  AND   EVAPORATION   OF  LIQUIDS. 


461 


small  difference  in  evaporating  power  of  about  a  twentieth  or  a  thirtieth 
in  favour  of  the  copper:  results  which  are  corroborative  of  M.  Peclet's 
deductions. 

Professor  Rankine  states  that  in  all  experiments  of  this  kind  the  con- 
dition of  the  heating  surface  is  important,  whether  smooth  or  rough,  and 
whether  perfectly  clean  or  incrusted  to  any  extent. 

But,  the  rate  of  transmission  of  heat  through  metallic  plates  also  differs 
very  much  according  to  the  substances  in  contact  with  the  plate,  between 
which  the  heat  is  transmitted : — as  between  water  or  steam  and  water,  or 
between  water  and  air,  or  gaseous  matter  and  water,  and  so  on.  Mr. 
Thomas  Craddock,  at  an  early  period,  proved  that  the  rate  of  cooling 
by  transmission  of  heat  through  metallic  surfaces,  was  almost  wholly 
dependent  upon  the  rate  of  circulation  of  the  cooling  medium  over  the 
surface  to  be  cooled,  and  that  water  was  enormously  more  efficient  than  air 
for  the  abstraction  of  heat.  He  suspended  a  tube  filled  with  hot  water 
having  a  thermometer  suspended  in  the  water.  The  water  was  cooled  from 
a  temperature  of  180°  to  100°  F.,  in  still  air,  in  25  minutes,  and  in  still  water 
in  one  minute.  Again,  when  he  moved  the  tube  filled  with  hot  water,  by 
rapid  rotation,  at  the  rate  of  40  miles  per  hour  through  air,  it  lost  as  much 
heat  in  i  minute  as  it  did  in  still  air  in  1 2  minutes.  In  water,  at  a  velocity 
of  two  miles  per  hour,  as  much  heat  was  abstracted  in  half  a  minute  as 
was  absorbed  in  one  minute  when  at  rest  in  the  water.  Mr.  Craddock 
concluded  that  the  circulation  of  the  cooling  fluid  becomes  of  greater 
importance  as  the  difference  of  temperature  on  the  two  sides  of  the  plate 
becomes  less. 

HEATING    AND    EVAPORATION    OF     LIQUIDS     BY    STEAM 
THROUGH   METALLIC   SURFACES. 

Mr.  John  Graham  heated  water  in  a  square  wooden  cistern,  having  a 
double  iron  bottom,  into  which  steam  of  16^  Ibs.  per  square  inch,  abso- 
lute pressure,  having  a  temperature  of  218°  F.,  was  admitted.  When  the 
water  in  the  cistern  stood  at  60°  F.,  the  steam  was  admitted,  and  the 
following  were  the  successive  temperatures  at  equal  intervals  of  time,  as- 
reported  by  Mr.  Graham : — 

Time  from  the 

commencement. 

seconds. 

O 
IO 

20 

40 

50 
60 
70 
80 
90 

IOO 

It  was  found  to  be  difficult  to  raise  the  temperature  above  210°  F.  The 
increased  activity  in  the  rise  of  temperature  towards  the  end,  was  no  doubt 


ie                                  Temperature 
it.                                of  the  water. 
Fahrenheit. 
60° 

Increments  of 
temperature. 
Fahrenheit, 
o       , 

IOO 

40         : 

1*8 

34 

m" 

16 

108 

6 

210 

4. 

462 


APPLICATIONS   OF   HEAT. 


due  to  the  increased  movement  in  the  water  as  it  approached  the  boiling 
point.  To  show  the  rate  of  the  passage  of  heat  with  respect  to  the  mean 
difference  of  the  temperatures  of  the  steam  and  the  water  during  each 
interval  of  ten  seconds,  the  mean  temperature  of  the  water  during  each 
interval  is  given  in  the  second  column  below,  the  difference  of  these  mean 
temperatures  and  that  of  the  steam  in  the  third  column,  the  increments  of 
temperature  in  the  fourth  column;  and,  in  the  last  column,  the  rise  of 
temperature  per  degree  of  difference  of  temperature  is  given : — 


Times. 

Mean  tempera- 
tures of  the 
water. 

Difference  of 
temperatures  of 
the  water  and 
the  steam. 

Increments  of 
temperature. 

Increments  of 
temperature 
per  degree  of 
difference. 

seconds. 

Fahrenheit. 

Fahrenheit. 

Fahrenheit. 

Fahrenheit. 

IO 

80° 

138° 

40° 

.290 

2O 

117 

101 

34 

.336 

30 

I46 

72 

24 

•333 

40 

166 

52 

16 

.308 

5° 

178 

40 

9 

.225 

60 

187 

31 

9 

.290 

70 

i95 

23 

6 

.261 

80 

199-5 

18.5 

3 

.162 

90 

203-5 

14-5 

5 

•345 

IOO 

208 

10 

4 

.400 

The  quantity  of  heat  transmitted  per  degree  of  difference  of  temperature 
is  in  proportion  to  the  increments  of  temperature  in  the  last  column. 
Though  irregular,  they  are,  taken  together,  practically  uniform  per  degree 
of  difference  of  temperature.  At  the  same  time,  the  quantity  per  degree  in 
the  middle  stages  appears  to  be  slightly  reduced  as  the  total  difference  of 
temperature  is  reduced. 

M.  Clement  found  that  a  sheet  of  copper,  i  metre  square  and  about 
Y%  inch  thick,  when  heated  on  one  face  by  steam  of  212°  R,  and  cooled  on 
the  other  face  by  water  at  8 2°. 4  R,  making  an  excess  of  temperature  of 
i29°.6,  condensed  20.5  Ibs.  of  steam  per  square  foot  per  hour,  equivalent 
to 

20.5-4-  i29°.6  =  o.i6o  Ibs.  of  steam 

condensed  per  square  foot  per  degree  of  difference  of  temperature  per  hour. 
The  total  heat  of  steam  at  212°  R  is  io95°.6  above  82°.4,  and  for  20.5  Ibs. 
of  steam  there  are  1095.6  x  20.5  -  22,460  units  of  heat,  and 

22, 460 -f- 129°.6=  173  units  of  heat, 

which  is  the  quantity  of  heat  passed  through  the  plate  per  square  foot  per 
degree  of  difference  of  temperature  per  hour. 

M.  Peclet  gives  the  performance  of  a  copper  boiler  with  double  bottom 
for  boiling  beet-root  juice  by  steam  of  three  atmospheres,  or  275°  R, 
admitted  into  the  bottom.  The  area  exposed  to  steam  amounted  to  25.82 
square  feet.  A  quantity  of  juice  weighing  1984.5  Ibs.  was  delivered  into 
the  copper  at  a  temperature  of  39°  F.,  and  heated  to  212°  F.,  through  173° 


HEATING  AND   EVAPORATION   OF   LIQUIDS.  463 

in  sixteen  minutes,  equivalent  to  a  rise  of  (173  x  60  -f-  16)  =  649°  of  tempera- 
ture per  hour.  The  total  heat  transmitted  per  square  foot  per  hour  was 

649°  x  1984.5  -r  25.82  =  49,880  units  of  heat. 

The  mean  temperature  of  the  juice  was  (2  12  +  39)  -^  2  =  126°,  and  the  mean 
difference  of  temperature  was  275°-  126°=  149°;  then  the  total  heat  trans- 
mitted per  square  foot  per  degree  of  difference  of  temperature  per  hour 
was 

49,880+  1  49°  =  335  units  of  heat. 

The  total  heat  of  the  steam  was  1071°  above  126°,  the  mean  temperature 
of  the  juice,  and  the  quantity  of  steam  condensed  per  square  foot  per 
degree  per  hour  was 

=.313  lb.  of  steam. 


1071 

M.  Peclet  quotes  the  results  of  experiments  made  by  Laurens  and 
Thomas  on  the  heating  power  of  steam  operating  through  coils  of  pipe. 
In  the  first  experiment,  the  pipe  was  137.8  feet  long  and  1.36  inches  in 
diameter  externally,  presenting  48.20  square  feet  of  surface.  Steam  of 
three  atmospheres,  or  275°  F.,  was  freely  admitted  into  the  pipe,  and  it 
raised  the  temperature  of  882  Ibs.  of  water  from  46°  F.  to  212°  through 
166°  in  four  minutes,  equivalent  to  a  rise  of  (166°  x  60  -4-  4)  =  2490°  in  an 
hour.  The  mean  temperature  of  the  water  was  (212  +  46)  -r-  2  =  129°,  and 
the  difference  of  temperature  was  275  -  129  =  146°.  Hence  the  total  heat 
transmitted  per  square  foot  per  hour  was 

2490°  x  882  -f-  48.20  =  45564  units  of  heat, 
and  the  total  heat  per  square  foot  per  degree  per  hour  was 
45564  -f  146  =  312.1  units  of  heat. 

The  total  heat  of  the  steam  was  1068°  above  129°,  and  the  quantity  of 
steam  condensed  per  square  foot  per  degree  per  hour  was,  therefore, 

'*  =  .292  lb.  of  steam. 
1068 

Next,  551.25  Ibs.  of  water  was  evaporated  from  212°  by  the  same  steam, 
in  ii  minutes,  being  at  the  rate  of  3007  Ibs.  per  hour,  or  62.38  Ibs.  per 
square  foot  per  hour.  The  total  heat  of  the  atmospheric  vapour  was  966° 
above  2  1  2°,  and  the  heat  transmitted  per  square  foot  per  hour  was 

966  x  62.38  =  59,710  units  of  heat. 

The  difference  of  temperature  was  275-212  =  63°,  and  the  quantity  of 
heat  passed  per  square  foot  per  degree  per  hour  was 

59,7  10  H-  63  =  948  units  of  heat. 
The  quantity  of  steam  condensed  per  square  foot  per  degree  per  hour  was 

?4?  =  .  981  lb.  of  steam. 
900 


464  APPLICATIONS    OF    HEAT. 

In  another  experiment,  two  coils  of  steam-pipe  49.2  feet  long,  and  1.34 
inches  in  diameter,  presenting  a  surface  of  34.52  square  feet,  with  steam 
of  two  atmospheres,  or  25o°.4  F.,  evaporated  1587.6  Ibs.  of  water  at  212° 
per  hour;  or  46  Ibs.  per  square  foot  per  hour,  with  a  difference  of  tem- 
perature (250.4  -  212)  =  38°.4  F.  The  quantity  of  heat  passed  per  square 
foot  per  hour  was, 

966  x  46  =  44,430  units, 
and,  per  degree  per  square  foot  per  hour,  was, 

44,430  -f-  38.4  =  1  157  units. 
The  quantity  of  steam  condensed  per  degree  per  square  foot  per  hour  was 


.. 

966 

The  following  are  the  results  of  experiments  by  M.  P.  Havrez  in  heating 
water  by  steam  with  a  coil  of  copper  pipe,  given  in  Engineering,  vol.  vi. 
The  coil  was  14.21  feet  long,  and  1.57  inches  in  diameter;  superficial  area, 
5.85  square  feet.  The  pipe  was  incrusted  to  some  extent:  — 

i.  67  Ibs.  300°  F.      232.65  Ibs.  from  68°  to  212° 

2-89,,     319.4          232.65        „         „          „         to  122°,  4min.;  212°,  10  min. 

3.  89  „     319.4          217.80        „       104  to  212  „      3    „         „        7^  „ 

Actual  weight  of  steam        Weight  per  square  foot     Weight  per  i°  F.  differ- 
condensed.  per  hour.  ence  of  temperature. 

1.  41.25  Ibs.  42.25  Ibs.  .264lb. 

2.  44.00  „  45.00  „  .271  „ 

3.  28.60  „  39.00  „  .270  „ 


Averages,  42.08  „  .268  „ 

It  may  be  noted  that  the  water  was  heated  to  122°,  and  then  to  212°,  in 
the  second  and  third  experiment,  at  the  following  rates : — 

2d  Experiment,  to  122°  at  13°.  5  per  minute.     To  212°  at  15°  per  minute. 
3d         do.  „          „    6.0         „  „        „    20         „ 

In  continuation  of  the  experiments,  with  the  same  pressure  of  steam, 
portions  of  the  water  were  evaporated : — 


Water  evaporated 
in  10  minutes. 

Ebullition. 

Evaporated  per 

And  per  degree. 

I. 

2. 

3- 

9.9  Ibs. 
15-95   „ 
7-7     „ 

soft, 
violent, 
very  soft, 

10.  1  5  Ibs. 
16.35   » 
7.89  „ 

.115  lb. 

•174  „ 
.084  „ 

11.46  „  .126  „ 

There  are,  as  Engineering  remarks,  inconsistencies  in  these  results.  The 
scale,  no  doubt,  impeded  the  activity  of  the  heat. 

The  results  of  experiments  by  M.  Havrez,  with  a  cast-iron  boiler  having 
a  double  bottom,  are  also  given  by  Engineering.  The  boiler  was  18.5 
inches  in  diameter,  and  13.5  inches  deep,  and  had  a  jacketted  surface  of 
6.576  square  feet: — 


HEATING  AND   EVAPORATION   OF   LIQUIDS. 


465 


Total  pressure  and 
temperature  of  steam. 

1.  67  Ibs.  300°  F. 

2.  67    „      300 


Water  heated. 

229.7  Ibs. 
237-6  „ 


from  80°  to  212°,  in  25  minutes. 
„     68  to  212,   in  22        „ 


Actual  weight  of  steam 
condensed. 

3i.24lbs. 
i        34.10  „ 

Averages, 


Per  square  foot 
per  hour. 

10.14  Ibs. 

14.10    „ 


12.12 


And  per  degree  differ- 
ence of  temperature. 

.066  lb. 
.088  „ 

•077  „ 


In  evaporating  the  water  at  212°,  from  8.8  to  11.44  Ibs.  of  steam  was 
condensed  in  10  minutes;  being  at  the  rate  of  8.02  to  10.4  Ibs.  per  square 
foot  per  hour;  or  .091  to  .118  lb.  per  square  foot  per  degree  of  difference 
of  temperature  per  hour. 

The  quantities  of  heat,  in  M.  Havrez's  experiments,  transmitted  per 
square  foot  per  degree  of  difference  of  temperature  per  hour  are  found 
from  the  quantities  of  steam  condensed,  as  follows : — 


Heat  utilized  from 
i  lb.  of  steam. 


Coiled  pipe ;  heating  water : — 

ist  Experiment, 
2d          Do. 
3d          Do. 


Steam  condensed  per 

square  foot  per  degree 

per  hour. 


1005  units 

1071      „ 
1053      „ 


Averages, 

Coiled  pipe ;  evaporating  water : — 
961  units 


ist  Experiment, 
2d          Do. 
3d          Do. 


967 
967 


Averages, 
Cast-iron  boiler ;  heating  water  :- 


ist  Experiment, 
2d          Do. 


1059  units 
1063    „ 


Averages, 

Cast-iron  boiler ;  evaporating  water :  — 
961  units         x 


ist  Experiment, 
2d         Do. 


Averages, 


.264  lb. 
.271  „ 
.270  „ 


.iiSlb. 

•174,, 
.084  „ 

126 


.066  lb. 
.088  „ 


.077 


.091  lb. 
.118,, 

.105  „ 


Heat  transmitted  per 

square  foot  per  degree 

per  hour. 


265. 3  units. 
290.2     „ 
284-3     „ 


280.0 


1 10.5  units. 
168.2     „ 
81.2     „ 

120.0 


69.90  units. 
93-54     „ 

81.72     „ 


87.45  units. 
113.40     „ 


100.43 


Mr.  William  Anderson1  gives  the  results  of  experiments  on  the  power 
of  sugar-clarifiers  in  heating  water.  They  were  6  feet  6^  inches  in 
diameter,  and  2  feet  6  inches  deep;  containing  a  copper  pan  18  inches 
deep,  bolted  into  cast-iron  steam-jackets,  with  a  working  capacity  of 
450  gallons,  and  having  a  heating  surface  of  52.58  square  feet.  The 


1  "On  the  Aba-el- Wakf  Sugar  Factory,  Upper  Egypt."     Proceedings  of  the  Institution 
of  Civil  Engineers,  vol.  xxxv.,  1872-73. 

30 


466  APPLICATIONS   OF   HEAT. 

average  results  of  three  experiments  in  heating  water  to  212°  are  as 
follows : — 

Mean  duration  of  the  experiments 24  minutes. 

Mean  initial  temperature  of  water 67°  F. 

Mean  steam  pressure  above  atmosphere  ...42.1  Ibs.,  289°  F. 

Mean  weight  of  condensed  steam 742     „ 

Mean  weight  of  water  heated 4558     „ 

units. 

Units  of  heat  in  condensed  steam 742     „  x  990°  734,580 

Heat  spent  in  heating  copper 840     „  x  145°  x. 095=    11,571 

„  „  cast  iron 2828     „  x  145° x. 129=    52,900 

„  „  wrought  iron 567     ,,xi45°x.ii3=     9,200 

„  water 4558     „  x  145°  =660,910 

734,671 

units. 

Units  of  heat  per  square  foot  per  difference  of  i°  per  hour  in 

heating  water 2 10.2 

Loss  in  heating  clarifier,  radiation,  &c u.i  per  cent. 

The  mean  temperature  of  the  water  was  — -  =  140°,  and  the  heat 

utilized  per  pound  of  steam  was  io62°(=ii7o  +  32-  140).     Then, 

2ICX2  =  .i981b.  of  steam, 
1062 

condensed  per  square  foot  per  degree  per  hour. 

In  other  experiments,  with  a  smaller  clarifier,  similar  in  construction,  of 
1 2  gallons  of  capacity,  the  trials  were  carried  further,  and  the  rate  of  boiling 
was  ascertained,  both  for  water  and  for  sirup,  the  latter  consisting  of  a 
solution  of  9  Ibs.  of  molasses  and  4  Ibs.  of  sugar  in  90  Ibs.  of  water,  equal 
to  juice  at  about  8°  Beaume.  The  quantities  of  heat  passed  through  the 
metal  were  as  follows : — 

Water.        Juice, 
units.  units. 

In  heating,  per  square  foot  per  difference  of  i°  F.  per  hour 260        219 

In  evaporating,  „  „  „  606         521 

showing  a  greatly  accelerated  passage  of  heat  when  evaporating,  2  y$  times 
as  much  as  in  only  heating  the  water:  also,  that  the  addition  of  14^  per 
cent,  of  sugar  reduced  the  efficiency  of  the  surface  by  about  1 5  per  cent. 

Mr.  Anderson  made  similar  trials  to  test  the  efficiency  of  the  concen- 
trators, for  their  evaporating  powers.  It  is  only  necessary  to  state  here  that 
each  of  the  concentrators  consists  of  a  copper  tray  23  feet  long  by  6  feet  wide, 
%  of  an  inch  thick,  heated  by  a  steam-boiler  beneath  it,  and  forming  part 
of  it.  The  boiler  is  12^  inches  deep,  flat-bottomed,  and  stayed  to  the 
tray  at  6  inches  pitch.  The  heating  surface  of  the  tray  is  increased  by  495 
upright  hollow  nozzles  of  brass,  screwed  into  it,  very  thin,  and  slightly  taper; 
average  external  diameter  2^  inches,  vertical  projection  4^  inches.  The 
tray  is  inclosed  by  a  sheet-iron  cover.  The  heating  surface  of  the  tray 
consisted  of  138  square  feet  horizontal  surface,  and  187  feet  of  vertical 
surface,  together,  325  square  feet.  By  experiment,  it  was  found  that  surfaces 
similar  to  those  of  the  tray  performed  as  follows : — • 


HEATING  AND   EVAPORATION    OF   LIQUIDS.  467 

In  heating  water  to  the  boiling  point,  5.8  Ibs.  effective  pressure 
per  square  inch,  228°  F.,  per  square  foot  per  i°  F.  difference 
per  hour 368  units. 

In  evaporating 660     „ 

Here  the  passage  of  heat  for  evaporation  was  1.8  times  as  much  as  in  heating 
without  evaporation.  Applying  this  ratio  to  the  performance  of  the  tray 
itself,  Mr.  Anderson  calculates  that  the  efficiency  of  the  tray  by  experiment 
was — 

For  heating 271  units. 

For  evaporation 49 1     „ 

The  obviously  superior  efficiency  of  the  model  is  accounted  for  by  its 
having  been  fully  charged  with  steam  from  the  factory  boilers;  "whilst  in 
the  actual  tray  the  generator  was  evidently  unequal  to  the  work." 

The  mean  pressure  in  the  generator  was  47  Ibs.  effective,  with  the 
temperature  294°  F.;  and  that  in  the  tray  was  5.8  Ibs.,  temperature  228°  F. 
The  total  heat  of  the  first  steam  was  1171°  from  32°  F.,  or  975°  above  228°; 
and  the  quantity  of  steam  condensed  per  square  foot  per  degree  per  hour 
was — 

Condensed. 

For  heating 271  —  975  =  0.278  pound  of  steam. 

For  evaporating 491-975  =  0.504       „  „ 

Mr.  F.  J.  Bramwell,  in  discussing  Mr.  Anderson's  paper,  gave  particulars 
of  similar  experiments  made  by  him  with  a  jacketed  copper  pan,  having  a 
working  capacity  of  100  gallons,  and  a  heating  surface  of  25  square  feet. 
The  pan  had  been  at  work  for  eight  or  nine  years,  and  probably  was 
incrusted  on  the  steam  side.  He  tried  the  performance  of  the  pan  with 
steam  successively  of  5  Ibs.,  10  Ibs.,  15  Ibs.,  and  20  Ibs.  effective  pressure, 
raising  the  temperature  of  the  water  from  58°  to  212°,  and  evaporating  it. 
In  the  first  experiment,  with  5  Ib.  steam,  he  found  that  the  rates  of  trans- 
mission of  heat  per  square  foot  per  degree  of  difference  per  hour,  taking 
observations  every  five  minutes,  in  raising  the  temperature  to  200°,  were 
successively  161,  151,  176,  160  units  of  heat,  whilst  in  heating  from  200° 
to  212°  the  rate  advanced  to  327  units;  and,  when  ebullition  commenced, 
to  427  units.  The  observed  rates  at  the  different  pressures  are  subjoined 
for  comparison: — 

Effective  pressure  Initial  temper-  Average  rate  of  trans-    Average  rate  of  transmission, 

of  steam.  ature  of  water.  mission  up  to  212°.  evaporating  at  212°  F. 

5  Ibs 58°  F —  427  units. 

10  Ibs 58         1 86  units  435     „ 

15  Ibs 58         —  458     » 

20  Ibs 58         205     „  488     „ 


Averages, 196     „      452     » 

From  these  results  it  appears  that  the  rate  of  transmission  for  evapora- 
tion is  more  than  double  the  rate  for  heating;  and  the  detailed  observa- 
tions, at  5  Ibs.  pressure,  show  a  marked  acceleration  of  transmission  when  the 
water  was  within  12°  of  the  boiling  point.  These  experiments  confirm 
those  of  Mr.  Anderson,  and  it  is  very  probable  that  the  greater  agitation 


468 


APPLICATIONS   OF   HEAT. 


and  quicker  circulation  of  the  water  as  it  neared  the  boiling  point,  and 
whilst  boiling,  was  the  cause  of  the  increased  rate  of  transmission  of  the  heat. 
The  average  rates  above  given  show  that,  per  square  foot  per  degree  per 
hour,  the  quantities  of  steam  condensed  were : — 

Condensed. 

In  heating  up  to  212°, 201  Ib. 

In  evaporating  at  212°, 463   „ 

The  various  results  of  performance  above  detailed  are  numbered  and 
collected  in  table  No.  160,  and  the  averages  for  copper-plate  surfaces, 
copper-coil  surfaces,  and  cast-iron  surfaces  are  given  in  the  lower  part  of 
the  table. 

Table  No.    160. — RESULTS  OF  PERFORMANCE  OF  COILED  PIPES  AND 
BOILERS  IN  HEATING  AND  EVAPORATING  WATER  BY  STEAM. 


AUTHORITY. 

APPARATUS. 

Steam  condensed 
per  square  foot,  for 
i°  F.  difference  of  tem- 
perature per  hour. 

Heat  transmitted 
per  square  foot,  for 
ic  F.  difference  of  tem- 
perature per  hour. 

Heating. 

Evapo- 
rating. 

Heating. 

Evapo- 
rating. 

i.  Clement  .... 
2.  Peclet  

Copper  plate 

Ibs. 
.l6o 

.313 
.292 

.268 
.077 
.198 
.278 

.2OI 

Ibs. 

.981 
1.  2O 
.126 
.105 

•5°4 
•463 

units. 
173 

335 
312 

280 
82 

210 
271 

368 
196 

units. 

Copper  boiler. 

948 
1120 
120 
100 

49  1 
660 

452 

3.  Laurens  
4.      Do  
5.  Havrez  
6.     Do  

7.  Anderson... 
8.       Do.      ... 

9.       Do.      ... 
10.  Bramwell... 

Copper  coil     

2  Do     do  

Copper  coil  

Cast-iron  boiler  

Copper  clarifier 

Copper  concentrator.. 
(  Copper   concentra-  ) 
\      tor  (model)  J 
Copper  pan  

Averages    for    c 
Nos.  2,  7,  8, 
Averages     for 
Nos.  3,  4... 

:opper-  plate    surface,  ) 

9,   10.                                       f 

.248 

.292 

.077 

•483 

1.090 
.105 

276 

3I2 

82 

534 
1034 

100 

copper-pipe     surface,  ) 

Cast-iron  plate  s 

urface,  No.  6  

Note. — Nos.  i  and  5  are  omitted  from  the  averages  as  the  information  is  incomplete, 
and  for  No.  5,  the  results  are  not  consistent. 

It  appears  that  the  efficiency  of  copper-plate  surface  for  evaporation  is 
double  its  efficiency  for  heating  water ;  for  copper-pipe  surface  the  efficiency 
is  more  than  three  times  as  much;  and  for  cast-iron  plate  surface,  a  fourth 
more. 

That  the  efficiency  of  pipe-surface  is  a  fifth  more  than  that  of  plate- 
surface  for  heating,  and  more  than  twice  as  much  for  evaporation. 


COOLING   OF   HOT  WATER   IN   PIPES.  469 

That,  in  round  numbers,  copper-plate  surface  condenses  half  a  pound  of 
steam,  copper  pipe  condenses  a  pound  of  steam,  and  cast-iron  plate-surface 
a  tenth  of  a  pound,  per  square  foot  per  degree  of  difference  of  temperature 
per  hour,  for  evaporation. 

That  the  quantity  of  heat  transmitted  is  at  the  rate  of  about  1000  units 
per  pound  of  steam  condensed. 

These  are  the  results  to  be  expected  when  the  surfaces  are  in  good 
condition. 

COOLING   OF   HOT  WATER    IN    PIPES. 

M.  Darcy  states  that  the  water  from  the  artesian  wells  at  Crenelles 
passed  underground  through  cast-iron  pipes  of  from  6%  to  10  inches 
diameter,  for  a  length  of  2530  yards,  in  8^  hours,  equivalent  to  an  average 
velocity  of  3  inches  per  second,  discharging  about  50  gallons  per  minute. 
The  water  was  cooled  from  80°  to  69°.$  F.,  or  io°.5;  being  at  the  rate  of 
i°.24  per  hour.  The  loss  of  heat  amounted  to  307,600  units  per  hour, 
which  passed  through  16,424  square  feet  of  surface:  at  the  rate  of  18.7 
units  per  square  foot  per  hour,  for  a  mean  temperature  of  75°.  When  at 
rest  in  the  pipe,  the  water  was  cooled  at  the  rate  of  10°  F.  in  7  hours,  or 
21.6  units  per  square  foot  per  hour.  Taking  the  temperature  of  the  ground 
at  62°,  the  mean  difference  of  temperature  was  13°  F.,  and  the  heat  trans- 
mitted per  square  foot  per  degree  per  hour  was  18.7  +  13  =  1.44  units  when 
the  water  was  in  motion,  and  21.6-^13  =  1.66  units  when  the  water  was 
at  rest. 

Taking  the  results  of  experiments  by  Mr.  Tredgold  on  the  rate  of  cool- 
ing of  water  in  pipes,  in  air,  as  corrected  by  Mr.  Hood,  a  cast-iron  pipe 
30  inches  long,  2*4  inches  in  diameter  internally,  and  %  inch  thick,  was 
filled  with  water  at  152°  F.  It  exposed  a  surface  of  2  square  feet,  with 
a  surrounding  temperature  of  67°  F.;  and  the  quantity  of  water,  including 
an  equivalent  for  the  heated  iron,  was  172  cubic  inches,  or  6  Ibs.  weight. 
The  water  was  cooled  at  a  nearly  uniform  rate,  from  152°  to  140°  F.,  in 
the  following  times,  to  which  are  added  the  cooling  and  the  units  of  heat 
passed  per  minute : — 

State  of  cast-iron  surface.  Cooled  «?  F.  in     ^.^  ^Sf^fm^ute. 

1.  Ordinary  brown  (rusty), 15  minutes  ...  o°.8  F....0.8    x  3  =  2.4units. 

2.  Black  varnished, T4-53  »       ...0.83     ...0.83x3  =  2.5     „ 

3.  White,  two  coats  of  lead  paint,  15.33  „       ...0,78     ...0.78x3  =  2.34  „ 

To  reduce  these  results  to  the  general  standard  for  comparison : — 


Heat  passed  off 

Mean  tem- 
perature of 
water. 

Mean  difference 
of  temperature  of 
water  and  air. 

per  square  foot 
per  hour. 

per  square  foot  per 
degree  of  differ- 
ence per  hour. 

Fahr. 

Fahr. 

units. 

units. 

I 

146°    . 

.    7Q° 

2.4    x  60=  144 

1.823 

2 

..146 

.    70 

2   C      X  60=  I  HO 

.     I.QOO 

1.    , 

..146 

/  7 

.    70 

0                w           *Jfw 

2.^4  X  60=140.4 

•  y  wv- 
.     1.778 

From  other  experiments  by  Tredgold,  hot  water  was  cooled  in  vessels 
made  of  tinned  plate,  sheet  iron,  and  glass  from  180°  to  159°  F.,  in  a  room 
at  56°,  showing  an  average  excess  of  temperature  of  114°  F.  They  con- 


4/0  APPLICATIONS   OF   HEAT. 

tained  2.2  Ibs.  of  water,  including  an  equivalent  for  the  metal.     The  results 
were  as  follows  :  — 

g     r  Area  of          Time  to  cool          Cooled  per  Heat  passed  per  square 

surface.  30°  F.  minute.  foot  per  minute. 

square  feet.  minutes.  Fahr.  units. 

4.  Tinplate,  ......  55  ......  46       ......  o°.65  .......  65x2.2-    -55  =  2.60 

5.  Sheet  iron,..  .533  ......  29       ......  i°.c>3  ......  1.03  x  2.2  -  .533  ^4.26 

6.  Glass,  .........  500  ......  S1/^  ......  °°-94  .......  94  x  2.2  -    .50  =  4.14 


The  heat  passed  off  per  hour  was, 

4.   156.0  units  per  square  foot,  and  1.37  units  per  degree  of  difference. 

5-  255-6     „  »  »    2-24     >?  »  >, 

6.   248.4     „  „  „    2.18     „ 

To  group  the  experimental  results  adduced  for  the  transmission  of  heat 
from  hot  water  in  iron  pipes  and  vessels  to  the  external  air  :  — 

Per  square  foot  per 
degree  difference  of 
temperature  per  hour. 

2^  inch  cast-iron  pipe,  ^  inch  thick,  naked,  .....   1.82  units. 
Sheet-iron  vessel,  ........................................  2.24     „ 


Mean, 2.03     „ 

COOLING  OF  HOT  WORT  ON  METAL  PLATES  IN  AIR. 

The  results  of  experiments  on  the  cooling  of  wort  at  Trueman's  brewery 
are  recorded  in  Engineering,  vol.  vi.  Two  coolers,  no  feet  by  25  feet, 
made  of  thin  copper,  No.  15  wire-gauge,  or  T/i3  inch  thick,  were  supported 
on  open  joists,  and  air  was  free  to  circulate  above  and  below  the  coolers. 
The  total  cooling  surface  amounted  to  5500  square  feet.  The  wort  was  run 
over  the  coolers  in  a  thin  stream,  of  which  50  barrels  of  360  Ibs.  each  were 
cooled  from  212°  to  no0  F.  per  hour.  The  total  heat  passed  off  by 
evaporation  and  by  conduction  through  the  metal  was  50  x  360  x  (212°- 
iio°)  =  1,836,000  units  per  hour;  being  at  the  rate  of  334  units  per  square 
foot  per  hour. 

When  the  wort  was  left  to  stand  on  the  coolers,  from  2  to  2  ^  inches  deep, 
it  was  cooled  140°  in  from  six  to  eight  hours.  Taking  10  Ibs.  per  square 
foot  as  the  weight  of  the  water,  the  quantity  of  heat  passed  off  was 

140  x  10  .  - 

—  =  200  units  per  square  foot  per  hour. 


The  mean  temperature  of  the  wort  was,  in  the  first  case,  -       -1^  -  161 


and  in  the  second  case  212°-  =  142°.  The  mean  differences  of  tem- 
perature, taking  that  of  the  air  at  62°,  were  99°  and  80°,  and  the  heat 
passed  off  per  square  foot  per  degree  of  difference  of  temperature  per 
hour  was — 

For  the  flowing  wort, 334^99  =  3-37  units. 

For  the  still  wort, 200-^-80  =  2.50     „ 


COOLING   OF   HOT  WORT   BY  COLD  WATER. 


471 


COOLING  OF  HOT  WORT  BY  COLD  WATER  IN  METALLIC  REFRIGERATORS. 

From  the  instructive  discussion  of  the  principles  of  brewery  engineering 
in  Engineering,  vol.  vi.,  the  following  particulars  are  derived  of  the  perform- 
ance of  tubular  refrigerators,  in  which  cold  water  is  passed  through  thin 
metallic  tubes,  which  are  surrounded  by  the  wort  to  be  cooled.  The  water 
and  the  wort  are  moved  in  opposite  directions  in  such  a  manner  that 
the  cold  water,  on  its  entrance  into  the  refrigerator,  meets  the  cooled  wort 
just  before  it  leaves  the  refrigerators,  and  the  warmed  water  passes  away 
from  the  refrigerator  where  the  hot  wort  enters.  The  following  are  parti- 
culars of  the  performance  in  five  experiments : — 

Table  No.  161. — RESULTS  OF  PERFORMANCE  OF  METALLIC   REFRIGERA- 
TORS IN  COOLING  HOT  WORT  WITH  COLD  WATER. 


WORT. 

WATER. 

Area  of  cool- 
ing surface  of 
refrigerator. 

Specific 
gravity. 

Quantity 
passed 
through 
per  hour. 

Initial 
tempera- 
ture. 

Final 
tempera- 
ture. 

Cooled 
down. 

Quantity 
passed 
through 
per  hour. 

Initial 
tempera- 
ture. 

Final 
tempera- 
ture. 

Warmed 
up. 

Square  feet. 

Barrels. 

Fahr. 

Fahr. 

Fahr. 

Barrels. 

Fahr. 

Fahr. 

Fahr. 

i.   881 

-i 

33-9 

212° 

72° 

I400 

61.1 

6S° 

I69° 

104° 

2.   514 

I.IO4 

36.1 

155 

59 

96 

75-5 

54 

IOO 

46 

3-   5H 

1.  088 

36.6 

IQI 

59 

132 

99-5 

54 

100 

46 

4.   514 

1-035 

47-3 

193 

59 

134 

90.7 

54 

IOO 

46 

5-   5H 

I.OI8 

48.0 

I78 

59 

119 

102.0 

54 

IOO 

46 

Note  i. — A  barrel  contains  36  gallons,  or  360  Ibs.  of  water. 

2. — The  temperature  of  the  air  in  Nos.  2  and  4  was  44°  F.,  and  in  Nos.  3  and  5,  40°. 

Dealing  with  the  data  of  this  table,  the  following  are  the  mean  tempera- 
tures and  differences  of  temperature  of  the  wort  and  the  water,  with  the 
quantities  of  heat  transmitted  per  unit  of  surface,  temperature,  and  time : — 


Mean  temperatures 

Heat  transmitted  per  square 
foot  per  degree  per  hour. 

No.  of 
experiment. 

Mean  differ- 
ence of 
temperature. 

Of  wort. 

Of  water. 

Measured  by 
reduction  of 
temperature 

Measured  by 
increase  of 
temperature 

of  wort. 

of  water. 

Fahr. 

Fahr. 

Fahr. 

units. 

units. 

I 

I42° 

H7° 

25° 

78 

IO4 

2 

107 

77 

30 

81 

81 

3 

125 

77 

48 

7i 

67 

4 

126 

77 

49 

91 

59 

5 

118.5 

77 

41-5 

96 

79 

Averages, 

83-4 

78 

472  APPLICATIONS   OF    HEAT. 

To  show  how  the  quantities  of  heat  in  the  last  two  columns  are  calculated, 
take  the  first  example.  The  quantity  of  wort  passed  through  per  hour  was 
33.9  barrels  of,  say,  360  Ibs.  each,  neglecting  the  extra  specific  gravity; 
cooled  down  through  140°  F.,  the  cooling  surface  was  88 1  square  feet,  and 
the  mean  difference  of  temperature  of  the  wort  and  the  water  was  25°  F. 
Then, 

2?.o  x  -?6o  x  140       0  r , 

•££z — J ±-  =  78  units  of  heat, 

881  x  25 

passed  from  the  wort,  per  square  foot  per  i°  F.  difference  of  temperature 
per  hour.  Again,  61.1  barrels  of  water  were  warmed  up  through  104°  F. 
Then, 

61.1  x  160  x  104  •       r, 

— ^ *=  104  units  of  heat, 

881  x  25 

absorbed  by  the  water,  per  square  foot  per  i°  F.  per  hour,  and  similarly  for 
the  other  examples.  There  is  an  inconsistency  in  the  excess  of  heat  taken 
up  by  the  water,  as  calculated,  above  that  which  was  passed  from  the  wort 
in  the  first  example,  indicating  that  there  was  an  error  of  observation.  For 
the  second  example,  the  quantities  are  equal.  The  remaining  observations 
show,  reversely,  that  more  heat  passed  from  the  wort  than  was  taken  up  by 
the  water.  The  averages  of  all  the  examples  show  that  83.4  units  of  heat 
were  passed  from  the  wort,  and  78  units  were  absorbed  by  the  water,  per 
square  foot  per  i°  F.  difference  of  temperature  per  hour. 

It  is  well  to  note,  as  observed  in  Engineering,  that  the  rate  at  which  the 
wort  parts  with  its  heat  increases  generally  as  the  specific  gravity  is  less. 
This  acceleration  points  to  the  conclusion,  that  if  water  be  substituted  for 
wort,  the  rate  of  transmission  would  be  100  units  per  square  foot  per  i°  F. 
difference  per  hour;  although,  conversely,  the  cooling  water  would  absorb 
only  80  units.  The  difference,  20  units,  would  be  passed  off  by  radiation 
and  conduction. 


CONDENSATION   OF   STEAM   IN   PIPES   EXPOSED   TO   AIR. 

Tredgold  found,  by  experiment,  that  steam  of  an  absolute  pressure  of 
17.5  Ibs.  per  square  inch,  temperature  221°  F.,  produced  one  cubic  foot  of 
water  per  hour  by  condensation  in  iron  pipes  exposing  182  square  feet  of 
surface  in  a  room  at  60°  F.  The  difference  of  temperature  was  161°,  and 
the  condensation  per  square  foot  per  hour  was  .352  Ib.  of  water;  or,  per 
degree  of  difference  of  temperature,  .0022  Ib. 

Experiments  made  in  1859  by  M.  Burnat,  on  the  efficiency  of  coating 
for  cast-iron  steam  pipes,  afford  valuable  data  in  this  connection.1  The 
pipes  were  4.72  inches  in  diameter  externally,  and  ^  inch  thick;  they  were 
arranged  in  five  groups  of  four  pipes  each,  each  group  presenting  an  aggre- 
gate surface  of  58^  square  feet.  The  groups  were  placed  at  40  inches 
apart,  and  inclined  at  an  angle  of  i  in  20,  in  a  large  unheated  hall  free  from 
air-currents.  The  pipes  of  the  first  group  were  covered  with  straw  laid 
lengthwise  to  the  thickness  of  0.6  inch,  bound  with  straw  rope  laid  closely 
round  it.  The  second  group  were  left  bare  as  they  came  from  the 

1  Reported  in  Proceedings  oj  'he  Institution  of  Civil  Engineers,  vol.  xli.,  1874-75. 


CONDENSATION    OF   STEAM    IN    PIPES. 


473 


foundry.  In  the  third  group,  each  pipe  was  laid  in  a  pottery  pipe,  with  an 
air-space  between  the  two,  and  coated  with  a  mixture  of  loamy  earth  and 
chopped  straw,  covered  with  tresses  of  straw.  In  the  fourth  group,  the  pipes 
were  covered  with  cotton  waste  to  a  thickness  of  an  inch,  wrapped  in  cloth 
bound  with  string.  In  the  fifth  group,  the  pipes  were  coated  with  a  com- 
position of  clay  and  cow's  hair  to  a  thickness  of  2.36  inches.  Finally,  trials 
were  made  with  the  second  group  of  pipes  by  coating  them  with  some  old 
felt  which  had  been  treated  with  caoutchouc;  and  a  second  trial  of  the 
fifth  group,  after  the  composition  had  received  a  coat  of  white  paint.  The 
pipes  were  supplied  with  steam  of  from  16^  Ibs.  to  30  Ibs.  absolute  pressure 
per  square  inch;  and  each  experiment  lasted  from  40  to  56  minutes.  The 
results  of  the  experiments  are  given  in  the  annexed  table  No.  162. 

Table  No.  162. — RESULTS  OF  EXPERIMENTS  ON  THE  CONDENSATION  OF 
STEAM  IN  CAST-IRON  PIPES. 

(M.  Burnat.) 


Temperatures. 

Steam  condensed  per  square  foot  of  external  surface 
of  pipes  per  hour. 

Absolute  pres- 

sure of  steam 

per  square  inch. 

Steam. 

Air. 

Differenc'", 

Straw 
coat, 

Bare, 

Pottery 
coat, 

Waste 
coat, 

Plaster 
coat, 

ISt. 

ad. 

3d. 

4th. 

5th. 

Ibs. 

Fahr. 

Fahr. 

Fahr. 

lb. 

lb. 

lb. 

lb. 

lb. 

16-5 

2l8°.0 

46°.4 

i7i°.6 

•139 

.496 

.170 

.217 

-254 

I6.5 

218.0 

33-3 

184.2 

.152 

.485 

.166 

.205 

.262 

18.4 

223.4 

33-7 

189.7 

.164 

-555 

.186 

.229 

.287 

18.4 

223.4 

27.1 

196.4 

.182 

•571 

.264 

.287 

•344 

22.  0 

233-2 

4i.5 

191.7 

.246 

-576 

.258 

.244 

.320 

22.  0 

233-2 

36.5 

196.7 

.164 

.158 

.250 

22.  0 

233-2 

36.1 

197.1 

.162 

•557 

.178 

.260 

— 

22.0 

233-2 

28.9 

204.3 

.2OI 

.586 

.264 

.328 

.346 

25-7 

241.6 

43-3 

198.4 

.244 

-645 

.301 

•375 

-389 

25.7 

241.6 

36.5 

205.1 

.274 

— 

.285 

•369 

29.4 

249.1 

43-3 

205.8 

.252 

.721 

.270 

•342 

•379 

29.4 

249.1 

30.6 

218.4 

.225 

.621 

.250 

.328 

.336 

Averages, 

22.0 

233-1 

36.5 

196.6 

.2OO 

.581 

.229 

.286 

•324 

When  the  plaster  coat  'of  the  fifth  group  was  painted  white,  an  average  of 
0.307  lb.  of  steam  was  condensed  per  square  foot  of  pipe  per  hour;  and 
the  second  group,  with  the  felt  coating,  condensed  0.313  lb.  of  steam  per 
square  foot  per  hour. 

From  these  data  the  following  constants  have  been  derived,  for  an  absolute 
pressure  of  steam  of  22  Ibs.  per  square  inch;  for  the  quantity  of  steam  con- 
densed, and  the  quantity  of  heat  passed  off,  per  square  foot  of  external 
surface  of  pipe  per  hour  for  i°  F.  difference  of  temperature.  The  quantity 
of  heat  transmitted  per  pound  of  steam  is  the  difference  of  the  total  and 
sensible  heats  of  the  steam,  or  (1152.5  -f  32)  -  233.1  =  951.4  units: — 


4/4  APPLICATIONS   OF   HEAT. 

Steam  Condensed 
CoNomoN  o,  S™FACB.  ?Sf JSST  Payoff. 

Ib.  units. 

Bare,  or  uncovered  pipe 00300  2.812 

Coated  with  straw 00102  0.968 

Cased  in  pottery  pipes,  with  air  space 00115  1.108 

Coated  with  cotton-waste,  i  inch  thick 00146  1.384 

Coated  with  old  felt 00159  I-5I5 

Coated  with  plaster  of  loamy  earth  and  hair 00165  1.568 

The  same,  painted  white 00156  1.486 

The  most  effective  coat  for  the  prevention  of  condensation  was  the  straw 
coat,  and  that  it  had  the  effect  of  reducing  the  loss  by  condensation  to  one- 
third  of  that  which  took  place  with  the  naked  pipe.  With  the  naked  pipe, 
2.812  units  of  heat  were  transmitted  per  square  foot  per  degree  per  hour. 

In  experiments  by  Mr.  B.  G.  Nichol,  a  wrought-iron  pipe  3^  inches  in 
diameter  outside,  ^  inch  thick,  and  lagged  to  half  an  inch  thick  with  felt 
and  spun  yarn,  condensed  steam  at  245°  F.  at  the  rate  of  .262  Ib.  per  square 
foot  per  hour,  in  an  external  temperature  of  60°,  equivalent  to  1.26  units  of 
heat  per  square  foot  per  i°  difference  of  temperature. 

According  to  M.  Clement's  experiments,  the  quantities  of  steam  given 
in  the  second  column  below,  were  condensed  per  square  foot  of  pipe-surface 
per  hour,  in  a  temperature  of  7  7°  F.  Assuming  that  the  steam  condensed 
was  of  20  Ibs.  absolute  pressure,  the  difference  of  temperature  was  151°  F., 
and  the  weight  of  steam  condensed  per  i  °  F.  is  given  in  the  third  column. 

c.TD.  Steam  condensed  per  square  foot 

per  hour, 
total.  per  i°  F. 

Bare  cast-iron  pipe,  horizontal 328  Ib.  .002 1 7  Ib. 

Blackened      do.  do 308  „  .00204   „ 

Bare  copper  pipe,          do 267   „  .00177    „ 

Blackened      do.  do 308  „  .00204   » 

Do.  do.        upright 359   „  .00238   „ 

Here  it  appears  that  the  blackened  surfaces  of  iron  and  of  copper  were 
equally  active;  and  that  the  upright  pipe  condensed  more  steam  than  the 
same  pipe  laid  horizontally. 

Mr.  Grouvelle  found  that,  in  a  temperature  of  60°  F.,  a  square  foot  of 
pipe  heated  by  steam  condensed  0.328  Ib.  of  steam  per  square  foot  per 
hour.  Assuming  that  the  steam  was  of  20  Ibs.  absolute  pressure,  the  differ- 
ence of  temperatures  was  228°  —  60°  =  168°  F.;  and  0.0020  Ib.  of  steam  was 
condensed  per  square  foot  per  i°  F. 

Summarizing  the  several  results  above  for  bare  cast-iron  pipes: — 

Difference  of  temperature.      Steam  condensed  per  square  foot  per  hour, 
total.  per  i°  F. 

Tredgold.... 161°  F.  0.352  Ib.  .0022    Ib. 

Burnat 196.6  0.581    „  .0030     „ 

Cle'ment  151  0.328  „  .00217  „ 

Grouvelle 168  0.328  „  .0020     „ 


Average,  say,  for  steam  1 

of  20  Ibs.  absolute  >  169  0.400  „         .00235   »    Sa7  x/42o 

pressure j 


CONDENSATION    OF  VAPOURS   IN   PIPES.  475 

To  find  the  quantity  of  heat  dissipated  by  the  condensation  of  J/42o  lb. 
of  steam:  —  the  difference  of  the  sensible  and  total  heats  of  one  pound  of 
steam  of  20  Ibs.  absolute  pressure,  is  the  latent  heat,  954  units;  and 
954^-420  =  2.27  units,  the  heat  dissipated  per  square  foot  of  surface  per 
i°  F.  difference  of  temperature  per  hour. 

To  compare  the  condensing  power  of  still  air  with  that  of  still  water,  and 
referring  to  the  contents  of  table  No.  160,  page  468,  in  the  absence  of 
records  of  experiments  made  under  exactly  the  same  conditions,  it  may  be 
inferred  that  the  rate  of  condensation  in  thin  pipes  in  air  is  to  that  in 
water  below  the  boiling  point,  per  unit  of  surface,  temperature,  and  time,  as 
2.26  units  to  312  units,  or  as  i  to  138.  M.  Peclet  takes  the  ratio  as  i  to 
200,  though  so  high  a  ratio  is  scarcely  warranted  by  the  evidence. 

Condensation  of  Steam  in  a  Boiler  Exposed  in  Open  Air.  —  Messrs.  Fox, 
Head,  &  Co.,  Middlesborough,  made  comparative  experiments  with  a 
steam-boiler  on  their  premises,  in  two  conditions  —  naked,  and  covered 
with  non-conducting  cement.  From  an  account  of  the  experiments  in 
Engineering,  vol.  vi.,  it  appears  that  steam  of  50  Ibs.  absolute  pressure  per 
square  inch  was  maintained,  and  that  the  effect  of  removing  the  covering 
was  that  one  cubic  foot  of  water  converted  into  steam  was  condensed  by 
50  square  feet  of  exposed  boiler-surface  per  hour.  This  is  equivalent  to 
1.25  Ibs.  of  steam  per  square  foot  per  hour.  The  weather  was  fine,  and 
taking  the  temperature  of  the  open  air  at  62°  F.,  that  of  the  steam  was 
298  -  62°  =  236°  above  the  atmospheric  temperature;  and  the  rate  of  con- 
densation per  square  foot  per  degree  of  difference  of  temperature  per  hour 
was, 

i.  25-^236  =  .0053  lb. 


The  latent  heat  of  one  pound  of  the  steam  was  904°  and  904  xi.25 
units  of  heat  transmitted  per  square  foot  per  hour.  The  quantity  of  heat 
transmitted  per  square  foot  per  degree  of  difference  of  temperature  per 
hour  was, 

1130-^  236  =  4.79  units. 

This  is  more  than  three  times  as  much  as  was  found  to  be  transmitted  in 
the  still  air  of  a  room. 

CONDENSATION  OF  VAPOURS  IN  PIPES  OR  TUBES  BY  WATER. 

The  condensation  of  vapours  by  the  application  of  cold  water  or  air, 
is  in  principle  the  same  as  the  heating  of  water  or  air  by  steam  ;  and  the 
same  proportions  for  condensing  surface,  when  steam  is  to  be  condensed, 
are  applicable  in  the  two  cases. 

The  surface-condenser  of  a  steam-engine  is  a  case  in  point.  To  educe 
the  constant  quantity  of  heat  transmitted  per  unit  of  surface,  temperature, 
and  time,  close  analysis  of  the  indicator-diagram  would  be  required,  to 
follow  exactly  the  variations  of  pressure  and  temperature  of  the  condensing 
steam.  From  the  investigations  of  M.  Audenet,1  of  the  action  of  the 
surface-condensers  on  board  the  transport  ship  Dives,  it  appears  that  500 
English  units  of  heat  were  transmitted  per  square  foot  per  i°  F.  difference 
of  temperature  per  hour.  These  condensers  were  arranged  in  three  groups 

1  Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xxxix.,  1874-75,  p.  399. 


476  APPLICATIONS   OF   HEAT. 

of  tubes,  successively  traversed  by  the  water.  For  the  condensers,  arranged 
in  two  groups,  on  board  the  Rochambeaii,  the  constant  was  only  from  220 
to  240  English  units. 

A  valuable  series  of  experiments  on  the  surface-condensation  of  steam  was 
made,  in  1875,  by  Mr.  B.  G.  Nichol,  at  the  Ouseburn  Engine  Works,  New- 
castle.1 A  brass  tube,  ^  inch  in  diameter  outside,  and  No.  18  wire-gauge 
in  thickness,  was  inclosed  in  an  iron  pipe  3^  inches  in  diameter  outside, 
y±  inch  thick,  and  5  feet  5  ^  inches  long  between  the  ends.  The  brass 
tube  exposed  an  external  condensing  surface  of  1.0656  square  feet.  Steam 
was  admitted  into  the  pipe,  and  was  condensed  by  cold  water  passed 
through  the  tube.  The  pipe  was  lagged  with  felt  and  wrapped  with  white 
spun  yarn  to  a  diameter  of  4^  inches.  It  was  tested  for  the  radiation  of 
heat  from  its  external  surface,  which  had  an  area,  including  the  ends,  of 
5.48  square  feet;  the  inner  tube  having  been  sealed  up  during  the  test.  It  was 
found  that  steam  of  an  average  temperature  of  245°  F.  was  condensed  in 
the  pipe  at  the  rate  of  1.4375  Ibs.  per  hour,  equivalent  to  .262  Ib.  per 
square  foot  of  surface.  The  heat  transmitted  was  (total  heat  1154  +  32) 
-  245  =  941  units  per  pound  of  steam  condensed;  and  it  was  (941  x  .262) 
=  246.5  units  per  square  foot  per  hour.  The  external  temperature  in  the 
workshop  was  60°  F. ;  the  difference  of  internal  and  external  temperatures 
was  245°  — 60°  =  195°;  thence  the  radiation  per  degree  of  difference  of 
temperature  was  246.5  ^-195  =  1. 26  units  per  square  foot. 

The  temperature  oif  the  steam  introduced  for  experiment  into  the  pipe, 
was  about  255°  F.,  for  a  total  pressure  of  32.5  Ibs.  per  square  inch,  and  the 
initial  temperature  of  the  condensing  water  was  58°.  Two  series,  of  three 
experiments  each,  were  made  with  the  pipe  in  a  vertical  and  in  a  horizontal 
position.  The  following  are  the  principal  results  of  the  six  experiments : — 

Vertical  Position.  Horizontal  Position. 

I23  456 

Steam  condensed  per  square  foot  of  tube  per  hour, — 

52.32,     78.18,     84.34,                   67.8,       104.6,       121.3  pounds. 
Condensing  water  passed  through  tube  per  square  foot  per  hour, — 

659,       2272,      3184,                    633,        2505,        3390  pounds. 
Condensing  water  per  pound  of  steam  condensed, — 

12.6,         29,        37.7,                     9.3,           24,           27.9  pounds. 
Velocity  of  water  through  the  tube  in  feet  per  minute, — 

81,         278,       390,                      78,         307,          415  feet. 
Final  temperature  of  condensing  water, — 

140°,      93°.5,       85°,                     165°,        101°,        94°.5  F. 
Rise  of  temperature  of  condensing  water, — 

82°,       35°-5,       27°,                     107°,         43°,          36°-5  F. 

1  An  excellent  account  of  these  experiments  was  published  in  Engineering,  of  December 
10,  1875,  fro™  which  the  principal  data  are  derived  for  this  notice. 


WARMING  AND   VENTILATION.  477 

Vertical  Position.  Horizontal  Position. 

123  456 

Mean  temperature  of  condensing  water, — 

99°,       75°-7,      7i°-5,  m°.5,      79°-5,        76°.2    F. 

Mean  difference  of  temperature  of  steam  and  condensing  water, — 

Heat  transmitted  from  steam,  reckoned  from  its  temperature,  per  square 
foot  per  hour, — 

45,960,  68,670,  74,040,  59,650,    91,950,    106,700  units. 

Heat  transmitted  from  steam,  reckoned  from  its  temperature,  per  square 
foot  per  hour,  per  i°  F.  difference  of  temperature, — 

295>        383>        4oi?  422,         530,          600     units. 

Heat  absorbed  by  the  water  per  square  foot  per  hour, — 

54,038,  80,656,  85,968,  67,731,  107,715,  123,735  units. 

Heat  absorbed  by  the  water,  per  square  foot  per  hour,  per  i°  F.  difference 
of  temperature, — 

346,        449,        466,  479,         621,          696      units. 

The  condensing  tube  acted  more  efficiently  in  the  horizontal  position 
than  in  the  vertical  position :  a  result  the  reverse  of  what  was  found  by  M. 
Clement,  condensing  in  air  (p.  474).  There  is  a  large  excess  of  heat  as 
carried  off  by  the  water,  above  the  heat  as  calculated  from  the  quantity  of 
steam  condensed.  In  this  calculation,  it  is  assumed  that  the  condensed 
steam  left  the  pipe  at  the  temperature  of  the  steam;  but  very  probably 
the  water  was  reduced  within  the  pipe  more  nearly  to  the  temperature  at 
which  it  was  discharged — about  200°  F. 

It  appears,  further,  that  the  efficiency  of  the  condensing  surface  was  very 
much  increased  by  an  increase  of  velocity  of  the  water  through  the  tube. 

When  other  vapours,  as  those  of  alcohol,  are  to  be  condensed,  it  may  be 
assumed  for  purposes  of  general  comparison,  that  the  weight  of  vapour 
that  may  be  condensed  per  unit  of  surface,  temperature,  and  time,  will  be 
inversely  as  the  total  heat  of  the  vapour.  The  total  heat  of  vaporized 
alcohol,  by  table  No.  125,  page  372,  is  461.7  units,  which  is  about  4/10ths 
of  that  of  steam  at  one  atmosphere;  and  the  relative  weights  of  steam  and 
alcoholic  vapour,  at  this  pressure,  that  may  be  condensed  per  unit  of 
surface,  temperature,  and  time,  are  as 

461.7  to  1146.1,  or  as  i  to  2.5  nearly. 

WARMING  AND   VENTILATION. 
VENTILATION. 

Mr.  Hood  finds  that  in  winter  from  3^  to  5  cubic  feet  of  air  per  head 
per  minute  are  sufficient,  under  ordinary  conditions,  for  the  proper  ventila- 
tion of  apartments;  and  in  summer,  from  5  to  io  cubic  feet  per  minute. 
With  these  proportions  the  wholesomeness  and  purity  of  the  atmosphere 
are  maintained. 

These  proportions  agree  with  those  deduced  by  M.  Peclet;  according  to 


4/8 


APPLICATIONS   OF    HEAT. 


his  deductions,  3^  cubic  feet  of  air  per  head  per  minute  is  the  minimum 
that  should  be  provided,  in  ordinary  circumstances.  When  the  ventilation 
takes  place  by  numerous  apertures  from  below  upwards,  from  4  to  6^ 
cubic  feet  maintains  the  air  of  the  room  sufficiently  pure.  In  peculiar 
cases,  as  in  hospitals,  from  30  to  60  cubic  feet  of  air  per  bed  per  minute 
are  admitted. 

Ventilation  is  produced  by  natural  draft,  or  by  artificial  draft  produced 
by  mechanical  means.  The  second  method  will  be  considered  in  a 
subsequent  section.  With  respect  to  the  first,  the  ascensional  force  is 
measured,  as  it  is  with  hot  water,  page  484,  by  the  difference  in  weight 
of  two  columns  of  air  of  the  same  height,  the  height  being  measured  by 
the  total  difference  of  level  between  the  inlets  for  warm  air  and  the  outlets 
into  the  atmosphere.  The  difference  of  weight  is  ascertained  from  the 
difference  of  the  temperatures  of  the  ascending  warmer  air  and  the 
external  atmosphere,  by  the  aid  of  table  No.  115,  page  351;  or  for  inter- 
mediate temperatures,  by  the  formulas  (9),  page  350,  and  (2),  page  347. 
The  reasoning  that  is  applied  to  the  question  of  the  circulation  of  water- 
columns,  page  485,  is  applicable  to  that  of  air-columns.  Suffice  it  for  the 
present  to  reproduce  the  following  table,  No.  163,  by  Mr.  Hood,  showing 
the  rate  of  discharge  through  a  ventilating  opening  one  foot  square,  for 
various  heights  and  differences  of  temperature,  calculated  by  a  rule  like 
that  for  water  at  page  485;  and  subjected  to  a  reduction  of  one-fourth 
the  calculated  quantities,  to  comprise  the  necessary  corrections  for  the 
contaminations,  chiefly  carbonic  acid,  which  go  to  increase  the  specific 
gravity  of  the  current,  for  frictional  resistance,  and  for  the  resistance  of 
angular  deviations  : — 

Table  No.  163. — AIR  DISCHARGED  THROUGH  A  VENTILATOR  PER  SQUARE 
FOOT  OF  OPENING,  FOR  VARIOUS  HEIGHTS  AND  DIFFERENCES  OF 
TEMPERATURE. 


Excess  of  Temperature  of  the  Room  above  that  of  the  External  Air, 

Height  of  Ventilator 

in  Fahrenheit  degrees. 

5° 

10° 

15° 

20° 

25° 

30° 

feet. 

cubic  feet. 

cubic  feet. 

cubic  feet. 

cubic  feet. 

cubic  feet. 

cubic  feet. 

10 

116 

I64 

200 

235 

260 

284 

15 

142 

2O2 

245 

284 

318 

348 

20 

164 

232 

285 

330 

368 

404 

25 

184 

260 

318 

368 

410 

450 

30 

2OI 

284 

347 

403 

45° 

493 

35 

218 

306 

376 

436 

486 

53i 

40 

235 

329 

403 

465 

5i8 

570 

45 

248 

348 

427 

493 

551 

605 

50 

260 

367 

45° 

5i8 

579 

635 

The  velocity  of  the  draft  having  been  found  for  any  particular  case, 
together  with  the  quantity  of  air  to  be  supplied  per  minute,  the  sectional 
area  of  the  air  passages,  inlet  and  outlet,  may  be  simply  calculated  from 
those  data. 


VENTILATION   OF   MINES  BY  HEATED  AIR.  479 

"  In  all  methods  of  ventilation,"  says  Mr.  Hood,  "it  is  advisable  to  make 
the  aggregate  area  of  the  openings  that  admit  the  fresh  air  larger  than  the 
aggregate  openings  for  the  efflux  of  the  vitiated  air.  This  becomes  necessary 
notwithstanding  the  increase  of  volume  which  takes  place  in  the  heated  and 
vitiated  air.  If  the  opposite  course  be  adopted,  and  the  eduction-tubes 
be  larger  than  the  induction-tubes,  then  a  counter-current  takes  place  in  the 
hot-air  or  ventilating  tubes,  and  the  cold  air  descends  through  them;  but 
by  making  the  induction-tubes  numerous,  and  of  a  large  total  area,  the 
velocity  of  the  entering  current  is  reduced,  and  unpleasant  drafts  are 
avoided.  It  is  also  expedient  to  divide  the  entering  current  as  much  as 
possible;  for  by  so  doing,  it  prevents  the  dangerous  effects  of  cold  draughts, 
when  the  entering  current  is  colder  than  the  air  of  the  room;  and  when  it 
is  hotter  than  the  air  of  the  room  it  prevents  the  air  from  rising  too  rapidly 
towards  the  ceiling,  and  therefore  distributes  it  more  equally  throughout  the 
apartment.  Provided  the  aggregate  openings  for  the  admission  of  cold  air 
be  not  less  in  size  than  those  for  the  emission  of  the  heated  air,  the  quantity 
of  air  which  enters  a  room  depends  less  upon  the  size  or  number  of  the  open- 
ings which  admit  the  fresh  air  than  upon  the  size  of  those  by  which  the 
vitiated  air  is  carried  off." 

In  very  hot  weather  and  with  crowded  assemblies,  the  draft  is  assisted  in 
theatres  and  some  other  large  buildings,  by  heating  the  air  in  the  upper 
part  of  the  ventilating  tube,  which  materially  accelerates  the  upward 
current,  and  increases  the  influx  of  fresh  air.  The  heat  of  the  large  gasa- 
lier  in  the  centre  of  the  house  near  the  ceilings  of  theatres  is  thus  utilized 
for  ventilation. 

Another  mode  of  accelerating  the  draft  is  to  conduct  the  spent  air  into 
the  lower  part  of  a  vertical  shaft,  where  a  furnace  is  maintained  in  active 
combustion,  and  a  very  hot  column  of  air  is  maintained. 

VENTILATION  OF  MINES  BY  HEATED  COLUMNS  OF  AIR. 

Reserving  for  a  subsequent  section  the  consideration  of  mechanical  venti- 
lation, the  ventilation  of  a  mine  by  the  assistance  of  a  furnace  placed  at  the 
bottom  of  the  upcast  shaft  is  effected  by  the  heating  of  the  ascending  column 
of  air  and  other  gases  discharged  from  the  mine,  just  before  entering  the 
shaft,  by  burning  fuel.  The  furnace  should  be  as  low  down  as  possible, 
so  as  to  afford  the  longest  column  of  heated  air  that  may  be  got,  since  the 
velocity  of  draft  increases  as  the  square  root  of  the  height  of  the  column. 
The  furnace  should  be  so  constructed  that  all  the  air  from  the  mine  should 
pass  freely  under  and  over  the  grate.  The  grate  may  be  six  feet  in  length 
from  front  to  back,  but  only  the  first  four  feet  of  bar-surface  are  covered 
with  fuel;  and  with  air-space  round  the  arch,  the  radiant  heat  of  the 
furnace  is  economized.  There  is  a  great  loss  of  heat  by  lateral  conduction 
through  the  rock  and  the  walls  of  the  shaft.  When  shafts  are  dry  and  bricked 
throughout,  a  temperature  of  200°  F.  is  the  greatest  that  can  be  had 
economically.  Even  in  such  shafts  the  loss  of  heat  laterally  often  amounts 
to  a  fifth;  and  in  shafts  which  are  wet  and  unwalled,  the  loss  amounts  occa- 
sionally to  four-fifths  of  the  whole  of  the  heat  communicated.  According 
to  Mr.  Mackworth,  100°  F.  should  be  a  sufficiently  high  temperature  for 
good  ventilation;  it  is  relatively  economical,  and  does  not  do  much  injury 
to  machinery.  With  a  powerful  furnace,  and  in  the  absence  of  obstructions, 


480  APPLICATIONS   OF   HEAT. 

the  greatest  velocity  of  the  current  is  30  feet  per  second;  but  when  there  is 
machinery  in  the  shaft,  the  velocity  seldom  exceeds  10  feet  per  second. 
The  first  object  in  ventilation  is  to  produce  a  slow  perceptible  motion  in 
the  whole  of  the  air  of  the  mine.  At  a  velocity  of  30  feet  per  minute,  the 
flame  of  a  candle  is  just  perceptibly  deflected.  The  air  should  not,  if 
possible,  be  made  to  travel  faster;  for  the  resistance  of  the  sides,  and  leak- 
ages, increase  rapidly  as  the  velocity  is  increased.  The  air  should  be  heated 
uniformly,  but  slightly;  and  that  it  may  not  be  impeded,  the  furnace-drift 
should  be  5  or  10  fathoms  in  length,  and  should  rise  at  an  inclination 
of  i  in  4. 

One  pound  of  coal  of  average  composition,  when  completely  burned,  is 
capable  of  raising,  in  round  numbers,  600,000  cubic  feet  of  air  i°  F.  in 
temperature.  At  Hetton  Colliery,  where  there  are  three  furnaces,  of  which 
one  is  9  feet  wide,  and  two  are  8  feet  wide,  one  pound  of  coal  raises  the 
temperature  of  11,066  cubic  feet  of  air  62°  F.,  equivalent  to  the  raising  of 
1 1, 066  x  62  =  686,092  cubic  feet  i°  in  temperature. 

One  of  the  best  examples  of  furnace-ventilation  is,  or  was,  to  be  found 
at  Morfa  Colliery,  South  Wales.  The  furnace  is  6  feet  2  inches  wide,  at 
the  base  of  a  shaft  10  feet  in  diameter,  and  60  fathoms  deep:  it  delivers 
62,000  cubic  feet  of  air  per  minute,  raised  to  a  temperature  of  198°  F.  by 
the  combustion  of  5^  Ibs.  of  coal.  The  average  temperature  of  the 
ascending  column  at  a  depth  of  25  yards  down  the  shaft  was  observed 
to  be  1 88°,  just  before  coals  were  charged  on  the  grate;  two  minutes  after 
charging  the  temperature  was  196°;  three  minutes  after  charging,  196°;  and 
eight  minutes  after,  191°.  The  "drag"  or  draft  was  3^  Ibs.  per  square 
foot,  not  including  the  shafts.  The  useful  effect  was,  therefore, 

62,000  x  3.  s      .,    Oi 

— > ^  =  6.58  horse-power, 

33,000 

or  6^  horse-power,  as  estimated  by  Mr.  Mackworth;  from  which  he  infers 
that  i^  horse-power  was  obtained  by  one  pound  of  coal  per  minute. 
This,  reduced  to  the  ordinary  form  for  comparison,  is  equivalent,  for 
(5.25  Ibs.  x  60  =  )  315  Ibs.  of  coal  consumed  per  hour,  to  48  Ibs.  of  coal 
per  horse-power  per  hour.  At  Hetton  Colliery  it  is  found,  by  a  similar 
calculation,  that  40  Ibs.  of  coal  was  consumed  per  horse-power  per  hour, 
in  a  shaft  150  fathoms  deep. 

It  is  stated  that  a  consumption  of  one  pound  of  coal  per  minute  for 
furnace-ventilation  is  sufficient  for  a  mine  employing  300  men,  in  the 
hottest  summer  day. 

In  collieries  at  Wrexham,  the  waste-steam  of  the  engine  is  employed  to 
heat  the  air  in  the  upcast  shaft.  A  cage,  consisting  of  150  gas-pipes  united 
at  top  and  bottom  by  hollow  cast-iron  rings,  is  placed  in  the  lower  part  of 
the  shaft,  or  in  the  return  drift,  the  exhaust  steam  is  condensed  in  the  cage, 
and  a  temperature  of  80°  F.  is  thereby  maintained.1 

COOLING  ACTION  OF  WINDOW  GLASS. 

Mr.  Hood  states  that  one  square  foot  of  window  glass  will  cool  1.28 
cubic  feet  of  air  (say  at  62°  F.)  i°  F.  per  minute,  or  76.8  cubic  feet  per 

1  The  data  contained  in  the  above  notice  of  the  ventilation  of  mines  are  derived  from  a 
lecture  by  Mr.  H.  Mackworth,  reported  in  the  Colliery  Guardian,  in  1858. 


HEATING  ROOMS  BY   HOT  WATER.  481 

hour,  per  degree  of  difference  of  temperatures  of  the  internal  and  external 
air.  One  unit  of  heat  will  raise  the  temperature  of  55^  cubic  feet  of  air 
at  62°  F.  by  i°  F.,  from  which  it  follows  that  heat  is  transmitted  through 
window-glass  from  the  air  of  a  room  to  the  external  air,  at  the  rate  of 

1—L-  =  1.40  units, 
55-5 

per  square  foot  per  degree  of  difference  of  temperature  per  hour. 

The  relative  cooling  influence  of  wind,  or  air  in  motion,  on  glass,  was 
tested  by  exposing  the  bulb  of  a  thermometer,  which  was  raised  to  a  maxi- 
mum temperature  of  120°  F.,  to  a  current  of  air  at  68°,  moving  at  various 
velocities.  The  time  required  to  cool  the  thermometer  20°,  varied  inversely 
as  the  square  root  of  the  velocity. 

HEATING  ROOMS  BY  HOT  WATER. 

The  effect  of  hot  water  in  heating  air  is  a  function  of  the  respective 
specific  heats. 

The  average  specific  heat  of  water  between  32°  and 

212°  F.  is 1.005 

The  specific  heat  of  air  is ; .2377 

Ratio  of  densities  of  water  and  air  at  62°  & i  to    819.4 

Ratio  of  the  volumes  of  water  and  air  raised  i°  F. 

by  equal  quantities  of  heat  (i  to  819.4^.2377).  i  to  3465. 

From  this  it  appears  that  one  cubic  foot  of  water  will,  by  parting  with  i°  F. 
of  heat,  raise  the  temperature  of  3465  cubic  feet  of  air  at  62°  by  i°  F.;  or 
one  unit  of  heat  will  raise  55}^  cubic  feet  of  air  at  62°  by  i°  F. 

Mr.  Hood  estimates,  from  experiments  made  by  Tredgold,  that  the  water 
contained  in  an  iron  pipe  of  4  inches  diameter  internally  and  4^/2  inches 
externally,  loses  0.851°  F.  of  heat  per  minute  when  the  excess  of  its 
temperature  is  125°  F.  above  that  of  the  surrounding  air,  and  that  one  foot 
in  length  of  the  pipe  will  heat  222  cubic  feet  of  air  one  degree  per  minute 
when  the  difference  of  temperature  is  125°  F.  This  estimate  is  too  low,  as 
it  is  based  upon  too  high  a  value  for  the  specific  heat  of  air,  namely,  .2767. 
If  the  quantity  be  increased  in  the  inverse  ratio  of  the  assumed  and  the 
actual  specific  heat  of  air,  the  volume  of  air  raised  i°  by  one  foot  length  of 
four-inch  pipe,  when  the  excess  of  temperature  is  125°  F.,  will  be 

222  x  '27  ?  =  258  cubic  feet. 
.2377 

Assuming  that  the  rate  of  cooling  of  a  hot-water  pipe  is  proportional  to 
the  excess  of  temperature,  it  would  follow  from  the  observation  above 
recorded  that  when  the  temperature  of  the  pipe  is  147°  F.  above  that  of 
the  air  in  the  room,  it  falls  i°  in  a  minute. 

Let  /  =  the  temperature  of  the  pipes,  f  =  the  required  temperature  of 
the  room,  t"  =  the  temperature  of  the  external  air,  V  =  the  volume  of  air  in 
cubic  feet  to  be  warmed  per  minute,  and  /=  the  length  of  the  pipe  in  feet 
Then,  according  to  the  preceding  data, 

31 


482 


APPLICATIONS   OF   HEAT. 


222     (t-f) 
f  -t" 


/=.56V 
using  Mr.  Hood's  divisor  222.     But 

.rt'-t' 


(O 


t-t' 


..   (i  a) 


using  the  divisor  258.     Whence  the  rule: — 

RULE. — To  find  the  length  of  four-inch  pipe  required  for  heating  the  air  in 
a  building.  Multiply  the  volume  of  air  in  cubic  feet  to  be  warmed  per 
minute,  by  the  difference  of  temperature  in  the  room  and  the  external 
temperature,  and  by  0.56  (Mr.  Hood),  or  by  0.50  (the  author),  and  divide 

Table  No.   164. — LENGTH  OF   FOUR-INCH    PIPE   TO  HEAT    1000  CUBIC 
FEET  OF  AIR  PER  MINUTE. 

Temperature  of  the  Pipe,  200°  F. 


EXTERNAL 
TEMPERA- 
TURE. 

•              TEMPERATURE  OF  THE  ROOM. 

45° 

5o° 

55° 

60° 

65- 

7o° 

75° 

80° 

85° 

90° 

Fahrenheit. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

feet. 

10° 

126 

J5o 

174 

2OO 

229 

259 

292 

328 

367 

409 

12 

119 

142 

166 

192 

220 

251 

283 

318 

357 

399 

14 

112 

135 

159 

184 

212 

242 

274 

309 

347 

388 

16 

105 

127 

J51 

I76 

204 

233 

265 

300 

337 

378 

18 

98 

120 

143 

168 

195 

225 

256 

290 

328 

368 

20 

91 

112 

135 

160 

I87 

216 

247 

28l 

3i8 

358 

22 

83 

i°5 

128 

152 

179 

207 

238 

271 

308 

347 

24 

76 

97 

120 

144 

170 

199 

229 

262 

298 

337 

26 

69 

90 

112 

136 

l62 

190 

220 

253 

288 

327 

28 

61 

82 

104 

128 

154 

181 

211 

243 

279 

3i7 

30 

54 

75 

97 

1  20 

145 

173 

202 

234 

269 

307 

32 

47 

67 

89 

112 

137 

164 

193 

225 

259 

296 

34 

40 

60 

81 

IO4 

129 

155 

184 

215 

249 

286 

36 

32 

52 

73 

96 

120 

147 

175 

2O6 

239 

276 

38 

25 

45 

66 

88 

112 

138 

166 

196 

230 

266 

40 

18 

37 

58 

80 

104 

129 

157 

I87 

220 

255 

42 

10 

30 

50 

72 

95 

121 

148 

I78 

210 

245 

44 

3 

22 

42 

64 

87 

112 

i39 

168 

200 

235 

46 

— 

15 

34 

56 

79 

I03 

130 

159 

I9O 

225 

48 

— 

7 

27 

48 

70 

95 

121 

i5° 

181 

214 

50 

— 

— 

19 

40 

62 

86 

112 

140 

171 

204 

52 

— 

— 

ii 

32 

54 

77 

I03 

131 

161 

194 

HEATING   ROOMS   BY   HOT  WATER. 


483 


the  product  by  the  difference  of  the  internal  temperature  and  that  of  the 
pipes.     The  quotient  is  the  length  of  pipe  in  feet. 

Mr.  Hood.      Author. 

Note. — For  three-inch  pipes,  use  the  multiplier  0.75,  or  0.67. 
For  two-inch  pipes, 


do.         do. 


1. 12,  or  i. oo. 


The  table  No.  164,  composed  by  Mr.  Hood,  shows  the  length  of  four- 
inch  pipe  required  to  heat  1000  cubic  feet  of  air  per  minute,  when  the 
temperature  of  the  pipe  is  200°  F. 

Total  Quantity  of  Air  to  be  Warmed  per  Minute. — In  habitable  rooms  the 

Table  No.  165. — LENGTH  OF  FOUR-INCH  PIPE  REQUIRED  TO  WARM 

ANY  BUILDING. 


(fl 

Building. 

Length  of  Pipe 
per  1000  cubic 
feet. 

Temperature 
maintained. 

Remarks. 

Churches   and   large  ) 
public  rooms             J 

feet. 

5 

Fahrenheit. 

55° 

{In    very    cold    weather. 
If  the  air  is  regularly 
changed,  from  50  to  70 

Dwelling-rooms  

I  2 

6c 

per  cent,  more  pipe  is 
required. 

Do  

14. 

v  j 

7O 

Halls,  shops,  waiting-  ) 
rooms  &c                 j 

10 

55 

Do.         do. 
Work-rooms,    manu-  ) 
factories,  &c  J 

12 

6 

60 

5°  to  55 

Do.         do. 

Schools  and  lecture-  ) 
rooms.    ...         ..      J 

8 
6  to  7 

60 
55  to  58 

Drying-rooms  for  wet  \ 
linen,     &c.—  When  V 
empty...                   .  I 

150  to  180 

120 

Do.,  when  filled  

80 

Drying-rooms  for  cur-  \ 
ing    bacon,    drying  > 
paper,  leather,  hides  ) 
Greenhouses  and  con-  ) 
servatories                 J 

20 

35 

70 

55 

In  coldest  weather. 

Graperies  and  stove-  ) 
houses  j 

45 

65  to  70 

Do.,         do. 

Do.          do. 
Pineries,   hot-houses,  ) 
and  cucumber  pits  .  J 

5o 
55 

70  to  75 
80 

Do.,         do. 

Note  to  Table. — The  lengths  of  pipe  are  only  suitable  for  buildings  on  the  usual  plan 
and  of  ordinary  proportions. 


484 


APPLICATIONS   OF   HEAT. 


total  quantity  is  equal  to  from  3^  to  5  cubic  feet  per  minute  for  each 
person,  plus  the  equivalent  of  i  ^  cubic  feet  for  each  square  foot  of  glass. 

For  conservatories,  forcing-houses,  and  like  buildings,  the  quantity  of 
air  to  be  warmed  is  i  ^  cubic  feet  per  square  foot  of  glass  per  minute. 
The  radiation  of  heat  from  frames  and  sashes  made  of  metal  is  as  great  as 
from  glass.  The  surfaces  of  these  are  to  be  included  in  the  calculation. 
For  wood  frames,  deduct  one-eighth  from  the  gross  area  of  surface. 

Approximate  Rules  for  the  Length  of  Four-inch  Pipe  required  to  Warm 
any  Building. — Rules  are  deduced  by  Mr.  Hood  from  the  results  of  experi- 
ence, and  they  are  generally  useful  in  practice.  The  multipliers  are 
collected  in  the  table  No.  165. 

Proper  Diameter  of  Pipe. — The  four-inch  pipe  is  of  the  best  size  for  all 
horticultural  purposes.  For  most  other  purposes,  smaller  pipes  may 
generally  be  more  advantageously  employed. 

Loss  by  Sinking  Heating  Pipes  in  Trenches. — When  pipes  are  placed  in 
trenches  covered  with  grating,  the  loss  of  heat,  as  estimated  by  Mr.  Hood, 
amounts  to  from  5  to  7  per  cent.,  which  passes  into  the  ground. 


Motive  Power  of  Water  in    Circulation 


through 


Heating  Pipes. — The 

ascensional  force  is  measured  by  the  difference  in  weight  of  the  two 
columns  of  water  of  the  same  height,  ascending  and  descending  from  and 
to  the  boiler.  The  difference  of  weight  is  ascertained  from  the  difference 
of  the  average  temperatures  of  the  columns  from  which  the  respective 
densities  are  deduced  by  the  aid  of  table  No.  109,  page  339. 

The  following  table  showing  the  difference  of  weight  of  two  columns  of 
water  one  foot  high  at  various  temperatures,  which  is  calculated  by  Mr. 
Hood  by  Dr.  Young's  formula,  and  gives  practically  the  same  results  as 
Rankine's  formula,  table  No.  109,  page  339. 


Table  No.  166. — DIFFERENCE  OF  WEIGHT  OF  Two  COLUMNS  OF  WATER, 
EACH  ONE  FOOT  HIGH,  AT  VARIOUS  TEMPERATURES. 

Assumed  actual  Temperatures  from  170°  to  190°  F. 


i 

Difference  of 
Temperature  of 
the  two  Columns. 

Diameter  of  Pipe. 

Difference  of  weight 
per  square  inch. 

i  Inch. 

2  Inches. 

3  Inches. 

4  Inches. 

Fahrenheit. 

grains. 

grains. 

grains. 

grains. 

grains. 

2° 

1.5 

6-3 

14-3 

25.4 

2.028 

4 

3.1 

12.7 

28.8 

51-1 

4.068 

6 

4-7 

I9.I 

43-3 

76.7 

6.108 

8 

6-4 

25.6 

57-9 

102.5 

8.160 

10 

8.0 

32.0 

72.3 

128.1 

10.200 

12 

9.6 

38.5 

87.0 

154.1 

12.264 

14 

II.  2 

101.7 

180.0 

14.328 

1.6 

12.8 

51.4 

116.3 

205.9 

16.392 

18 

14.4 

57-9 

131.0 

231.9 

18.456 

20 

16.1 

64.5 

145-7 

258.0 

20.532 

HEATING  ROOMS  BY  HOT  WATER. 


485 


The  velocity  of  circulation  is  that  of  a  falling  body  due  to  the  difference 
of  height  of  two  columns  of  water  of  equal  weights  or  pressures  on  the  base, 
and  it  varies  as  the  square  root  of  the  difference  of  height.  The  velocity 
may  be  found  by  the  aid  of  table  No.  85,  page  280.  The  difference  of 
height  is  proportional  to  the  difference  of  volumes,  table  No.  109;  and  if 
the  mean  height  be  increased  in  the  same  proportion,  the  increase  will  be 
the  height  from  which  the  velocity  is  to  be  calculated.  For  example,  let 
the  mean  height  be  10  feet,  and  the  difference  of  average  temperatures  of 
the  two  columns  10°  F.,  say  between  170°  and  180°.  The  respective 
volumes  are  as  1.0269  an^  I-°3I> 


io  feet  x 


1.0269 


-  10.04  feet. 


Then  10.04  -  10  =  .04  f°ot> the  difference  of  height;  and  the  velocity  due  to 
this  height  is  1.61  feet  per  second,  or  96.6  feet  per  minute. 

If  the  height  be  20  feet,  the  difference  is  .08  foot,  for  which  the  velocity 
due  is  136.20  feet  per  minute. 

In  practice,  of  course,  the  velocities  due  are  not  attained,  nor,  at  least 
in  the  more  complex  forms,  nearly  attained.  The  actual  velocities  are,  in 
some  cases,  not  more  than  a  half  or  even  a  ninth  of  the  velocities  due  to 
gravity. 

Quantity  of  Coal  Required  to  Heat  the  Pipes. — Mr,  Hood  gives  the  fol- 
lowing table,  No.  167,  showing  the  quantities  of  coal  consumed  in  heating 
100  feet  of  pipe  for  various  differences  of  temperatures.  These  quantities 
are  based  on  the  results  of  experiments  by  Rumford  and  others  in  heating 
water  with  coal  as  fuel,  and  are  no  doubt  approximately  correct. 


Table  No.  167. — COAL  CONSUMED  PER  HOUR  TO  HEAT  100  FEET 

OF  PIPE. 

For  given  differences  of  temperature  of  the  pipe  and  the  air. 


Diameter 
of 
Pipe. 

Difference  of  Temperature  of  the  Pipe  and  the  Air  in  the  Room  in 
Fahrenheit  Degrees. 

ISO 

H5 

140 

135 

I30 

125 

120 

"5 

no 

105 

100 

95 

90 

85 

80 

inches. 
4 

3 

2 

I 

Ibs. 
4-7 
3-5 
2-3 
i.i 

Ibs. 

4.5 

3-4 

2.2 
I.I 

Ibs. 
4-4 
3-3 

2.2 
I.I 

Ibs. 
4.2 
3-i 

2.1 
I.O 

Ibs. 
4.1 

3-o 

2.0 
I.O 

Ibs. 

3-9 
2.9 

1-9 
0.9 

Ibs. 

3-7 

2.8 

1.8 

0.9 

Ibs. 

3-6 

2.7 

0.9 

Ibs. 

3-4 
2-5 

oi 

Ibs. 
3-2 
2.4 
1.6 
0.8 

Ibs. 

3-i 

2.3 

i-5 
jo.; 

Ibs. 
2.9 
2.2 
1.4 
0.7 

Ibs. 
2.8 
2.1 
1.4 
0.7 

Ibs. 
2.6 
2.0 

J-3 
0.6 

Ibs. 
2.5 

1.2 

0.6 

Boiler-Power. — One  square  foot  of  boiler-surface  exposed  to  the  direct 
action  of  the  fire,  or  three  square  feet  of  flue-surface,  will  suffice,  with  good 
coal,  for  heating,  in  round  numbers,  50  feet  of  pipe.  Mr.  Hood  fixes  the 
proportion  at  40  feet  of  four-inch  pipe  for  all  purposes.  The  usual  rate  of 
combustion  of  coal  is  about  10  Ibs.  or  n  Ibs.  of  coal  per  square  foot  of 
fire-grate,  and  at  this  rate,  20  square  inches  of  grate  suffice  for  heating 
40  feet  of  four-inch  pipe. 


486  APPLICATIONS   OF   HEAT. 

Four  square  feet  of  boiler-surface  exposed  to  the  direct  action  of  a  good 
fire  are  capable  of  evaporating  one  cubic  foot  of  water  per  hour.  The  best 
form  of  boiler  for  heating  purposes  is  shown  in  Fig.  126  annexed.  It  is 


Fig.  126. — Boiler  for  heating  purposes.  Fig.  127. — Boiler  for  heating  purposes. 

generally  made  of  wrought-iron  plates  rivetted  together.      Another  good 
form  is  shown  in  Fig.  127. 

French  Practice. — M.  Claudel  states  that  to  warm  a  factory  13  metres 
wide  by  3.25  metres  high  (43  feet  by  10.5  feet),  a  single  line  of  hot-water 
pipe  6^  inches  in  diameter  along  the  room  appears  to  be  sufficient,  the 
temperature  in  the  pipe  being  from  170°  to  180°  F.  He  adds  that,  in  prac- 
tice, the  water  being  at  180°  F.,  and  the  air  at  60°  F.,  making  a  difference  of 
120°  F.,  it  is  convenient  to  reckon  from  1.5  to  1.75  square  feet  of  water- 
heated  surface  as  equivalent  to  one  square  foot  of  steam-heated  surface,  and 
to  allow  from  8  to  9  square  feet  of  hot- water  pipe-surface  per  1000  cubic 
feet  of  room. 

M.  Grouvelle  affirms  that  four  square  feet  of  cast-iron  pipe-surface, 
whether  heated  by  steam  or  by  water  at  80°  or  90°  C.,  or  176°  to  194°  F., 
will  warm  1000  cubic  feet  of  workshop,  maintaining  a  temperature  of  60°  F. 
Steam  is  condensed  at  the  rate  of  0.328  Ib.  per  square  foot  per  hour. 

Perkins1  System. — This  system  consists  of  the  continuous  circulation  of 
water  through  endless  wrought-iron  tubes  of  ^-inch  bore  and  i  inch  out- 
side diameter,  proved  under  a  pressure  of  200  atmospheres.  The  tempera- 
ture of  the  water  at  the  upper  part  of  the  circuit,  varies  from  300°  to  400°  F., 
corresponding  to  pressures  of  from  4^  to  15  atmospheres.  The  tubes 
become  red-hot  in  the  furnace.  The  length  of  tube  in  the  furnace  is  a 
sixth  of  the  total  length  of  the  circuit.  Twenty  feet  of  length  are  allowed 
for  heating  1000  cubic  feet  of  capacity.  Taking  the  mean  diameter  ^  inch, 
this  gives  four  square  feet  of  surface  per  1000  cubic  feet.  Though  the 
heater  is  apparently  water-tight,  the  larger  sizes  are  subject  to  a  loss  of 
about  a  pint  of  water  in  eight  or  ten  days,  which  is  restored  by  means  of  a 
force-pump. 

M.  Gaudillot,  in  France,  manufactures  heaters  on  this  system  with  tubes 
of  from  1.20  to  1.60  inches  in  external  diameter.  They  support  a  pressure 
of  40  atmospheres  very  well. 

HEATING  ROOMS  BY  STEAM. 

To  find  the  length  of  pipe  required  for  heating  a  room  by  steam,  the 
temperature  of  the  steam,  which  varies  with  the  pressure,  and  may  be  found 
in  table  No.  128,  page  387,  is  to  be  employed  for  the  value  of  t  in  the 
formulas  ( i )  and  ( i  a\  page  482.  The  length  of  pipe  required  for  heating 
by  steam,  is  of  course  less  than  that  required  with  water,  as  the  temperature 


HEATING  ROOMS   BY   STEAM.  487 

is  much  higher.  Taking  a  standard  absolute  pressure  of  steam  of  20  Ibs. 
per  square  inch,  the  temperature  is  228°;  and  if  the  room  is  to  be  heated 
to  60°,  the  difference  is  168°,  and  the  formula  (  i  a),  page  482,  becomes 


'         f" 

(2) 


- 

336 

RULE.  —  To  find  the  length  of  fotir-inch  pipe  required  for  heating  the  air  in 
a  building  by  steam  of  20  Ibs.  absolute  pressure  per  square  inch.  Multiply 
the  volume  of  air  in  cubic  feet  to  be  warmed  per  minute,  by  the  difference 
of  the  external  and  internal  temperatures,  and  divide  the  product  by  336. 
The  quotient  is  the  length  of  pipe  in  feet. 

Note.  —  For  three-inch  pipes  use  the  divisor  ............  252 

For  two-inch  pipes          „          „      ............  168 

For  one-inch  pipes          „          „      ............  84 

The  boiler  for  a  steam-heating  apparatus  should  be  capable  of  evapor- 
ating as  much  water  per  hour  as  the  pipes  would  condense  in  the  same 
time.  Mr.  Hood  recommends  that  six  square  feet  of  direct  surface  of 
boiler  should  be  provided  to  evaporate  a  cubic  foot  per  hour.  Now,  adopt- 
ing the  mean  weight  of  steam  of  20  Ibs.  absolute  pressure  condensed  per 
square  foot  of  pipe  per  degree  of  difference  of  temperature  per  hour, 
namely  .00235  Mb.,  the  quantity  of  pipe-surface  that  would  form  a  cubic 
foot  of  condensed  water  per  hour,  taking  the  weight  of  this  volume  of 
water  at  62.4  Ibs.,  would  be,  per  i°  difference  of  temperature, 

62.4  -f-  .00235  =  26,550  square  feet. 
For  a  difference  of  1  6  8°  the  required  surface  would  be 

26,550  -T-  168°=  158  square  feet,  say  160  square  feet. 

Four  square  feet  of  direct  boiler-surface,  or  its  equivalent  of  flue-surface, 
should,  therefore,  be  provided  for  every  160  square  feet  of  steam-pipe  con- 
taining steam  of  20  Ibs.  absolute  pressure  per  square  inch,  and  maintaining 
a  temperature  of  60°  F.  in  a  room. 

The  following  lengths  of  pipe  are  required  to  present  160  square  feet  of 
surface  :  — 

Length  for  Length  for 

i  square  foot.          160  square  feet. 

4-inch  pipe,  ^  inch  thick,  .............  10.2  inches,       136  feet. 

3         „          »  »          .............  13-°      »  J73    » 

2         »          >,  „          .............  18.3      „  244    „ 

*     „     rt      „     .............  36.6    „       488  „ 

French  practice.  —  According  to  M.  Grouvelle,  one  square  metre  of  pipe- 
surface,  heated  by  steam,  sufficed  to  heat  and  maintain  at  15°  C.,  or  say 
60°  F.,  a  room  with  ordinary  proportions  of  walls  and  windows,  such  as 
a  library  or  an  office,  of  from  66  to  70  cubic  metres  of  capacity,  or  a  work- 
shop of  from  90  to  100  cubic  metres.  If  the  workshop  is  to  be  maintained 
at  a  high  temperature,  a  square  metre  of  surface  is  allowed  for  70  cubic 
metres.  The  Exchange  at  Paris  is  sufficiently  heated  by  one  square  metre 


488  APPLICATIONS   OF   HEAT. 

for  67  cubic  metres.  The  allowance  of  one  square  metre  for  70  cubic 
metres  is  equivalent  to  4.35  square  feet  per  1000  cubic  feet  of  capacity;  or 
to  5.11  lineal  feet  of  four-inch  pipe  per  1000  feet. 

For  heating  workshops,  8  metres  wide  by  3  metres  high,  having  260 
square  feet  of  section,  with  a  window-surface  one-sixth  of  the  total  surface, 
engineers  in  France  allow  an  iron  pipe  of  16  inches  in  circumference,  or 
5  inches  in  diameter,  passing  once  through  the  shop,  presenting  1.33  square 
feet  of  surface  per  foot  run,  or  5.2  square  feet  per  1000  cubic  feet,  the 
same  as  has  just  been  calculated. 

According  to  the  observations  of  M.  Peclet  on  steam-heating  apparatus, 
particularly  in  a  large  factory,  for  a  maximum  difference  of  36°  F.  between 
the  interior  and  exterior  temperatures,  it  was  necessary  to  reckon  on  a 
delivery  of  26  units  of  heat  per  hour  per  square  foot  of  wall  of  13  or  14 
inches  in  thickness,  and  30  units  of  heat  per  square  foot  of  glass. 

HEATING  BY  ORDINARY  OPEN  FIRES  AND  CHIMNEYS. 

M.  Claudel  says  that  the  quantity  of  heat  radiated  into  an  apartment  from 
a  fireplace  is  about  one-fourth  of  the  total  heat  radiated  by  the  combustible. 
The  heat  radiated  into  an  apartment  from  wood  when  burned  amounts 
to  only  6  or  7  per  cent,  of  the  total  heat  of  combustion.  For  coal  and  for 
coke,  the  heat  thus  utilized  amounts  to  about  13  per  cent. 

In  burning  wood,  ordinary  chimneys  draw  about  1600  cubic  feet  of  air 
per  pound  of  fuel;  and  better  constructed  chimneys  about  1000  cubic  feet. 
A  sectional  area  of  from  50  to  60  square  inches  is  sufficient  for  the  chim- 
neys of  ordinary  apartments.  For  apartments  designed  to  hold  a  great 
number  of  persons,  a  section  of  400  square  inches,  say  32  by  13  inches,  is 
usually  employed. 

From  experiment  it  appears  that  the  proportions  of  fuel  required  to 
heat  an  apartment  are  as  100  for  ordinary  fire-places,  63  for  metal  stoves, 
and  from  13  to  1 6  for  apparatus  similar  to  stoves,  with  open  fires. 

HEATING  BY  HOT  AIR  AND  STOVES. 

Sylvester's  cockle-stove  is  constructed  of  wrought-iron,  %  inch  thick, 
formed  with  an  arch  and  two  sides,  closed  at  the 
ends,  through  one  of  which  the  furnace-mouth  is 
made.  The  furnace  is  formed  of  fire-brick  within 
the  case,  and  the  products  of  combustion  are 
drawn  off  by  flues  below  the  furnace.  The  case 
is  inclosed  in  fire-brick,  with  about  5  inches  clear 
space  for  the  circulation  of  the  air  to  be  heated. 
The  air  is  introduced  through  the  brickwork  at 
the  lower  part  of  the  sides,  through  numerous  iron 
tubes,  which  are  laid  to  within  an  inch  clear  of  the 
sides  of  the  case,  and  cause  the  fresh  air  to 
impinge  upon  the  heated  surface.  The  air  thus 

Fig.  128.-syivester»s  cockle-stove.  brought  in  Passes  over  the  entire  surface  of  the 
cockle  into  the  upper  part  of  the  envelope,  whence 

it  is  led  away  through  any  required  number  of  pipes  to  the  different  rooms 
to  be  warmed.  The  ends  of  these  exit  pipes  are  placed  within  an  inch 


HEATING  BY   OPEN   FIRES,   HOT  AIR,   AND   STOVES.          489 

above  the  top  of  the  case.  One  of  these  cockle-stoves  is  illustrated  by 
Fig.  128;  the  wrought-iron  case  is  5  feet  square  by  5  feet  high.  There  are 
from  150  to  200  air  pipes  2  inches  in  diameter,  or  2  inches  square,  at  the 
sides.  The  grate  contains  about  5  square  feet  of  area,  and  the  flues  at  the 
bottom  are  9  by  6  inches. 

From  the  results  of  Mr.  Sylvester's  experiments  with  a  smaller  cockle- 
stove,  it  was  found  that  with  a  consumption  of  5  Ibs.  of  coal  per  hour,  and  a 
heating  surface  of  17  square  feet,  the  temperature  of  344,600  cubic  feet  of 
air  was  raised  56°  F.  in  twelve  hours,  with  60  Ibs.  of  coal;  being  equivalent 
to  the  heating  of  95,000  cubic  feet  of  air  i°  F.  per  square  foot  of  surface 
per  hour;  or  to  the  heating  of  321,626  cubic  feet  of  air  i°  F.  by  one  pound 
of  coal.  It  thus  appears  that  each  square  foot  of  cockle-surface  is  equal 
to  7  square  feet  of  hot-water  pipe. 

French  Practice. — From  results  obtained  by  M.  Peclet,  it  is  ascertained 
that  when  the  flue-pipes  of  stoves,  conveying  hot  products  of  combustion, 
heat  directly  the  air  of  a  room,  the  quantities  of  heat  passed  off  per  square 
foot  per  hour  for  i°  F.  difference  of  temperature,  vary  according  to  the 
material  of  the  pipe  as  follows : — 

Cast-iron, 3.65  units  of  heat. 

Sheet-iron, 1.45     „  ,, 

Terra-cotta,  0.4  inch  thick, 1.42     „  „ 

It  may  be  noted  that  the  great  difference  here  observable  between  cast 
and  wrought  iron  in  passing  heat  from  a  flue  to  the  outer  air,  does  not  exist 
when  the  pipe  is  occupied  by  steam  or  hot  water.  If  an  excess  of  temper- 
ature equal  to  800°  F.  be  assumed,  as  between  the  inside  and  outside  of 
the  pipe,  the  quantities  of  heat  given  off  per  square  foot  per  hour  would  be, 

For  cast-iron, 3.65  x  450=  1642  units  of  heat. 

For  sheet-iron, 1.45x450=    652     „  „ 

For  terra-cotta, 1.42x450=    639     „  „ 

Yet,  in  practice,  the  same  surface  is  allowed  for  cast  and  for  sheet  iron; 
at  the  rate  of  one  square  foot  for  328  cubic  feet  of  space  to  be  heated. 
The  diameters  of  stove-pipes  vary  from  4  to  8  inches. 

The  air  thus  heated  receives  a  degree  of  humidity  from  a  vase  full  of 
water  placed  on  the  stove.  The  water  so  dissipated  amounts  to  a  little 
more  than  2^  pints  per  day  for  a  room  of  from  2500  to  3000  cubic  feet  of 
capacity. 

House-stoves  placed  in  the  Room  to  be  Warmed. — M.  Claudel  says  that 
inside  stoves  are  employed  in  schools  and  hospital-wards;  they  consist  of 
an  upright  column,  square  or  cylindrical,  from  5  to  7  feet  high,  inclos- 
ing the  furnace;  surmounted  by  a  pipe  which  rises  vertically,  and  is  then 
carried  nearly  horizontally  through  the  apartment  to  a  chimney.  The 
column  is  inclosed  in  an  outer  casing  of  sheet  iron  or  brickwork,  with  an 
interspace  into  which  the  external  air  is  admitted,  and  from  the  upper  part 
of  which  the  air  passes  into  the  room.  The  temperature  of  the  furnace 
does  not  exceed  from  1100°  to  1300°  F.  In  practice,  it  is  found  conven- 
ient to  assume  that  the  products  of  combustion  leave  the  stove  at  a  temper- 
ature of  about  950°  F.,  or  500°  C.,  that  they  are  completely  cooled  in  their 
course,  that  the  temperature  of  the  room  is  60°  F.,  and  that  the  quantity  of 


4QO  APPLICATIONS  OF  HEAT. 

heat  emitted  is  the  same  as  if  the  pipe  had  an  average  temperature  of 
480°  F.,  or  250°  C.  A  heating  surface  of  from  20  to  30  square  feet  is 
allowed  per  pound  of  coal  burned  per  hour,  not  reckoning  the  surface  of 
the  stove.  Large  grates  are  preferred,  with  slow  combustion. 

House-stoves  placed  Outside  the  Room  to  be  Heated. — The  useful  effect  of 
these  stoves  may  be  taken  at  from  60  to  70  per  cent,  of  the  heating 
power  of  the  fuel.  The  surface  of  grate  should  be  15  square  inches  per 
pound  of  coal  consumed  per  hour.  The  heating  surface  is  two  square 
metres  per  kilogramme  of  coal,  or  per  two  kilogrammes  of  wood :  equivalent 
to  10  square  feet  per  pound  of  coal  per  hour.  From  2^  to  3}^  pints 
of  water  are  consumed  per  1000  cubic  metres;  or  i  pint  for  from  1000  to 
1400  cubic  feet  of  space. 

The  spent  air  of  the  room  is  passed  off  into  a  chimney. 


HEATING  OF  WATER  BY  STEAM  IN  DIRECT  CONTACT. 

The  heating  of  water  by  steam,  when  the  elements  are  brought  into 
direct  contact,  is  practically  instantaneous.  The  author  made  experiments 
on  this  subject  by  admitting  steam  at  90  Ibs.  effective  pressure  from  a 
locomotive-boiler  into  a  body  of  cold  water  contained  in  a  cylindrical 
reservoir,  3  feet  6  inches  in  diameter,  and  15  feet  long,  made  of  ^-inch 
iron  plate,  having  a  total  capacity  of  144  cubic  feet.  The  steam  was  con- 
veyed from  the  boiler  to  the  reservoir  by  a  i-inch  pipe,  from  which  it  was 
freely  discharged  into  a  2-inch  iron  pipe,  open  at  the  extremity,  laid  in  the 
water  along  the  bottom  of  the  reservoir.  The  reservoir  lay  horizontally, 
without  any  covering,  in  a  factory.  Fifty-five  and  a  half  cubic  feet,  or  3464 
Ibs.  of  cold  water  at  60°,  were  delivered  into  the  reservoir,  and  the  water 
was  heated  by  the  steam  blown  into  it  to  a  pressure  of  85  Ibs.  effective 
per  square  inch,  in  two  hours,  with  a  temperature  of  328°  F.,  or  through 
328-60  —  268°  F.  The  quantity  of  heat  communicated  in  two  hours  was, 
therefore,  3464  x  268  =  928,352  units,  at  the  rate  of  464,176  units  per  hour. 
Taking  the  initial  temperature  of  the  steam  of  the  boiler,  331°  F.,  and  the 

mean  temperature  of  the  heated  water  *—  -  -  =  194°;  the  mean  difference 

2 

of  temperature  was  331  -  194=  137°,  and  the  quantity  of  heat  communi- 
cated per  i°  F.  of  difference  per  hour  was, 


464176  -3388  units  of  heat. 


To  communicate  the  whole  of  this  quantity  X)f  heat  through  the  surface  of 
a  pipe  at  the  rate  of  300  units  per  foot  per  i°  F.  per  hour,  there  would 
have  been  required  3388  -f-  300  =  1  1.3  square  feet  of  surface.  It  is  probable, 
as  a  matter  of  fact,  that,  though  the  2-inch  pipe  was  open  to  the  water  at 
the  end,  the  most  of  the  steam  was  condensed  within  the  pipe  before  it 
could  reach  the  end.  The  surface  of  the  pipe  had  about  8  square  feet  of 
area. 

There  was,  of  course,  a  loss  of  heat  by  radiation  from  the  surface  of  the 
reservoir;  but  it  is  not  material  to  the  purpose  of  this  notice. 


EVAPORATION   IN   OPEN   AIR.  491 


EVAPORATION  (SPONTANEOUS)  IN  OPEN  AIR. 

So-called  "  spontaneous  "  evaporation  from  water  exposed  to  air  proceeds 
at  all  temperatures,  when  the  conditions  are  suitable.  The  total  rate  of 
evaporation  is  in  proportion  to  the  extent  of  the  surface  exposed  to  the  air. 
An  increase  of  the  temperature  of  the  liquid  is  attended  by  an  increase  of 
the  rate  of  evaporation,  though  not  in  direct  proportion.  The  rate  of  evapora- 
tion is  greater  when  the  air  is  in  motion  over  the  surface  of  the  water  than 
when  it  is  at  rest.  The  rate  of  evaporation  is  also  greater  in  proportion  as 
the  air  is  dryer,  or  the  less  the  moisture  previously  existing  in  the  air;  and 
on  the  contrary,  when  the  air  is  saturated  with  moisture,  the  evaporation 
is  reduced  to  nothing. 

When  the  atmosphere  is  perfectly  dry,  the  rapidity  of  evaporation  is 
proportional  to  the  pressure  of  the  vapour  due  to  the  temperature  of  the 
water,  for  which  reference  may  be  made  to  tables  No.  127,  page  386,  and 
No.  130,  page  396.  This  law  was  discovered  by  Dr.  Dalton,  who  gives 
the  following  illustration : — 

At  the  temperatures  212°,  180°,  164°,  152°,  144°,  138°, 
The  pressures  are       30,       15,       10,      7^,     6,        5    inches  of  mercury ; 
And  the  weights  of  water  evaporated  at  these  temperatures  are  proportional 
to  30,      15,       10,      7^,     6,        5. 

But  the  atmosphere  impedes  the  diffusion,  and,  consequently,  the  genera- 
tion of  vapour;  although,  ultimately,  the  full  charge  of  saturated  vapour 
due  to  the  temperature  is  absorbed  by  it.  When  vapour  is  present  in  the 
air,  which  it  usually  is  to  a  greater  or  less  degree,  the  pressure  of  this 
vapour  is  to  be  deducted  from  that  of  the  vapour  due  to  the  temperature 
of  the  water;  and  the  residual  force  is  the  active  "evaporating  force." 
Dr.  Dalton  found  that  with  the  same  evaporating  force,  thus  determined, 
the  same  rapidity  of  evaporation  is  maintained,  whatever  be  the  tempera- 
ture of  the  air. 

But  when  a  current  of  air  blows  over  the  surface  of  the  water,  the  rapidity 
of  evaporation  is  greater  than  when  the  air  is  still,  because  the  air  in  motion 
sweeps  away  the  vapour  as  it  rises,  and  a  continuous  supply  of  compara- 
tively dry  air  is  secured.  With  the  same  evaporating  force,  a  strong  wind 
will  double  the  production  of  vapour,  compared  with  the  quantity  produced 
in  a  still  atmosphere. 

Dr.  Dalton's  experiments  were  made  with  an  evaporating  surface  of 
6  inches  in  diameter,  in  still  air  and  in  wind,  and  he  gives  a  table  of  the 
rates  of  evaporation  in  grains  per  minute,  for  temperatures  up  to  85°  F.,  on 
the  assumption  that  the  air  is  perfectly  dry.1  The  following  table,  No.  168, 
is  calculated  to  show  the  rate  of  evaporation,  in  pounds  per  square  foot  per 
hour,  extended  up  to  212°  F.,  and  for  three  states  of  the  air: — when  still, 
when  there  is  a  gentle  wind,  and  when  there  is  a  brisk  wind.  The  pressures 
are  given  in  inches  of  mercury,  and  are  those  adopted  by  Dr.  Dalton,  which, 
for  the  purpose  of  the  table,  do  not  materially  vary  from  those  given  in 
table  No.  127. 

1  Memoirs  of  the  Literary  and  Philosophical  Society  of  Manchester,  vol.  v.  p.  579. 


492 


APPLICATIONS   OF    HEAT. 


Table  No.  168. — "SPONTANEOUS"  EVAPORATION  OF  WATER  IN  STILL 
AIR  AND  IN  WIND,  ASSUMING  THE  AIR  TO  BE  PERFECTLY  DRY, 
FOR  TEMPERATURES  FROM  32°  TO  212°  F. 

(Founded  on  Dr.   Dalton's  tables.) 


Tem- 
perature 
of  the 
water. 

Pressure 
of  the 
vapour. 

Water  evaporated  per  square 
foot  of  surface  per  hour. 

Tem- 
perature 
of  the 
water. 

Pressure 
of  the 
vapour. 

Water  evaporated  per  square 
foot  of  surface  per  hour. 

Air  still. 

Gentle 
wind. 

Brisk 
wind. 

Air  still. 

Gentle 
wind. 

Brisk 
wind. 

Fahr. 

inches  of 

Ibs. 

Ibs. 

Ibs. 

Fahr. 

inches  of 

Ibs. 

Ibs. 

Ibs. 

mercury. 

mercury. 

32° 

.200 

.0349 

.0448 

•°55° 

125° 

3-79 

.6619 

.8494 

•043 

35 

.221 

.0386 

•0495 

.0608 

I30 

4-34 

.7580 

.9727 

.194 

40 

.263 

•0459 

.0589 

.0723 

!35 

5.00 

.8730 

1.  121 

.376 

45 

.316 

•°552 

.0708 

.0869 

140 

5-74 

1.003 

1.286 

•579 

5o 

•375 

•0655 

.0841 

.1032 

i45 

6-53 

I.I40 

1.463 

.796 

55 

•443 

.0774 

•0993 

.1218 

150 

7.42 

1.296 

1.663 

2.043 

60 

.524 

.0917 

.1175 

.1441 

J55 

8.40 

1.467 

1.882 

2.310 

62 

.560 

.0979 

•1255 

.1540 

1  60 

9.46 

1.652 

2.120 

2.602 

65 

.616 

.1076 

.1381 

.1694 

165 

10.68 

1.865 

2-394 

2.938 

70 

.721 

•1257 

.l6l6 

.1983 

170 

12.13 

2.118 

2.719 

3-336 

75 

.851 

.1486 

.1907 

.2341 

i75 

13.62 

2.378 

3-°53 

3.746 

80 

1.  000 

.1746 

.2241 

•2751 

180 

15-15 

2.646 

3-395 

4.167 

35 

1.17 

.2043 

.2622 

.3218 

185 

17.00 

2.969 

3.8io 

4.676 

90 

1.36 

•2375 

.3048 

•3745 

190 

19.00 

3.318 

4.258 

5.226 

95 

1.58 

.2760 

•3541 

.4346 

!95 

21.22 

3.706 

4.758 

5.837 

100 

1.86 

.3248 

.4169 

.5116 

200 

23.64 

4.128 

5.298 

6.502 

105 

2.18 

.3807 

.4886 

•5996 

205 

26.13 

4-563 

5-856 

7.187 

no 

2-53 

.4418 

.5670 

•6959 

210 

28.84 

5.034 

6.464 

7-933 

IJ5 

2.92 

.5IOO 

.6544 

.8030 

212 

30.00 

5-239 

6.724 

8.252 

I2O 

3-33 

.5815 

•7463 

.9160 

It  appears  from  the  table  that  the  rates  of  evaporation,  for  each  of  the 
three  conditions  of  the  air,  when  perfectly  dry,  are  in  simple  proportion  to 
the  pressure  of  the  steam ;  and,  as  affected  by  the  stillness  or  the  motion  of 
the  air,  they  are — 


for  still  air, 
as  i, 


a  gentle  wind, 
1.28, 


a  brisk  wind, 


It  is  to  be  understood  that  the  temperature  212°  is  a  limiting  tempera- 
ture, which  cannot  be  actually  reached  without  displacing  the  air  entirely. 

It  is  also  to  be  remarked  that,  though  Dr.  Dalton  lays  down  the  proposi- 
tion that  the  rapidity  of  the  evaporation  is  the  same,  whatever  may  be  the 
temperature  of  the  air,  yet  it  is  clear  that  water  is  evaporated  more  rapidly 
when  a  warm  current  blows  over  it  than  when  it  is  traversed  by  a  cold 
current.  Such  increase  of  evaporation  is  probably  the  result  of  the  reflex 
action  of  heat  imparted  by  the  air  to  the  superficial  water,  the  "  evaporative 


DESICCATION.  493 

force  "  of  which  is  increased  by  the  rise  of  temperature  due  to  the  heat 
abstracted  from  the  air.  The  cooling  of  air  by  passing  it  over  or  through 
water  is  a  well-known  expedient.  In  India,  the  air  of  apartments  is  cooled 
by  passing  it,  as  it  enters,  through  and  over  the  "  tatta,"  a  bamboo  frame  or 
trellis,  over  which  water  is  suffered  to  trickle. 

From  what  has  been  stated,  with  respect  to  "mixtures  of  gases  and 
vapours,"  page  392,  it  appears  that  the  condition  of  "  saturation,"  attributed 
to  the  mixture  of  vapour  and  air,  properly  belongs  to  the  vapour  itself,  as 
vapour,  when  it  has  arrived  at  its  maximum  density  and  pressure  for  the 
temperature  of  the  air. 

Use  of  the  Table  No.  168.  —  Dr.  Dalton  gives  the  solution  of  the  problems 
based  upon  the  original  tables,  of  which  the  first  is  here  rendered  into  a 
rule  in  relation  with  the  table  No.  168. 

RULE.  To  find  the  quantity  of  water  exposed  to  air  that  would  be  evaporated 
per  square  foot  of  surface  per  hour,  at  a  given  temperature  of  air,  with  a  given 
dew-point.  Subtract  the  tabulated  weight  of  water  corresponding  to  the 
dew-point  from  the  weight  corresponding  to  the  temperature  of  the  air;  the 
remainder  is  the  weight  of  water  that  would  be  evaporated  per  square  foot 
of  surface  per  hour. 

The  weights  of  water  are  to  be  selected  from  the  3d,  4th,  or  5th  columns, 
according  to  the  state  of  the  wind. 

To  find  the  dew-point,  Dr.  Dalton  used  a  very  thin  glass  vessel,  into 
which  he  poured  cold  water,  of  which  he  noted  the  temperature.  If  the 
vapour  in  the  atmosphere  was  instantly  condensed  on  the  glass,  he  changed 
the  water  for  warmer  water,  and  so  proceeded  until  he  ascertained  the 
proper  temperature  —  the  dew-point  —  when  he  could  just  perceive  a  slight 
dew  deposited  on  the  glass.  The  dew-point  may  be  found  with  much 
greater  precision  by  means  of  hygrometers,  described  at  page  393. 

Dr.  Pole  has  constructed  an  empirical  formula  which  roughly  represents 
the  results  of  Dr.  Dalton's  experiments.  Let  T  =  the  temperature  of  the 
atmosphere  in  degrees  Fahr.,  t=  the  dew-point,  V  =  the  velocity  of  the  wind 
in  miles  per  hour,  E  =  the  rate  of  evaporation  in  inches  per  day  from  a 
water  surface,  and  A  =  a  numerical  coefficient.  Then, 


m 

A(ioo-  V) 

The  value  of  A  =  80  for  high  or  summer  temperatures,  and  A—  100  for 
low  or  winter  temperatures.  Dr.  Pole  remarks  that  Dalton's  tables  do  not 
provide  for  cases  where  the  temperature  of  the  water  differs  materially  from 
that  of  the  air;  and  that,  probably,  in  such  cases,  T  should  be  made  to 
represent  the  temperature  of  the  water-surface,  and  not  that  of  the  air.1 

DESICCATION. 

The  drying  of  wet  or  moist  materials,  by  means  of  currents  of  air,  is 
based  on  the  principles  already  announced  which  regulate  the  evaporation 
of  water  from  the  surface.  If  a  current  of  air  be  saturated  with  moisture 

1  Minutes  of  Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xxxix.  page  36. 


494  APPLICATIONS   OF   HEAT. 

or  vapour,  its  efficiency  for  drying  out  moisture  from  bodies  with  which  it 
comes  in  contact,  is  exactly  nothing.  To  act  as  a  dryer,  in  other  words,  to 
assist  in  evaporating  and  carrying  off  moisture,  it  must  be  either  perfectly  dry, 
or,  at  the  least,  sub-saturated ;  and  inasmuch  as  its  capacity,  in  the  conven- 
tional language  already  explained,  for  absorbing  moisture — in  the  state  of 
vapour,  of  course — increases  with  its  temperature,  it  is  obvious  that  the 
higher  the  temperature  of  the  air,  the  greater  is  its  efficiency.  If,  then,  the 
air-current  be  surcharged  with  heat,  it  stimulates  evaporation  in  two  forms 
— by  imparting  a  portion  of  its  heat  to  the  wet  or  moist  surface,  which  is 
utilized  in  the  evaporation  of  the  moisture,  and  by  tolerating  the  presence 
of  a  greater  quantity  of  moisture  in  mixture  with  it,  which  is  carried  away 
as  it  rises  from  the  surface  by  the  current. 

The  drying,  or  vaporization  of  moisture,  by  such  means,  involves,  of 
course,  a  lowering  of  the  temperature  of  the  air,  or  the  moist  body,  or 
both;  and  the  problem  arises:  What  is  the  initial  temperature  of  dry  air 
required? 

The  first  problem  for  solution  is  twofold: — Given  the  final  temper- 
ature at  which  the  saturated  mixture  is  to  be  discharged,  what  is  the 
quantity  of  dry  air  required  for  a  given  weight  of  vapour  in  saturated  mix- 
ture? and  to  what  initial  temperature  is  the  air  required  to  be  raised  in 
order  to  supply  heat  for  the  evaporation  of  the  given  weight  of  steam? 
The  answer  is  to  be  found  in  the  sub-section  on  the  "  Properties  of  Satur- 
ated Mixtures  of  Air  and  Aqueous  Vapour,"  with  table  No.  130,  page  394. 

But,  in  ordinary  practice,  the  artificially  heated  air-current  does  not 
arrive  at  the  condition  of  saturation  before  it  is  discharged;  and  a  large 
surplus  of  air  is  therefore  to  be  provided,  the  proportional  amount  of  which 
varies  with  the  circumstances  under  which  the  current  is  applied.  The 
standard  of  perfect  efficiency  is  presented  in  the  table  No.  130.  M.  Peclet 
notices  a  process  employed  by  M.  Montgolfier  for  drying  the  skins  of 
grapes  after  having  been  pressed,  by  means  of  a  forced  current  of  air.  It 
was  found  that,  in  autumn,  5340  cubic  feet  of  air,  moving  at  a  velocity  of 
about  1 6  feet  per  second,  were  required  for  the  evaporation  of  one  pound 
of  water  from  the  pressed  grapes.  Let  the  initial  temperature  of  the  air  be 
assumed  at  64°  R,  then,  by  the  table  No.  130,  a  volume  of  dry  air  equal 
to  2526  cubic  feet  would  have  sufficed  to  evaporate  one  pound  of  water; 
but  if,  as  is  probable,  the  air  had  already  been  loaded  with  half  the  quantity 
of  moisture  it  could  carry,  then  at  least  double  the  tabular  quantity  would 
have  been  necessary,  or  2526  x  2  =  5 05 2  cubic  feet,  which  is  nearly  equal  to 
the  quantity  actually  employed. 

In  the  design  of  a  drying-chamber,  it  is  of  the  first  importance  that  the 
air-current  should  be  admitted  at  the  highest  point  of  the  chamber  and 
discharged  at  the  level  of  the  floor.  The  reverse  process,  of  admitting  it  at 
or  near  the  floor,  and  discharging  it  at  the  upper  part,  is  vicious  practice. 
In  the  latter  case  the  circulation  is  imperfect,  for  the  hot  air  seeks  the 
most  direct  route  to  the  points  of  egress;  in  the  former  case  the  hot  air 
is  uniformly  distributed,  and  if  the  points  of  discharge  are  properly  placed, 
the  descending  current  is  applied  equally  over  the  area  of  the  chamber. 
In  a  drying-chamber,  noticed  by  M.  Peclet,  for  drying  vermicelli,  at  Saint 
Ouen,  having  a  capacity  of  upwards  of  6000  cubic  feet,  and  heated  by  air 
of  from  86°  to  104°  F.,  the  following  were  the  results  of  heating  by  ascend- 
ing and  by  descending  currents : — 


DESICCATION.  495 

DISCHARGE  OF  CURRENT.  VERMICELLI  PRODUCED. 

First  Quality.  Second  Quality.  Fermented. 

Above  (mean  of  5  trials) 540  Ibs 400  Ibs 4.5  Ibs. 

Below       „  „        3,200   „     143  „     86.0    „ 

These  results  prove  decisively  the  superiority  of  the  descending  current. 
The  consumption  of  fuel  with  the  descending  current  was  also  much 
the  less. 

Dry  ing-house  for  Calico  (Peclet). — In  a  drying-house  used  by  M.  Rene  Duvoir, 
the  pieces  of  calico  were  suspended  from  bars  ranged  horizontally  across  the 
upper  part  of  the  house.  Air  heated  to  250°  F.  was  admitted  through  a 
number  of  openings  from  a  brick  flue  at  the  floor,  regulated  by  dampers, 
from  which  it  rose  to  the  upper  part,  and  thence  descended  to  the  floor, 
where  it  was  discharged.  The  external  temperature  was  77°  F.,  and  the 
temperature  of  the  discharged  current  in  the  chimney  was  100°  F.  In  six 
hours,  150  pieces  of  calico,  holding  2490  Ibs.  of  water,  were  dried,  with  a 
consumption  of  706  Ibs.  of  coal,  corresponding  to  an  evaporation  of 
3.52  Ibs.  of  water  per  pound  of  coal.  The  quantity  of  air  heated  to 
250°  F.,  for  this  duty,  was  1,943,000  cubic  feet,  the  weight  of  which,  at 
the  rate  of  13.52  cubic  feet  to  the  pound  (by  Rule  9,  page  350),  was 
143,713  Ibs.  Then  143713  x  .2377  =  34160  units  of  heat  for  i°  F.  eleva- 
tion of  temperature,  and  for  250  -  77  =  173°,  the  total  elevation  of  temper- 
ature, the  total  heat  consumed  was  34160  x  173°  =  5,909,256  units,  being 
at  the  rate  of 

5'9°9>25°  =  8370  units  per  pound  of  coal. 
706 

The  water  evaporated  per  pound  of  coal  was  3.52  Ibs.,  for  which  1067 
units  of  heat,  reckoned  from  77°  F.,  were  absorbed  per  pound  of  water; 
and  for  3.52  Ibs., 

1067  x  3.52  =  3756  units  of  heat, 

was  the  quantity  of  heat  utilized  for  evaporation  per  pound  of  coal,  being 
45  per  cent,  of  the  total  heat  communicated  to  the  air. 

The  temperature  at  which  the  air  was  discharged  being  100°,  it  was  23° 

above  the  external  temperature,  or  the  loss  by  the  excess  was  — ^-  x  100  =  13 

per  cent,  of  the  whole  heat  communicated  to  the  air. 

The  distribution  of  the  heat  communicated  to  the  air  was,  therefore, 
approximately  as  follows : — 

In  evaporating  moisture, 45  per  cent. 

Carried  off  by  the  air, 13       „ 

Loss  by  radiation  and  conduction, 42       „ 

100 

It  is  easy  to  show  that  the  air  when  discharged  was  not  nearly  saturated. 
At  1 00°  F.,  the  temperature  of  discharge,  the  proportions  of  moisture  and 
air  in  one  pound  of  a  saturated  mixture,  by  table  No.  130,  are  as  .283  to 
6.641,  or  as  i  to  23.5.  In  143,713  Ibs.  of  air,  therefore,  when  in  a  state  of 
saturation,  there  would  have  been 

1 43, 713 -=-23. 5  =  6, 115  pounds  of  moisture. 


496  APPLICATIONS   OF   HEAT. 

But  there  was  only  2490  Ibs.  of  moisture  in  the  air,  or  about  two-fifths 
of  the  proportion  for  saturation. 

The  single  good  feature  in  this  drying-house,  is  the  extraction  of  the 
spent  current  at  the  level  of  the  floor.  No  provision  was  made  to  effect 
the  distribution  of  the  heat  uniformly  through  the  room;  and  there  can  be 
no  doubt  that  the  condition  of  sub-saturation  of  the  air  was,  for  the  most 
part,  the  result  of  the  absence  of  such  provision. 

Drying  Linen. — The  maximum  evaporative  performance  of  coal  in  drying 
linen,  does  not  exceed  an  evaporation  of  3  Ibs.  of  water  per  pound  of  fuel; 
and  it  is  sometimes  as  low  as  1.36  Ibs. 

Drying  Various  Stuffs. — According  to  M.  Rouget  de  Lisle,  in  one  pound 
of  wet  cloth,  after  having  been  wrung  or  pressed,  or  passed  through  the 
hydro-extractor,  there  remained  the  respective  quantities  of  water  as 
follows  : — 

Water  Left  in  One  Pound  of 
Flannel.  Calico.  Silk.  Linen. 

When  twisted, 2.00  Ibs.     i.o  Ib.      .95  Ib.       .75  Ib. 

When  pressed, i.oo  .6  .5  .40 

When  passed  through  the  )      , 
hydro-extractor, j 

In  these  instances,  the  centrifugal  machine  was  26  inches  in  diameter, 
and  made  from  500  to  600  turns  per  minute. 

M.  Penot  made  experiments  on  drying-houses  at  Mulhouse,  and  found 
that  one  pound  of  coal  evaporated  from  1.02  to  2.86  Ibs.  of  water:  the 
latter  under  favourable  circumstances. 

According  to  M.  Royer,  in  a  drying-house  31^  feet  long,  26  feet  wide, 
and  62%  feet  high,  the  heating  surface  of  the  stove  amounted  to  758 
square  feet,  with  a  consumption  of  55  Ibs.  of  coal  per  hour.  There  were 
during  these  trials,  lasting  fifteen  days  each,  evaporated  successively  2.37, 
2.53,  and  2.18  Ibs.  of  water  per  pound  of  coal. 

Mr.  J.  R.  Napier,  in  drying  stuffs  by  air  heated  to  240°  F.,  with  a 
descending  draft,  evaporated  3  Ibs.  of  water  per  pound  of  coal. 

Drying  Stuffs  by  Contact  with  Heated  Metallic  Surfaces. — M.  Clement 
applied  a  piece  of  calico,  weighing  2^  Ibs.,  holding  an  equal  weight 
of  water,  to  a  plate  of  copper  of  the  same  extent,  heated  by  steam  at 
212°  F. ;  and  it  was  dried  in  one  minute.  The  evaporation  was  effected  at 
the  rate  of  1.42  Ibs.  of  water  per  square  foot  per  hour. 

When  stuffs  are  dried  by  passing  them  over  cast-iron  cylinders,  heated 
by  steam  internally,  it  appears  from  experiments  made  by  M.  Royer,  that 
in  drying  calico  which  held  its  weight  of  water,  74  Ibs.  of  water  were 
evaporated  by  the  condensation  of  102  Ibs.  of  steam.  In  other  experiments 
made  with  a  machine  of  six  cylinders,  the  efficiency  in  drying  was  only 
two-thirds  of  that  attained  in  the  first-described  experiment.  The  experi- 
ments were  made  in  winter  in  a  place  which  was  imperfectly  closed. 

Drying  Grain. — It  is  reported  in  the  Engineer,  that  Messrs.  Crighton 
&  Co.,  Abo,  dried  450  Ibs.  of  grain,  extracting  15  per  cent,  of  its  weight, 
or  67^  Ibs.  of  water,  by  the  consumption  of  18  Ibs.  of  birchwood,  being 
at  the  rate  of  3.75  Ibs.  of  water  per  pound  of  wood. 

Drying  Wood. — In  the  forges  of  Lippitzbach,  Carinthia,  according  to 
M.  Leplay,  wood  is  piled  and  dried  in  close  chambers  by  burning  a  part  of 


HEATING  OF   SOLIDS.  497 

the  wood,  averaging  a  fourth  of  the  total  quantity.  The  furnaces  are  below 
the  floor,  between  which  and  the  furnaces  a  space  is  provided  for  the 
circulation  of  the  products  of  combustion  under  the  floor.  Air  in  consider- 
able quantity  is  admitted  to  and  mixed  with  the  products  of  combustion  to 
moderate  their  temperature  to  350°  F.;  when  the  current  passes  into  the 
upper  chamber  amongst  the  wood  to  be  dried.  On  this  system,  there  is 
considerable  loss  of  heat  by  radiation  and  by  the  excessive  dilution  of  the 
products  of  combustion  with  air. 

At  the  Neuberg  factory,  the  products  of  combustion  circulate  in  a  species 
of  stove  constructed  of  thin  masonry,  and  pass  thence  through  cast-iron 
pipes  by  which  the  air  is  heated  for  drying  the  wood.  On  this  system  the 
wood  consumed  does  not  exceed  an  eighth  of  the  total  quantity. 

The  limit  of  temperature  at  which  wood  should  be  dried  ought  not 
to  exceed  340°  or  350°  F.  M.  Leplay  states  that  the  wood  to  be  dried 
contains  40  per  cent,  of  water;  whence  it  appears  that  one  pound  of  the 
fresh  wood  evaporates  1.20  Ibs.  of  water  in  the  first  of  the  above-described 
processes;  and  in  the  second  process,  2.80  Ibs.  of  water. 

In  a  system  of  drying-furnace  recently  adopted  in  France  for  wood,  peat, 
&c.,  a  chamber  62  feet  long  and  14  feet  wide  is  employed.  The  wood, 
in  billets,  is  loaded  into  waggons,  having  a  capacity  of  about  100  cubic 
feet  each,  on  rails.  Each  waggon-load  successively  is  introduced  at  one 
end  and  withdrawn  at  the  other,  whilst  the  mixture  of  hot  gases  and  air  is 
introduced  at  the  other  end,  and  passes  to  the  end  at  which  the  waggons 
are  introduced.  A  temperature  of  270°  F.  is  maintained  at  the  middle 
of  the  chamber.  The  maximum  temperature  is  320°  F.  The  wood 
remains  sixty-four  hours  in  the  chamber,  and  the  usual  quantity  of  moisture 
it  contains,  from  20  to  25  per  cent,  is  evaporated  by  the  combustion  of 
i^  cords  of  wood  for  every  16  cords  to  be  dried;  that  is,  i^  Ibs.  are 
burned  to  dry  16  Ibs.,  evaporating  4  Ibs.  of  water;  being  at  the  rate  of 
2.66  Ibs.  of  water  per  pound  of  fresh  wood. 

HEATING   OF   SOLIDS. 

Cupola  Furnace. — M.  Peclet  estimates  that,  in  melting  pig  iron  in  an 
ordinary  cupola,  by  the  combustion  of  30  per  cent,  of  its  weight  of  coke, 
14  per  cent,  only  of  the  heat  of  combustion  is  actually  utilized.  This 
estimate  is  based  on  the  result  of  an  experiment  by  Clement,  showing  that 
to  heat  and  melt  i  pound  of  pig  iron,  504  English  units  of  heat  are 
necessary. 

Plaster  Ovens. — He  also  states  that  to  dry  plaster,  the  heat  of  combustion 
of  7  per  cent,  of  its  weight  in  wood  is  absorbed,  whereas  the  actual 
consumption  of  wood  amounts  to  from  9  to  14  per  cent., — showing  that 
from  50  to  80  per  cent,  of  the  total  heat  generated  is  utilized. 

Metallurgical  Furnaces. — Dr.  Siemens  states  that,  in  an  ordinary  reheating 
furnace,  employed  in  metallurgical  operations,  one  ton  of  coal  is  consumed 
in  heating  1^3  tons  of  wrought  iron  to  the  welding  point,  2700°  F.;  whilst 
he  estimates,  in  terms  of  the  specific  heat  of  iron,  .114,  and  the  heating  power 
of  coal,  14,000  units  of  heat,  that  a  ton  of  coal  is  capable  of  heating  up 
39  tons  of  iron.  From  this  it  appears  that  only  4^  per  cent,  of  the  whole 
heat  generated  is  appropriated  by  the  iron.  Similarly,  he  estimates  that 

32 


498  APPLICATIONS   OF   HEAT. 

barely  i  %  per  cent,  of  the  whole  heat  generated  is  utilized  in  melting  pot- 
steel,  in  ordinary  furnaces ;  whilst,  in  his  regenerative  furnaces,  a  ton  of  steel 
is  melted  by  the  combustion  of  12  cwts.  of  small  coal,  showing  that  6  per 
cent,  of  the  heat  produced  is  utilized. 

Blast-Furnace. — Mr.  J.  Lothian  Bell1  has  formed  detailed  estimates  of 
the  appropriation  of  the  heat  of  Durham  coke  in  the  Cleveland  blast- 
furnaces ;  from  which  the  following  abstract  has  been  prepared : — 

Durham  coke,  it  is  assumed,  consists  of  92.5  per  cent,  of  carbon,  2.5  per 
cent,  of  water,  and  5  per  cent,  of  ash  and  sulphur.  To  produce  i  ton  of 
pig-iron,  there  are  required  1 1  cwts.  of  limestone,  and  49  cwts.  of  calcined 
iron-stone;  the  iron-stone  consists  of  18.6  cwts.  of  iron,  9  cwts.  of  oxygen, 
and  21.4  cwts.  of  earths.  There  is  formed  7.26  cwts.  of  slag,  of  which  i.i 
cwt.  is  formed  with  the  ash  of  the  coke,  and  6.16  cwts.  with  the  limestone. 
There  are  21.4  cwts.  of  earths  from  the  iron-stone,  less  .74  cwt.  of  bases 
taken  up  by  the  pig-iron  and  dissipated  in  fume;  say,  20.66  cwts.  Total 
of  slag  and  earths,  27.92  cwts. 

Mr.  Bell  assumes  that  30.4  per  cent,  of  the  carbon  of  the  fuel,  which 
escapes  in  a  gaseous  form,  is  carbonic  acid;  and  that,  therefore,  only  51.27 
per  cent,  of  the  heating  power  of  the  fuel  is  developed,  and  the  remaining 
48.73  per  cent,  leaves  the  tunnel-head  undeveloped.  He  adopts,  as  a  unit 
of  heat,  the  heat  required  to  raise  the  temperature  of  1 1 2  Ibs.  of  water 
i°  Centigrade. 

Distribution  of  the  heat  generated  in  the  blast-furnace  for  the  production 
of  i  ton  of  pig-iron: — 

UNITS.  PER  CENT. 

Evaporation  of  water  in  coke,  and  chemical  action, 

in  smelting,  48,354  54-i 

Fusion  of  pig-iron, 6,600  7.4 

Fusion  of  slag, I5,3S^>  I7-2 

Expansion  of  blast,  3,700  4.1 

Appropriated  for  the  direct  work  of  the  furnace,  74,010         82.8 

Loss  by  radiation  through  the  walls, 3,600  4.0 

Carried  away  by  tuyere-water, i ,800  2.0 

Sensible  heat  of  gaseous  products, 10,000  11.2 

Waste, 1 5,400        17.2 


Total  heat  generated  in  the  furnace, 89,410       100.0 

The  undeveloped  heat  of  the  fuel  amounts  proportionally  to  89,410  x 
=  84,980  units.    Add  to  this,  the  sensible  heat  of  the  gaseous  products, 
10,000  units,  and  the  sum,  94,980  units,  is  disposed  of  as  follows: — 

1  The  Journal  of  the  Iron  and  Steel  Institute,  1872,  1875.  Tne  abstract  given  in  the 
text  affords  but  a  meagre  notion  of  the  variety  and  extent  of  Mr.  J.  Lothian  Bell's  investi- 
gations, the  value  and  importance  of  which  are  highly  and  justly  appreciated  by  manufac- 
turers of  iron. 


HEATING  OF  SOLIDS.  499 

Distribution  of  the  waste  and  undeveloped  heat  of  the  fuel  required  for  the 
production  of  i  ton  of  pig-iron. 

UNITS.  PER  CENT. 

Generation   of   steam   for   blast-engine   and   various 

pumps  connected  with  the  work, 28,080         29.6 

Heating  the  blast  to  905°  F., 11,920         12.5 

Appropriated  for  direct  work, 40,000         42.  i 

Loss  by  radiation  from  the  gas  tubes, 3320  3.5 

Loss  of  heat  escaping  by  the  chimneys,  21,660  22.8 

(temperature,  770°  R,  from  boilers) 
(       Do.  640°  F.,  from  stoves) 

Radiation  at  boilers  and  stoves,  25  per  cent,    16,240  17.1 

Waste, 41,220         43.4 

Loss  of  gases  from  blast-furnaces,  in  charging,  5  per  cent,  4,740  5.0 

Sundry, 9,020           9. 5 

Total  waste  and  undeveloped  heat, 94,980       100.0 

For  the  performance  of  the  duty  according  to  these  analyses,  Mr.  Bell 
states  that  19.08  cwts.  of  carbon,  or  20.62  cwts.  of  coke,  are  required,  per 
ton  of  iron  produced  from  ore  yielding  41  per  cent,  of  iron.  In  a  furnace 
having  18,000  cubic  feet  of  capacity,  80  feet  high,  i  ton  of  No.  3  pig-iron 
was  produced  with  21^  cwts.  of  ordinary  Durham  coke,  from  Cleveland 
iron-stone. 

In  recent  years,  by  raising  the  temperature  of  the  blast  to  485°  C.,  or 
905°  F.,  the  consumption  of  coke,  with  a  furnace  48  feet  high,  was  reduced 
to  28  cwts.  per  ton  of  iron.  With  a  cold  blast,  more  than  60  cwts.  would 
probably  have  been  required. 

It  is  stated,  that  at  Barrow  works,  where  the  Siemens-Cowper  regenerative 
stove  is  employed  for  heating  the  blast  to  1100°  F.,  the  quantity  of  coke 
consumed  is  20.08  cwts.  per  ton  of  iron. 


THE   STRENGTH   OF   MATERIALS. 


The  strength  of  materials  is  measured  by  the  resistance  which  they 
oppose  to  alteration  of  form,  and  ultimately  to  rupture,  when  subjected  to 
force,  pressure,  load,  stress,  or  strain.  The  exigencies  of  scientific  precision 
have  caused  the  general  substitution  of  the  word  "stress"  for  the  good  old 
engineer's  word  "strain,"  as  expressive  of  force,  though  "strain"  may  still 
be  employed  to  express  alteration  of  form.1 

Stress  is  applied  in  five  recognized  modes : — 

i  st.  Tensile  stress,  tending  to  draw  or  pull  the  body  asunder.  The 
immediate  effect  is  elongation. 

2d.  Compressive  stress,  tending  to  crush  it.  The  immediate  effect  is 
compression. 

3d.  Shearing  stress,  tending  to  cut  it  through.  The  immediate  effect  is 
lateral  compression,  elongation,  and  deflection. 

4th.  Transverse  or  lateral  stress,  tending  to  bend  it  and  break  it  across, 
the  force  being  applied  laterally,  and  acting  with  leverage.  The  immediate 
effect  is  lateral  deflection. 

5th.  Torsional  stress,  tending  to  twist  it  asunder,  the  force  acting  with 
leverage.  The  immediate  effect  is  angular  deflection. 

Mr.  Callcott  Reilly  aptly  reduces  the  varieties  of  stress  to  three  kinds 
of  simple  stress: — Tensile  stress,  compressive  stress,  and  shearing  stress. 
These  are  the  ultimate  forms  of  stress;  they  are  combined  in  transverse 
stress,  and  the  third  is  substantially  the  form  of  torsional  stress,  where  the 
strain  is  applied  over  a  very  short  length.  Or,  where  torsional  stress  is 
applied  over  a  considerable  length,  the  tensile  form  of  stress  is  combined 
with  shearing  stress. 

When  stress  is  applied  gradually  to  a  solid  body,  the  strain,  or  alteration 
of  form,  is  proportional  to  the  intensity  of  the  stress,  so  long  as  the  inherent 
elastic  force  of  the  body  is  not  overbalanced  by  the  stress — so  long,  that  is 
to  say,  as  the  alteration  of  form  remains  within  the  elastic  limit,  the  stress, 
at  the  same  time,  remaining  within  the  limit  of  elastic  strength.  When  the 
elastic  limit  is  turned  and  exceeded,  the  body  begins  to  yield  under 
gradually  accumulating  stress,  and  the  strain  or  alteration  of  form  becomes 
proportionally  greater  and  greater  with  the  intensity  of  the  stress,  until, 
finally,  rupture  or  breakage  takes  place. 

1  As  Dr.  Pole  says,  in  his  lectures  on  Iron  as  a  Material  of  Construction,  this  word 
strain  "  appears  to  convey  its  idea  so  clearly  that  there  must  be  little  chance  of  expunging 
it  from  the  practical  mechanic's  vocabulary."  Mr.  Stoney  employs  it  exclusively  in  his 
work  on  The  Theory  of  Strains.  Mr.  Kirkaldy  uses  the  word  "stress  "in  his  reports  of 
his  experiments  on  the  strength  of  materials. 


WORK   OF   RESISTANCE   OF   MATERIAL.  5OI 

When  a  body  is  loaded  in  excess  of  the  elastic  limit,  without  breakage, 
it  returns,  when  unloaded,  towards  its  normal  form,  but  it  fails  to  regain  it. 
It  is,  in  so  far,  deformed,  and  it  has  acquired  a  permanent  set  or  a  set. 

There  are  five  data  of  importance  to  be  observed  in  the  measurement  of 
the  strength  of  materials. 

i  st.  The  limit  of  elasticity,  or  the  elastic  limit. 

2d.  The  greatest  stress  which  the  material  is  capable  of  sustaining  within 
the  elastic  limit,  or  the  elastic  strength. 

3d.  The  strain,  or  alteration  of  form — elongation,  compression,  deflec- 
tion, or  torsion — within  the  elastic  limit. 

4th.  The  total  extent  of  the  strain,  or  alteration  of  form,  with  the  set, 
before  rupture  takes  place. 

5th.  The  greatest  stress  which  the  material  is  capable  of  supporting 
before  rupture  takes  place ;  or,  the  absolute  strength. 

The  first  and  second  data  are  of  prime  importance;  the  others  are  sub- 
sidiary. For,  in  practice,  it  is  necessary,  in  order  to  insure  the  permanency 
of  a  structure,  that  its  proportions  should  be  such  that,  under  the  maximum 
stress  to  which  any  piece  is  to  be  subjected,  it  should  not  be  strained  be- 
yond the  elastic  limit  of  its  strength. 

WORK   OF    RESISTANCE   OF   MATERIAL. 

Under  a  Quiescent  Load. — Since  the  intensity  of  the  elastic  resistance  in- 
creases uniformly  with  the  total  space  through  which  the  action  of  the  stress 
takes  effect,  it  may  be  represented  by  the  triangular 
space  ABC,  Fig.  129,  in  which  AB  is  the  total  space  de- 
scribed, and  B  c  is  the  measure  of  the  stress  applied. 
Suppose  the  stress  BC=  10,000  Ibs.,  and  the  space  AB 
=  i  inch,  then  the  stress,  10,000  Ibs.,  which  has  been 
applied,  operates  through  a  space  of  i  inch,  and  has  been 
opposed  by  an  elastic  resistance  which  commenced  at 
A,  and  increased  uniformly,  from  o  at  A,  to  10,000  Ibs. 
at  B.  The  intensity  of  the  resistance  at  different  points 
along  the  space  AB,  is  measured  by  the  ordinates  of  the 
triangle  parallel  to  the  base  through  the  given  points; 
and  if  the  space  be  divided  into  four  parts,  for  example, 
at  the  points  a,  b,  c,  the  values  of  the  ordinates  a  a,  b  b', 
cc',  or  the  intensities  of  the  resistance  at  the  points 
of  elongation  a,  b,  c,  are  respectively  2500,  5000,  and  lgv 
7500  Ibs.  If  an  indefinitely  great  number  of  ordi- 
nates be  drawn,  they  will  occupy  the  whole  area  of  the  triangle,  and  the 
average  length  of  the  ordinates  will  be  half  the  base  B  c,  equivalent  to 
5000  Ibs. 

Hence  a  ready  means  of  calculating  the  quantity  of  work  necessary  to 
strain  a  piece  to  its  elastic  limit: — Multiply  half  the  elastic  strength  in 
pounds  by  the  space  in  feet  described  by  the  resistance  or  by  the  stress. 
The  product  is  the  work  expended.  If  R  =  the  elastic  strength  in  pounds, 


502  THE   STRENGTH   OF   MATERIALS. 

s  =  the  space  described  by  the  load  or  stress  in  feet,  and  w  -  the  work  done, 
the  rule  is  formulated  thus : — 

w^y2^s (i) 

For  example,  using  the  above  data,  if  10,000  be  the  elastic  strength,  and 
i  inch  or  .0833  foot  be  the  space  described,  then  the  work  done  in  strain- 
ing the  piece  to  the  limit  of  its  elastic  strength  is 

YZ  (10,000  x  .0833)  =  416. 7  foot-pounds. 

Under  a  Load  suddenly  applied. — In  these  calculations  of  stress  and  work, 
it  is  assumed  that  the  stress  is  applied  gradually,  so  that  no  appreciable 
velocity  and  momentum  be  generated  as  the  stress  is  applied.  If,  on  the 
contrary,  a  weight  equal  to  the  total  load  be  applied 
suddenly  and  all  together,  the  momentary  deflection 
under  the  load  amounts  to  twice  the  permanent  deflec- 
tion, or  twice  that  which  is  effected  by  the  load  when 
gradually  applied,  supposing  that  the  total  deflection 
-D'  does  not  exceed  the  elastic  limit.  Let  AB,  Fig.  130, 

Fig.  130.— Deflection  under  be  the  deflection  caused  by  the  gradual  application  of 
a  weight  w,  and  AB'  the  momentary  deflection  caused  by 
the  sudden  application  of  it.  Draw  the  ordinates  BC  and  B'C'  to  measure  the 
resistance  at  the  points  B  and  B',  and  complete  the  triangle  A  B'C'.  Through 
c  draw  the  vertical  D  D'.  Then  the  rectangle  A  B'  D'  D  measures  the  work 
done  by  the  load  in  falling  through  the  height  AB',  and  the  triangle  A  B'C' 
measures  the  work  of  resistance  to  deflection.  These  are  equal  to  each 
other;  and  as  B'C'  must  be  twice  B'D',  so  AB'  is  twice  AB;  that  is  to  say, 
the  momentary  deflection  under  a  load  suddenly  applied  is  twice  the  steady 
deflection  under  the  same  load  very  gradually  applied. 

It  follows  that  in  proportion  to  the  rapidity  with  which  loads  are  applied, 
as  when  railway  trains  run  upon  a  bridge,  of  course  in  the  absence  of  per- 
cussive action,  the  deflection  is  greater  than  that  due  to  the  same  load  at 
rest  on  the  bridge,  and  increases  with  the  speed  of  transit.  But  it  does  not 
amount  to  twice  the  deflection  due  to  a  quiescent  load,  though  it  approaches 
to  this  limit  as  the  speed  increases. 

Under  Stress  by  Percussion. — When  a  solid  material  is  exposed  to  percus- 
sive stress,  as,  for  instance,  when  a  heavy  weight  falls 
upon  a  beam  transversely,  the  work  of  resistance  is 
measured  by  the  product  of  the  weight  by  the  total 
fall — the  total  fall  being  equal  to  the  height  of  fall 
above  the  beam  plus  the  deflection.  To  exemplify 
percussive  action  within  the  elastic  limit,  reproduce 
Fig.  130  on  the  same  scale,  in  thick  lines,  in  Fig.  131, 
with  the  same  letters  of  reference,  and  let  A' A  be 
the  height  of  fall  above  the  beam,  and  B'  B"  the 
additional  deflection  under  the  weight.  A'B"  is  the 
total  fall,  and  AB"  the  total  deflection.  Draw  the 
horizontal  line  B"C",  and  produce  AC'  and  DD'  to 
^c"  meet  it  at  c"  and  D".  Then  the  rectangle  A'B^D^D'" 
measures  the  work  for  the  total  fall,  and  the  triangle 


D' 


Fig.  131.— Deflection  under    ABC    measures  the  work  of  resistance  to  deflection; 

and   these   quantities    of    work   are    equal   to   each 

other.     The  scale  of  the  diagram  above  the  normal  level  of  the  beam,  A  D,. 


TRANSVERSE  STRENGTH   OF   HOMOGENEOUS   BEAMS.         503 

is  the  same  as  that  of  the  portion  below  A  D,  for  the  sake  of  simplicity  of 
illustration,  and  for  ready  comparison  of  the  squares  representing  quantities 
of  work. 

COEFFICIENT  OF  ELASTICITY. 

The  elasticity  of  a  bar  of  any  solid  material  subjected  to  a  direct  tensile 
or  a  direct  compressive  force,  within  the  elastic  limits,  is  measured  by  a 
constant  fraction  of  the  length  per  unit  of  force  per  unit  of  sectional  area. 
The  unit  of  force  and  area  is  usually  taken  as  one  pound  per  square  inch, 
but  it  is  sometimes  taken  as  one  ton  per  square  inch.  E  is  used  to 
symbolize  the  denominator  of  the  fraction.  For  example,  if  a  bar  of  iron 
be  extended  V^.oooth  part  of  its  length  per  ton  of  stress  per  square  inch  of 
section, 

12,000     E' 

The  bar  would  therefore  be  stretched  to  double  its  normal  length  by  a  force 
of  12,000  tons  per  square  inch,  if  the  material  were  perfectly  elastic.  The 
supposition,  though  imaginary,  is  convenient;  and  the  coefficient  of  elasticity 
is  usually  denned  as  the  weight  which  would  stretch  a  perfectly  elastic  bar 
of  uniform  section  to  double  its  length.  It  is  represented  by  E,  which  may 
be  employed  to  express  pounds,  tons,  or  any  other  measure  of  weight. 

The  coefficient  of  elasticity  may  also  be  expressed  in  terms  of  the  length 
in  feet  of  a  bar  of  the  given  material,  the  weight  of  which  would  be  equal 
to  the  force  required  to  stretch  it  to  twice  its  normal  length.  For  example, 
a  i-inch  square  bar  of  iron  weighing  33^  pounds  per  lineal  foot  would  require 
to  be  (12,000  x  2240  -r  3^3  =  )  8,064,000  feet  in  length  to  stretch  it  at  the 
upper  end  to  twice  the  normal  length,  and  this  is  another  expression  for  the 
coefficient  of  elasticity. 

The  same  methods  of  expressing  the  coefficient  of  elasticity  are  applied 
to  the  elastic  resistance  to  compression.  That  is,  the  coefficient,  in  weight, 
is  expressed  by  the  denomination  of  the  fraction  of  its  length  by  which  a 
bar  is  compressed  per  unit  of  weight  per  square  inch  of  section. 

TRANSVERSE   STRENGTH   OF   HOMOGENEOUS   BEAMS. 

Tensile  resistance  is  selected  as  the  basis  of  the  following  formulas,  which 
are  constructed  on  the  assumption  that,  within  elastic  limits,  extension  is 
equal  to  compression,  under  equal  stresses;  and  their  strict  application  is 
confined  to  the  calculation  of  stress,  strain,  and  strength,  within  elastic 
limits.  At  the  same  time,  they  are  practically  applicable  for  calculating 
ultimate  strength. 

i.  SYMMETRICAL  SOLID  BEAMS. 

Let  a  homogeneous  beam,  AB,  Fig.  132,  of  rectangular  section,  be  freely 
supported  horizontally  at  both  ends,  at  a  and  b.  Bisect  the  depth  cd 
at  oy  and  draw  the  horizontal  line  n  op.  Let  the  beam  be  loaded  by  the 
weight  W,  applied  at  the  middle,  cd,  of  the  beam.  Then  the  beam  is 
deflected  under  the  load,  and  the  upper  half,  above  the  line  n  op,  is  com- 
pressed, and  the  lower  -half  is  extended,  in  such  a  manner  that,  having 
regard  to  the  vertical  section  cd,  the  compression  and  elongation  respectively 
increase  uniformly  from  zero  at  the  central  point,  0,  to  a  maximum  at  the 
upper  and  lower  surfaces,  c  and  d.  The  proportional  increase  may  be 


504 


THE   STRENGTH   OF   MATERIALS. 


represented  by  the  triangles  oef  and  ogh,  formed  by  the  lines  eh  an 
intersecting  at  the  central  point  o\  in  which  the  graduated  shortening  and 

lengthening  of  the  fibres 
at  any  given  height  are 
represented  by  horizontal 
lines  drawn  across  the  tri- 
angles. 

The  horizontal  com- 
pressive  stress  in  the 
upper  half,  and  the  ten- 
sile stress  in  the  lower 
half  of  the  beam,  with 
respect  to  the  section 
cd,  likewise  increase  uni- 
formly from  the  central 
point  o,  where  they  are 
zero,  to  the  upper  and 
lower  surfaces  at  c  and  d.  Such  is  the  ordinary  theory  of  transverse  stress 
in  a  rectangular  beam,  and  it  is  assumed  that,  throughout  the  whole  length  of 
the  beam,  as  at  the  section  at  the  middle,  there  is  no  horizontal  stress  in  the 
central  line  nopt  with  respect  to  any  vertical  section.  This  line  is  therefore 
called  the  neutral  line  or  neutral  axis;  and  it  is  a  line  of  demarcation  between 
the  directly  horizontal  compressive  stress  above  and  the  tensile  stress  below. 
Further,  the  sum  of  the  compressive  stress  above  the  neutral  axis  is  equal 
to  the  sum  of  the  tensile  stress  below,  and  each  may  be  replaced  by  its  re- 
sultant stress  at  the  resultant  centre  without  affecting  the  equilibrium.  If 
the  cross  section  of  the  beam,  Fig.  133,  No.  i,  be  divided  into  a  number  of 
strips  of  equal  thickness,  the  moment  of  stress  for  each  strip  in  the  upper 
and  in  the  lower  group,  with  respect  to  the  neutral  axis  n  op,  may  be  calcu- 
lated, and  the  sum  of  the  moments  for  each  group  divided  by  the  sum  of  the 


Fig.  132. — Transverse  Stress  on  a  Rectangular  Beam. 


fig-  133,  No.  i. — Longitudinal 
Resistance  in  Loaded  Beams. 


-  !33>  No.  2. — Diagonal 
Resistance  in  Loaded  Beams. 


Fig-  I33>  No.  3. — Combined  Resist- 
ance in  Loaded  Beams. 


resistances,  when  the  quotient  will  be  the  resultant  radius.  But  this  calcu- 
lation may  be  saved  by  drawing  the  diagonals  eh  and/^",  when  it  is  apparent 
that  the  shaded  triangles  formed  by  them  exhibit  the  relative  quantities  of 
stress  for  each  strip,  and  that  the  resultant  lines  of  stress,  m  m  and  //,  pass 
through  the  centres  of  gravity  of  the  triangles,  each  at  a  distance  from  the 
neutral  axis  equal  to  two-thirds  of  the  half-depth  of  the  beam.  (Fig.  133,1.) 
These,  the  moments  of  the  normal  stresses  or  resistances  due  to  the  abso- 


TRANSVERSE   STRENGTH   OF   HOMOGENEOUS   BEAMS.        505 


No.  i. 


Let   c  B  d, 
a  triangular 


lute  horizontal  compression  and  extension  of  the  beam,  are  supplemented 
by  diagonal  resistances,  by  which  each  of  them  is  augmented  75  per  cent. 

To  elucidate  the  origin  of  diagonal  resistance,  it  may  be  observed  that 
the  upper  and  lower  portions  of  a  beam — above  and  below  the  neutral 
axis — may  be  considered  as  two 
individual  members  of  a  frame, 
united  at  their  surface  of  contact 
— the  neutral  axis. 
Figs.  134,  No.  i,  be 
frame,  fixed  at  cd  and  loaded  at  B. 
The  pieces  c  c1  and  d  d'  of  the 
upper  and  lower  members  are 
respectively  extended  and  com- 
pressed, when  the  load  is  applied, 
to  the  lengths  c  c"  and  dd".  If 
the  members  of  the  frame  are 
placed  parallel  to  each  other,  in 
close  contact,  as  in  Figs.  134, 
No.  2,  extension  and  compres- 
sion take  place  as  before.  Let,  „. 
now,  the  two  members  be  united 
in  the  line  nop,  No.  3,  and  so 
consolidated  as  to  form  a  semi- 
beam;  the  extension  and  the 
compression  partially  neutralize 
each  other: — at  the  neutral  line 
nop  they  are  absolutely  neutral- 
ized, and  the  amounts  of  exten- 
sion and  of  compression  are 
represented  by  the  triangles  c' c" o 
and  d'  d"  o.  The  structure  is 
thus,  in  a  certain  sense,  crip- 
pled;  and  the  extension  and 


Figs.  134. — Stress  in  a  Loaded  Beam. 


compression,    instead    of    being 

rectilineal,  are  curvilineal,  and  the  semi-beam  is  deflected,  as  in  No.  4. 

The  counteraction  here  pointed  out  is  necessarily  exerted  diagonally,  at 
an  angle  of  45°  with  the  neutral  line;  as  in  the  line  o'c't  Fig.  135,  at  45° 
with  the  transverse  section  c'd'.  The  diagonal  forces,  as  applied  to  the 


d' 


Fig.  135.—  Diagonal  Stress  in  a  Beam. 


o    y 

Fig.  136. — Diagonal  Stress  in  a  Beam. 


transverse  section  c'd'  strained  into  the  position  c"d",  are  represented  by 
diagonals  at  45°  drawn  from  points  in  the  upper  half-depth  c"  o  to  the 
neutral  line  oo't  for  tensile  resistance;  and  from  points  in  the  lower  half- 


506  THE   STRENGTH   OF   MATERIALS. 

depth  od",  for  compressive  resistance.  Reproduce  the  upper  half-depth 
to  a  larger  scale,  in  Fig.  136.  Draw  the  perpendicular  c'n,  and  thence  the 
perpendicular  ns;  the  diagonal  extension  is  measured  by  nc",  and  the 
horizontal  component  sc"  is  the  horizontal  extension  at  the  upper  side  of 
the  beam,  due  to  the  length  of  the  portion  c'c'".  These  extensions  are 
also  measures  of  the  diagonal  force,  and  its  horizontal  component.  Take 
next  the  horizontal  line  t  u,  at  any  other  position  in  the  half-depth.  The 
diagonal  vu  becomes,  when  strained,  vu',  at  45°,  and,  by  the  same  con- 
struction as  before,  the  extension  of  the  diagonal  is  measured  by  z  u',  whilst 
its  horizontal  component  is  x  u1.  It  is  easily  deduced  from  the  similarity 
of  the  construction,  that  the  forces  measured  by  the  horizontal  components 
xu'  and  sc"  are  equal  to  each  other.  Similarly,  the  horizontal  components 
of  the  diagonal  forces  acting  throughout  the  whole  depth  of  the  section  c'  o, 
are  equal  to  each  other.  They  may,  therefore,  be  represented  in  their 
entirety  by  the  rectangle  c'syo,  of  which  the  length  c'  s  is  equal  to  half  the 
length  c'  c"  of  the  triangle  c'  c"  o\  and  the  areas  of  force  represented  by  the 
rectangle  and  the  triangle,  are  equal  to  each  other. 

By  a  similar  argument,  the  diagonal  compressive  resistance  in  the  lower 
section  of  the  semi-beam,  may  be  analyzed.  It  is  the  simple  converse  of 
the  diagonal  tensile  resistance  in  the  upper  section. 

The  horizontal  resistance  due  to  diagonal  stress,  is  represented  diagram- 
atically,  on  the  cross  section,  by  Fig.  133,  No.  2,  in  which  the  shaded  area 
inclosed  by  the  verticals  e'g'  and/'/#'  represents,  in  its  upper  half,  the  tensile 
stress;  and  in  its  lower  half,  the  compressive  stress;  for  which  the  resultant 
lines  of  stress,  m  m  and  //,  are  each  at  a  distance  from  the  neutral  axis 
equal  to  half  the  half-depth  of  the  beam.  The  two  elements  of  resistance 
(No.  i  and  No.  2)  are  combined  in  Fig.  133,  No.  3,  showing  in  deep  shad- 
ing the  combined  areas  of  stress  of  uniform  intensity;  the  amount  of  which 
is  equal  to  that  of  the  semi-rectangle.  This  investigation  for  a  semi-beam 
is  applicable  for  a  beam  supported  at  the  ends,  and  loaded  at  the  middle, 
like  Fig.  132,  the  tensile  and  compressive  stresses  being  inverted.  The 
tensile  and  compressive  stresses  act  at  a  resultant  radius,  measured  from  the 

neutral  line,  of  ((-  +  -)-=-  2  =)—  or  .5833,  taking  the  half-depth  as  i.     As 

\    3       2  /  12 

•5775  is  the  geometrical  radius  of  gyration  of  the  semi-rectangle  on  the 
neutral  axis,  when  the  half-depth  =i  (see  page  289),  the  moment  of  resist- 
ance will,  for  simplicity,  be  taken  as  .5775,  when  the  area  of  the  semi- 
rectangle  and  the  half-depth  are  each  represented  by  unity.  Then  the 
total  moment  of  either  resistance,  tensile  or  compressive,  with  reference  to 
the  neutral  axis,  is  expressed  by  the  product  of  half  the  sectional  area  of 
the  beam  by  half  the  depth,  and  by  .5775,  and  by  the  extreme  tensile  or 
compressive  stress  per  unit  of  sectional  area.  That  is  to  say,  by, 

................  (a) 


in  which  £  =  the  breadth,  ^=the  depth,  both  in  inches;  and  j=the  extreme 
tensile  stress  per  square  inch,  to  which  the  extreme  compressive  stress  is 
taken  as  equal.  The  sum  of  the  moments  of  the  tensile  and  the  com- 
pressive resistances  is,  therefore,  practically  twice  the  moment  (  a  )  round 
the  neutral  axis;  or 

.1444  £</2.rx2  =  .2888  bd*s  ................   (b) 


SYMMETRICAL  SOLID   BEAMS.  507 

When  the  beam  is  loaded  to  the  point  of  rupture,  the  extreme  tensile 
stress  in  the  lower  surface  of  the  beam  becomes  equal  to  the  ultimate 
strength  of  the  material.1 

To  express  the  moment  of  the  load  or  weight  W  :  —  Each  of  the  supports 
a  and  b  carries  half  the  weight,  and  presses  upwards  with  a  force  equal  to 

—  ;  the  leverage  of  each  of  these  pressures  on  the  section  of  the  beam  at 
2  / 

the  middle,  is  half  the  span  /,  equal  to  ad  or  db,  and  is  —  ;  therefore  the 

moment  of  the  weight  at  the  middle  is  expressed  by  — 

W      /      W/  . 


The  moment  of  the  weight  (c)  is  necessarily  equal  to  the  moment  of 

W/ 
resistance  (b);  or,  —  =.2888  bdzs,  and  W/=  1.155  bdzs\  whence, 


W/ 


That  is  to  say,  when  the  beam  is  supported  at  both  ends,  and  the  weight  is 
applied  at  the  centre,  the  breaking  weight  is  equal  to  the  product  of  the 
breadth  by  the  square  of  the  depth,  and  by  the  ultimate  tensile  strength 
per  square  inch,  and  by  1.155;  divided  by  the  span. 

Also,  the  ultimate  tensile  strength  per  square  inch  is  equal  to  the  product 
of  the  breaking  weight  by  the  span;  divided  by  the  product  of  the  breadth 
by  the  square  of  the  depth,  and  by  1.155. 

The  formula  (  i  )  signifies  that  the  breaking  weight  varies  directly  as  the 
breadth  of  the  beam,  directly  as  the  square  of  the  depth,  directly  as  the 
tensile  strength  of  the  material,  and  inversely  as  the  span. 

When  the  beam  is  fixed  at  both  ends  and  loaded  at  the  middle,  the  breaking 
weight  is  equal  to  2  times  that  of  a  beam  freely  supported,  or 

_  i-733  bd*s 


When  the  beam  is  fixed  at  one  end  only,  and  loaded  at  the  other  end,  the 
breaking  weight  is  one-fourth  of  that  of  a  beam  freely  supported  at  both 
ends;  or 


When  the  load  is  applied  at  any  other  point  than  the  middle  of  a  beam  freely 
suspended  at  both  ends,  let  m  and  n  denote  the  two  segments  into  which  the 

1  In  the  intervention  of  the  diagonal  stress  in  deflected  rectangular  beams,  according  to 
the  analysis  in  the  text,  is  found  the  solution  of  the  mystery  of  the  "  resistance  of  flexure," 
which  has  been  so  denominated,  and  has  been  experimentally  demonstrated  by  Mr.  W. 
H.  Barlow.  The  contrast  between  the  action  of  the  diagonal  bracing  of  lattice-girders 
and  that  of  the  solid  web  of  web-girders,  throws  a  flood  of  light  on  the  recondite  strains 
in  webs  and  in  solid  beams. 


508  THE   STRENGTH   OF   MATERIALS. 

length  is  divided  by  the  load.      Then  the  breaking  weight  is  inversely 
proportional  to  the  product  of  the  segments,  m  x  n.     At  the  middle,  in  n 

III2      TTTI  TW     1.  1  5«?  bd*s     I2 

=  —  x  —  =  —  .     Whence  W  =  —  ^-  -  x  —^mn-,  or, 
224  /  4 


When  the  weight  is  uniformly  distributed  along  the  beam,  the  total 
breaking  weight  is  equal  to  twice  the  breaking  weight  as  applied  at  the 
middle  of  a  beam  supported  at  both  ends,  or  at  the  end  of  a  beam  fixed  at 
the  other  end. 

When  the  beam  is  supported  at  both  ends  and  uniformly  loaded, 

.........................  (6) 


When  the  beam  is  fixed  at  both  ends  and  uniformly  loaded, 

_  3.466  bd*s 


When  the  beam  is  fixed  at  one  end  only,  and  uniformly  loaded, 


GENERALIZED  FORMULA  FOR  THE  BREAKING  WEIGHT  OF  SYMMETRICAL 

SOLID  BEAMS. 

Let  the  area  of  the  section,  b  d,  be  represented  in  the  expression  (#), 
by  a,  and  the  radius  of  gyration  by  r,  the  half-depth  of  the  beam  being  =  i  ; 
then,  by  substitution,  the  moment  of  resistance  of  the  half-depth  on  the 

neutral  axis,  is  expressed  by—  x—   •*  r  x  s  =         —  ;  and  twice  the  moment 

22  4 

is 


X    2     =  .  ....................     (d) 

2 
Then, 


4  2 

whence  the  general  formulas  — 

w  = 


...........................    do) 

2  adr 


W  =  the  breaking  weight  at  the  middle. 
a  -  the  sectional  area  of  the  beam,  in  square  inches. 
</=the  extreme  depth  of  the  beam,  in  inches. 
r  =  the  radius  of  gyration,  taking  the  half-depth  equal  to  i. 
/=  the  length  or  span  of  the  beam  between  the  supports,  in  inches. 
s  -  the  ultimate  tensile  strength  of  the  material,  per  square  inch. 


SOLID   BEAMS   OF   SYMMETRICAL   SECTION. 


509 


The  weight  and  the  tensile  strength  are  to  be  expressed  in  terms  of  the 
same  unit  of  weight,  whether  in  pounds,  hundredweights,  or  tons. 

The  formulas  (9)  and  ( 10)  are  applicable  to  solid  beams  of  any  symme- 
trical form  of  section,  no  part  of  which  is  overhung,  and  for  any  weight  and 
maximum  tensile  stress  not  exceeding  the  ultimate  weight  and  stress  for  the 
material. 

i.  SOLID  BEAMS  (WITHOUT  OVERHANG)  OF  SYMMETRICAL  SECTION. 

The  sections  of  beams  which  may  be  noticed  are  shown  by  Figs.  137,  in 
which  the  radius  of  gyration  above  and  below  the  neutral  axis  as  a  centre, 
is  marked  by  a  horizontal  line.  The  neutral  axis  passes  through  the  centre 
of  gravity  of  each  section  at  the  level  of  half  the  depth. 


No.  i. 


No.  2. 


No.  3. 


No.  4- 


No.  5. 


Figs.  137. — Symmetrical  Solid  Beams,  without  Overhang.     Sections. 

i.  Square  and  Rectangular  Sections. — Nos.  i  and  3.     The  radius  of  gyra- 
tion is  .5775  when  the  half-depth  is  i;  and  substituting  this  value  of  r  in 

formula  (9),  W  =  2  x  "577/5  x 

W=  — 

For  square  beams, 


_  I-I55 


(12) 


2.  Square  Section,  with  a  Diagonal  Vertical. — No.  2.  The  radius  of  gyra- 
tion is  .4083  when  the  half-diagonal  is  i ;  and,  by  substitution,  in  formula 
( 9 ),  and  reduction, 


The  diagonal  is  equal  to  1.414  times  the  side;  and,  calling  the  side  d\ 
'=  1.414  d'.     Substitute  this  value  in  formula  (  13  ),  and  reduce;  then, 

W=1^5S 


This  value  of  W  is  the  same  as  that  for  an  upright  square  section  ( 1 1 ), 
showing  that  a  square  beam  is  equally  strong  whether  placed  upright  or 
diagonally. 


5io 


THE   STRENGTH   OF   MATERIALS. 


3.   Circular  Section. — No.  4.     The  radius  of  gyration  is  half  the  radius  of 
the  section,  and,  by  substitution  in  formula  (9),  W=  2  x  '^  xa — -;  or, 


in  which  d  is  the  depth  or  the  diameter.     The  transverse  strength  of  circu- 
lar sections  varies  as  the  cube  of  the  diameter. 

4.  Elliptical  Section. — No.  5.      The  area  is  .7854  time  the  product  of 
the  breadth  by  the  depth,  and  the  radius  of  gyration  is  half  the  half-depth. 


Substituting  in  formula  (  9  ),  W  = 


_  2  x  .50  x  .7854  bd-x.  d*  s > 


or 


W  = 


.7854  b  d*s 

~~7        ' 


(16) 


Let  the  ratio  of  the  breadth  to  the  depth  =  <:,  then  b  =  cd,  and 


2.  FLANGED  OR  HOLLOW  BEAMS  OF  SYMMETRICAL  SECTION. 

Hollow  or  flanged  beams  may  be  generally  described  as  beams  of  over- 
hung section.  In  such  beams  the  diagonal  resistance  to  flexure  is  only 
excited  in  the  vertical  portions  of  the  section.  Figs.  138  are  examples 
of  overhung  sections;  and  the  neutral  axis  passes  through  the  centre  of 
gravity. 

No.  i.  No.  2.  No.  3.  No.  4. 


Figs.  138. — Symmetrical  Flanged  Beams.     Sections. 

i.  Hollow-rectangular  or  Double-flanged  Sections. — Nos.  i,  2,  3. 

When  the  depth  is  considerable  compared  with  the  thickness  of  the  flanges, 
calculate  for  the  flanges  and  for  the  web  separately.  The  separation  of 
the  web  from  the  flanges  is  shown  in  Fig.  139,  for  the  flanged  beam,  No.  i, 
and  it  is  to  be  done  in  the  same  manner  for  the  hollow  beam,  No.  3. 
The  moment  of  resistance  of  one  flange  is  sensibly  equal  to  the  product  of 
its  sectional  area  multiplied  by  the  distance  d"  between  the  centres  of  the 
flanges,  and  by  the  tensile  strength  per  square  inch;  or  to 


FLANGED  BEAMS   OF   SYMMETRICAL   SECTION. 


in  which   /  is  the  depth  or  thickness  of  a  flange,  and  a  its  sectional 
area. 

The  web  is  treated  as  a  rectangular  beam  of  the 
depth  d' ',  the  distance  between  the  centres  of  the 
flanges.  This  is  greater  than  the  actual  depth,  as 
between  the  flanges;  but  the  excess  is  compensated 
by  the  metal  rilled  in  at  the  angles.  Putting  t'  for 
the  thickness  of  the  web,  the  moment  of  resistance 
is,  by  (£),  page  506, 


.2888  t'd"2s=  2888  a" d" s,  (g) 

\  o    / 

in  which  a"  is  the  sectional  area  of  the  web.  The 
sum  of  the  moments  (f)  and  (g)  is  equal  to  the 
moment  of  the  weight ;  or, 

W  / 

4 
whence,  W/-4  ad" s+  1.155  a" d" s  =  d" s  (4  a+  1.155  a 


Fig.  139. — Calculation  of 
Strength  of  Beam. 


(18) 


That  is  to  say:  When  the  depth  is  considerable  compared  with  the  thickness 
of  the  flanges  —  multiply  the  sectional  area  of  one  flange  by  4  ;  and  multiply 
the  sectional  area  of  the  web  by  1.155.  Add  the  products  together,  and 
multiply  the  sum  by  the  reputed  depth  of  the  beam,  and  by  the  tensile 
strength  per  square  inch,  and  divide  the  product  by  the  span.  The  quotient 
is  the  breaking  weight. 

Note.  —  The  reputed  depth  of  the  beam,  and  also  that  of  the  web,  are 
taken,  for  calculation,  as  the  total  depth  minus  the  thickness  of  one  flange. 

2d  Method.  —  In  some  cases  the  strength  of  the  flanges  only  is  calculated, 

W/ 

when  the  web  is  comparatively  slight.     Then,  —  =  a  d"  s,  or  W  /  =  4  a  d"  s; 


4  ad's  _4  bid's 


(2o) 


That  is  to  say:    When  the  strength  of  the  web  is  neglected,  the  breaking 
weight  is  equal  to  four  times  the  sectional  area  of  one 
flange  by  the  distance  apart  between  the  centres  of  the 
flanges,  and  by  the  ultimate  tensile  strength  per  square 
inch,  divided  by  the  length. 

In  applying  the  formula  to  the  hollow  beam,  No.  3, 
Fig.  138,  t  is  taken  as  the  sum  of  the  thicknesses  of 
the  sides. 

When  the  thickness  of  the  flanges  is  considerable  com- 
pared with  the  depth  of  the  beam,  No.  4,  Figs.  138,  and 
Fig.  140. — In  the  double-flanged  section,  Fig.  140,  cal- 
culate the  strength  of  the  web  for  the  whole  depth, 
as  indicated  in  sectioning.  For  the  lateral  flange 
portions,  the  average  stress  is  less  than  s  in  the  ratio  of  the  total  depth  d 

d" 
to  the  reputed  depth  d",  or  it  is  s  —•  and  the  net  area  of  one  flange,  or  of 


Fig.  140. — Calculation 
of  Strength  of  Beam. 


512 


THE   STRENGTH    OF   MATERIALS. 


the  unshaded  parts,  Fig.  140,  being  put  equal  to  a',  the  moment  of  resist- 
ance of  one  flange  is,  ,„  ,,,a 

t   $  7//  tCL  /      i     \ 

~X~S  ...................... 


The  sum  of  the  moments  of  resistance  of  the  flanges  and  the  web  is  equal 
to  the  moment  of  the  weight,  or 


(21 


in  which  /  =  the  thickness  of  the  web,  and  ^= 

rpi 

'd2  'd~  + 


depth  of  the  section. 


1.155  *  '  d2};  and 


w  = 


(4     — 


1.155 


(22) 


That  is  to  say :  When  the  thickness  of  the  flanges  is  considerable  compared  with 
the  depth  of  the  beam. — Multiply  the  net  sectional  area  of  one  flange,  cal- 
culated for  its  width  minus  the  thickness  of  the  web,  by  the  square  of  the 
reputed  depth,  and  by  4,  and  divide  by  the  total  depth.  Multiply  the 
thickness  of  the  web  by  the  square  of  the  total  depth,  and  by  1.155.  Add 
together  the  quotient  and  the  product,  and  multiply  the  sum  by  the  tensile 
strength  per  square  inch,  and  divide  the  product  by  the  span.  The 
quotient  is  the  breaking  weight. 

Note. — The  reputed  depth  is  equal  to  the  total  depth  minus  the  thick- 
ness of  one  flange.  Double-headed  rails,  as  No.  4,  Fig.  138,  will  be  speci- 
ally treated. 

2.  Annular  Section,  Figs.  141,  No.  i. — The  hollo wness  of  the  section 
deprives  it  of  a  large  proportion  of  the  diagonal  resistance  exerted  in  the 


No.  i. 


No.  2. 


No. 


Figs.  141. — Symmetrical  Hollow  Beams.     Sections. 

solid  circular  section;  otherwise  the  strength  might  have  been  calculated 
from  the  section,  Fig.  143,  in  which  the  material  of  the  annular  section 
is  collected  about  the  vertical  centre  line.1  The  lateral  portions  of  the 

1  This  mode  of  aggregation  of  the  section  is  employed  in  Mr.  Edwin  Clark's  work  on 
the  Britannia  Bridge,  page  ill;  and  also  by  Mr.  Baker  in  his  excellent  work  on  the 
Strength  of  Beams,  page  26.  Mr.  Baker  very  properly  points  out  the  fallacy  of  the 
ordinary  mode  of  calculating  the  transverse  strength  of  a  beam  of  annular  section,  which 
does  not  take  cognizance  of  the  loss  of  "resistance  to  flexure"  in  a  hollow  beam. 


FLANGED   BEAMS   NOT   SYMMETRICAL   IN   SECTION. 


513 


section,  end,  c'  pd\  Fig  142,  are  by  their  position  subject  to  diagonal  stress, 
and  they  are  reproduced  in  darker  shading  in  Fig.  143.  The  breaking 
strength  may  be  approximated  to  by,  in  the  first  place,  reducing  the 
breadth  of  the  overhung  portions,  and  calculating  the  strength  of  the  re- 
duced section,  on  the  principle  to  be  explained  in  treating  of  beams  of 
unsymmetrical  sections. 


Figs.  142,  143. — Annular  Section  of  Beams. 


Fig.  144.  —Thin  Annular  Section  of  Beam. 


zd  Method. — When  the  section  is  thin,  as  in  Fig.  144,  the  matter  of  the 
section  may  be  assumed  to  be  collected  at  the  outer  circumference,  for 
which  the  radius  of  gyration  is  .7071,  when  the  radius  of  the  section  is  i; 
whence,  rating  the  stress  s  exerted  throughout  the  section  as  the  maximum 

stress,  the  resultant  point  of  resistance,  V,  of  the  half-section  is  •?°7I  =  .50, 

i 

when  the  radius  is  i ;  or  the  distance  of  the  centres  of  resistance,  V,  V,  is 
half  the  diameter.  The  sectional  area  is  equal  to  the  product  of  the 
circumference  by  the  thickness,  or  to  3.14  dxt;  and  the  half-section 
=  1.57  dt.  The  moment  of  the  half-section  is  1.57  dt^y^  ^=.785  d*t, 

and  — =  .785  d2  ts;  whence, 
4 


Hollow  Elliptical  Sections. — These  sections,  Nos.  2  and  3,  Fig.  141, 
page  512,  may  be  treated  on  the  same  principle  as  the  annular  sections, 
No.  i,  Figs.  141,  and  Fig.  143. 

2d  Method. — When  the  section  is  thin  the  breaking  strength  is,  by 
adapting  formula  (  23 ),  putting  b  =  the  breadth,  and  d  -=  the  depth, 


W=*.i4?_±fL, 


or, 


W=  1.57 


(24) 


3.  FLANGED  BEAMS  WHICH  ARE  NOT  SYMMETRICAL  IN  SECTION. 

For  such  beams,  of  which  the  sections,  Figs.  145,  are  examples,  it 
is  necessary  to  ascertain  the  quantity  of  longitudinal  tensile  resistance, 
and  the  distance  apart  of  the  resultant  centres  of  tensile  and  compressive 
stress,  for  a  given  section;  and  to  multiply  these  together  to  obtain  the 
moment  of  resistance  of  the  section;  whence  the  ultimate  transverse 
strength  may  be  calculated.  The  first  operation  is  to  find  the  neutral  axis 

33 


514 


THE   STRENGTH   OF   MATERIALS. 


of  the  section;  and  as  the  ultimate  longitudinal  resistance  in  the  web  is 
greater  than  that  of  a  flange,  the  neutral  axis  does  not  pass  through  the 
centre  of  gravity  of  the  section.  But,  if  the  area  of  the  flange  be  reduced 
in  proportion  to  the  potential  or  ultimate  unit-resistance  in  the  web  to  that 


No.  i. 


No.  2. 


No. 


_J 

No.  4. 


IT  3 


No.  5. 


0 


Figs.  145. — Sections  of  Unsymmetrical  Beams. 

of  the  flange,  or  as  1.73  to  i,  the  neutral  axis  will  pass  through  the  centre 
of  gravity  of  the  reduced  section. 

RULE. — To  find  the  neutral  axis  of  a  beam  of  unsymmetrical  section.  Divide 
the  section,  as  reduced,  into  its  simple  elements,  and  assume  a  datum-line 
from  which  the  moments  of  the  elements  are  to  be  calculated.  Multiply 
the  area  of  each  element  by  the  distance  of  its  own  centre  of  gravity  from 
the  datum-line,  to  find  its  moment.  Divide  the  sum  of  these  moments  by 
the  total  reduced  area;  and  the  quotient  is  the  distance  of  the  centre  of 
gravity  of  the  reduced  section,  or  of  the  neutral  axis  of  the  whole  section, 
from  the  datum-line. 

For  example,  the  _L  section,  No.  i,  Figs.  145,  and  shown  in  Fig.  146 
annexed,  is  12  inches  deep,  12  inches  wide,  and  i  inch  thick  throughout. 
Extend  the  web,  c  d,  to  the  lower  surface  at  d'  and  d",  leaving  5  ^  inches 
of  web,  a  d'  and  d"  b,  on  each  side.  Reduce  this  width  in  the  ratio  of  1.73 
to  i,  or  to  (5.5  -•- 1.73  = )  3.2  inches,  and  set  off  d'  a'  and  a"  b'  each  equal  to 
3.2  inches.  Then  the  reduced  flange  a' b' is  (6.4+1=)  7.4  inches  wide, 
and  the  reduced  section  consists  of  the  two  rectangles  a'  b'  and  cd.  Assume 
any  datum-line,  as  ef  at  the  upper  edge  of  the  section,  and  bisect  the 
depths  of  the  rectangles,  or  take  the  intersections  of  their  diagonals  at  g  and 
h,  for  their  centres  of  gravity.  The  distances  of  these  from  the  datum-line 
are  5  ^  and  1 1  ^  inches  respectively,  and  the  areas  of  the  rectangles  are 
ii  x  i  =  ii  square  inches,  and  7.4  x  i  =  7.4  square  inches.  By  the  rule, 


FLANGED   BEAMS   NOT   SYMMETRICAL    IN   SECTION. 


515 


Upper  rectangle, n     x    $%=   60.5 

Lower       do 7.4x11^  =    85.1 

18.4  x  7.91  =  145.6 

Showing  that  the  centre  of  gravity  of  the  reduced  section,  being  the  neutral 
axis  of  the  whole  section,  is  7.91  inches  below  the  upper  edge,  in  the  line 
ii.  The  centre  of  gravity  of  the  entire  section,  it  may  be  added,  is  8.63 
inches  below  the  upper  edge,  or  .72  inch  lower  than  that  of  the  reduced 
section. 

The  neutral  axes  of  the  other  sections,  Figs.  145,  found  by  the  same 
process,  are  marked  on  the  figures.  The  section  of  a  flange  rail,  No.  5, 
which  is  very  various  in  breadth,  may  be  treated  in  two  ways :  either  by 
preparatorily  averaging  the  projections  of  the  head  and  the  flange  into 
rectangular  forms;  or,  by  taking  it  as  it  is,  and  dividing  it  into  a  con- 
siderable number  of  strips  parallel  to  the  base,  for  each  of  which  the 
moment  with  respect  to  the  assumed  datum-line  is  to  be  found.  The  first 
mode  of  treatment  is  approximate ;  the  second  is  more  nearly  exact. 

Ultimate  Strength  of  Beams  of  Unsymmetrical  Section. — Resuming  the  _L 
section,  Fig.  146,  for  which  the  neutral  axis  has  been  ascertained,  to  find 
the  tensile  resistance,  divide  the  portion  below  the  neutral  axis  /,  Fig.  147. 
with  the  reduced  width  of  flange,  d  b',  into  parallel  strips,  say  ^  inch  deep, 


ED 


w       rid}'       b 

Figs.  146,  147,  148. — Beam  of  Unsymmetrical  Section. 

as  shown,  and  multiply  the  area  of  each  strip  by  its  mean  distance  from  the 
neutral  axis  for  the  proportional  quantity  of  resistance  at  the  strip.  Divide 
the  sum  of  the  products,  amounting  to  31.3,  by  the  extreme  depth  belov, 
the  neutral  axis — in  this  instance  4.09  inches,  and  multiply  the  quotient  b\ 
1.73  s,  the  ultimate  tensile  resistance  at  the  lower  surface.  The  final  pro- 
duct is  the  total  tensile  resistance  of  the  section;  or, 


31.3  x  i.73-y_ 


4.09 


=  13.24  s  total  tensile  resistance. 


Again,  multiply  the  area  of  each  strip  by  the  square  of  its  mean  distance 
from  the  neutral  axis,  and  divide  the  sum  of  these  new  products,  amount- 
ing to  104.64,  by  the  sum  of  the  first  products.  The  quotient  is  the  dis- 
tance of  the  resultant  centre  of  tensile  stress,  d',  from  the  neutral  axis. 
Or,  the  resultant  centre  is, 


104.64 


=  3.34  inches  below  the  neutral  axis. 


516  THE   STRENGTH   OF   MATERIALS. 

This  process  is,  in  fact,  the  process  for  finding  the  centre  of  gravity  of 
all  the  tensile  resistances. 

By  a  similar  process  for  the  upper  portion,  in  compression,  the  sum  of 
the  first  products  is  found  to  be  the  same  as  for  the  lower  part,  and  is  31.3. 
But  the  maximum  compressive  stress  at  the  upper  surface  is  greater  than 
the  maximum  tensile  stress  at  the  lower  surface,  in  the  ratio  of  their  dis 

tances  from  the  neutral  axis;  or  it  is  1.73  s  x  ZlZ_  =  3.34  j  and 

4.09 

3T-3  x  3-34  s  _  ^.24  s,  total  compressive  resistance, 
7.91 

which  is  the  same  as  the  total  tensile  resistance,  in  conformity  to  the 
general  law  of  the  equality  of  tensile  and  compressive  stress  in  a  section. 
The  sum  of  the  products  of  the  areas  of  the  strips,  divided  by  the  squares 
of  their  distances  respectively  from  the  neutral  axis,  is  164.90,  and  the 
resultant  centre,  c,  is 

—  4iz_  =  -  2y  inches  above  the  neutral  axis. 
3i-3 

The  sum  of  the  distances  of  the  centres  of  stress  or  of  resistance  from 
the  neutral  axis,  (3.34  +  5.27  =)8.6i  inches,  is  the  distance  apart  of  these 
centres,  as  represented  by  a  central  line,  c'd',  in  Fig.  147. 

Abbreviated  calculation,  —  As  the  upper  part  of  the  section  is  a  rectangle, 
detailed  calculation  is  not  necessary,  for  its  resultant  centre  is  known  to  be 
at  two-thirds  of  the  height,  or  (7.91  x  2/3  =  )  5.27  inches  above  the  neutral 
axis.  The  average  resistance,  too,  is  half  the  maximum  stress,  —  namely,  that 
at  the  upper  edge,  —  which  is,  as  above  explained,  3.34  ^  per  square  inch. 
The  area  of  the  rectangle  is  (7.91  x  i  =  )7.9i  square  inches;  and 


7.01  x  3.^4  s 

-  -  2_^±_  -  13.24  s  the  compressive  resistance, 

y  been  calculated. 

e:  the  moment  of  tensile  resistance  is  13.24^x8. 

114  s,  and  it  is  equal  to  —  ;  whence  W  =  1  14  s  x  4;  or,  in  a  general  form, 

* 


as  has  already  been  calculated. 

To  resume:  the  moment  of  tensile  resistance  is  13.24^x8.61  inches 


4 

(25) 


W  =  the  breaking  weight,  in  tons. 

S  =  the  total  tensile  resistance  of  the  section,  in  tons. 
dz  =  the  vertical  distance  apart  of  the  centres  of  tension  and  compres 
sion,  in  inches. 

/=the  span,  in  inches. 

Strength  of  the  Beam,  Fig.  147,  inverted.  —  When  inverted,  the  maximum 
tensional  resistance  of  the  beam  at  the  lower  surface  c,  in  Fig.  148,  is  1.73  s. 
The  area  of  the  rectangle  ic  is  7.91  square  inches,  and 

i-l?  —   ''3    _  5  g^  ^  total  tensile  resistance; 

which  is  only  about  half  the  tensile  resistance  offered  by  the  beam  in  its 
first  position,  Fig.  147.     The  breaking  weight  is,  therefore,  also  only  about 


FORMS   OF   BEAMS   OF   UNIFORM   STRENGTH.  517 

a  half;  and  the  reason  why  it  is  calculated  that  the  beam  bears  double  the 
breaking  weight  in  the  first  position  that  it  does  in  the  second  is,  that  the 
rectangular  portion,  c  i,  is  expected  to  oppose  at  least  twice  as  much  resist- 
ance, when  above,  to  compression  as  it  does,  when  below,  to  tension.  If 
the  effective  resistance  to  compression  were  only  equal  to  the  resistance  to 
tension,  the  beam  would  have  the  same  ultimate  strength  in  both  positions. 

RULE. — To  find  the  Ultimate  Strength  of  a  Homogeneous  Beam  of  Unsym- 
metrical  Section,  i.  Reduce  the  section  to  a  section  of  uniform  potential 
resistance,  as  explained  at  page  514.  2.  Find  the  position  of  the  neutral 
axis  of  the  reduced  section  by  the  previous  rule.  3.  Divide  the  section  into 
thin  strips  parallel  to  the  neutral  axis ;  if  such  division  has  not  already  been 
made,  for  No.  2.  4.  Multiply  the  areas  in  square  inches  of  the  strips, 
under  tensional  stress,  by  their  mean  distances  respectively  (that  is,  the 
distances  of  their  centres  of  gravity)  from  the  neutral  axis.  5.  Multiply 
the  same  areas  by  the  squares  of  their  mean  distances  respectively  from  the 
neutral  axis.  6.  Divide  the  sum  of  the  first  products  by  the  extreme  dis- 
tance of  the  surface  in  tension  from  the  neutral  axis,  and  multiply  the 
quotient  by  the  ultimate  tensile  resistance  per  square  inch;  the  product  is 
the  total  tensile  resistance  of  the  section.  7.  Divide  the  sum  of  the 
second  products  (  5  )  by  the  sum  of  the  first  products  ( 4 );  the  quo- 
tient  is  the  distance  of  the  resultant  centre  of  tension  from  the  neutral 
axis.  8.  Multiply  the  ultimate  tensile  resistance  by  the  distance  of  the 
neutral  axis  from  the  upper  surface,  and  divide  the  product  by  the  distance 
of  the  neutral  axis  from  the  lower  surface;  the  quotient  is  the  maximum 
compressive  stress  at  the  upper  surface.  9.  Make  the  calculations  4,  5, 
and  7  for  the  strips  under  compression.  10.  The  sum  of  the  distances  of 
the  resultant  centres  from  the  neutral  axis  is  the  distance  apart  of  the 
centres,  n.  Find  the  breaking  weight  by  formula  (25);  that  is,  multiply 
the  total  tensile  resistance  of  the  section  in  tons  by  the  distance  apart  of 
the  resultant  centres  of  tension  and  compression  in  inches,  and  by  4; 
and  divide  the  product  by  the  span  in  inches.  The  quotient  is  the  breaking 
weight  in  tons  at  the  middle. 

Note  to  Rule.—l\.  is  assumed  that  the  ability  of  the  beam  to  resist  com- 
pression is  sufficient  to  insure  that  fracture  shall  take  place  by  tension. 
Beams  of  comparatively  slender  dimensions  laterally  are  liable  to  cant 
under  the  thrust  of  compression  if  not  supported  laterally,  and  canting 
occasionally  takes  place  in  beams  which  are  tested  experimentally.  But 
beams,  when  in  their  destined  places,  are  in  general  so  supported  that 
any  liability  to  canting  is  removed. 

Elastic  Strength  of  Beams  of  Unsymmetrical  Section. — The  elastic  strength 
is  approximately  deducible  from  the  ultimate  strength,  according  to  the 
ordinary  ratio  of  one  to  the  other,  ascertained  experimentally.  The  elastic 
strength  and  deflection  of  a  homogeneous  beam  of  any  section  is  the  same, 
whether  in  its  normal  position  or  turned  upside  down. 

FORMS   OF   BEAMS   OF   UNIFORM   STRENGTH. 

A  beam  is  said  to  be  of  uniform  strength  when  its  capability  of  resistance 
to  transverse  stress  under  a  given  load,  applied  in  a  given  manner,  is  the 
same  at  all  parts  of  its  length. 


5i8 


THE  STRENGTH   OF   MATERIALS. 


r 


SEMI-BEAMS  OF  UNIFORM  STRENGTH  LOADED  AT  THE  END. 

By  a  semi-beam  is  meant  a  beam  fixed  at  one  end  and  free  at  the  other; 
as  it  represents  the  half  of  a  beam  supported  at  both  ends. 

The  moments  of  stress  due  to  the  weight  on  the  end,  at  any  section  of 
the  beam,  increase  directly  as  the  distance  of  the  section  from  the  end  of 
the  beam.  Let  cb,  Fig.  149,  be  a  rectangular  beam,  fixed  at  the  base  cd, 

and  loaded  at  the  end  b.  Draw  the  dia- 
gonal straight  line  bd,  then  the  ordinates 
to  the  triangle  b  cd,  represent  proportionally 
the  moment  of  stress  at  all  parts  of  the 
length;  and  the  moments  of  stress  vary 
directly  as  the  length.  Now,  the  ultimate 
moments  of  resistance  at  any  section  are 
as  the  square  of  the  depth,  when  the  breadth 
is  uniform;  and  it  follows,  conversely,  that 
the  depth  of  the  beam,  of  uniform  strength, 
must  vary  as  the  square  root  of  the  distance 
from  the  end  b.  Take,  for  instance,  the  sec- 
tion c' d'  at  the  half-length  of  the  beam;  the  moment  of  stress  at  cd  is  to  that 
of  the  stress  at  c'd',  as  i  to  ^ ;  and  the  required  depth  at  the  origin  cd,  is  to 
the  depth  at  c'  d',  as  */  i  to  */  }4,  or  as  i  to  .707.  The  depth  c'  c", 
equal  to  .707,  would  be  the  depth  of  a  beam  of  uniform  strength,  at  that 
section.  The  depth  for  uniform  strength  at  any  other  section  may  be 
calculated  in  the  same  way ;  and  the  form  of  the  lower  side  of  the  beam,  of 
uniform  strength,  is  that  of  a  parabola,  b  c"  d,  of  which  the  vertex  is  at  the 
end  b.  With  respect  to  transverse  resistance,  then,  the  semi-beam  would 
be  equally  strong  if  the  lower  portion  b  b'  d  were  removed. 

The  semi-beam,  rectangular  in  section,  of  uniform  strength,  fixed  at  one  end, 
and  loaded  at  the  other  end,  having  the  breadth  constant,  may  therefore  be 
moulded  in  depth  to  any  of  the  parabolic  outlines,  Figs.  150. 


Fig.  149. — Stress  in  a  Semi-beam 
loaded  at  the  end. 


No.  i. 


No.  2. 


No.  3. 


Figs.  150. — Semi-beams  loaded  at  one  end. 

When  the  depth  of  the  semi-beam,  rectangular  in  section,  is  constant,  the 
breadth  is  in  simple  proportion  to  the  distance  from  the  end  of  the  beam, 
as  in  Fig.  151,  and  the  beam  is  triangular  in  plan. 

When  the  section  of  the  semi-beam  is  double-flanged,  or  is  hollow  rectangular, 
and  the  breadth  is  constant,  the  flanges  are  assumed  to  be  of  a  constant 
sectional  area.  Leaving  out  of  the  calculation  the  strength  of  the  vertical 


SEMI-BEAMS   OF   UNIFORM   STRENGTH. 


519 


web,  and  calculating  only  for  the  flanges,  the  moment  of  resistance  at  any 
section  is  as  the  depth,  and  the  form  of  the  beam  is  triangular,  as  in  Fig. 
152,  which  shows  a  semi-beam  with  double  flanges. 

If  the  strength  of  the  vertical  web  be  taken  into  the  calculation,  the  form 
of  the  beam  is  intermediate  between  the  triangular  and  the  parabolic. 


Fig.  151. 


Fig.  152. 


Fig.  153- 


Semi-beams  loaded  at  one  end. 

When  the  section  of  the  semi-beam  is  double-flanged,  or  is  hollow-rectangular, 
and  the  depth  is  constant,  calculating  only  for  the  flanges,  Fig.  153,  their 
sectional  area  increases  uniformly  with  the  distance  from  the  end,  and  if 
their  thickness  be  uniform,  they  are  triangular  in  plan,  as  shown. 

If  the  web  be  taken  into  the  calculation,  it  is  calculated  as  a  solid  semi- 
beam  rectangular  in  section,  and  the  thickness  should  increase  as  the  dis- 
tance from  the  end.  The  web  would,  therefore,  be  triangular  in  plan. 

When  the  section  of  the  semi-beam  is  circular,  the  moment  of  resistance 
varies  as  the  cube  of  the  diameter,  and  the  cube  of  the  diameter  is  therefore 
as  the  distance  from  the  end;  or,  inversely,  the  diameter  is  as  the  cube 
root  of  the  distance,  and  the  outline  of  the  semi-beam  may  be  formed  by 
the  revolution  of  a  cubic  parabola  on  its  axis,  Fig.  154. 

When  the  section  of  the  semi-beam  is  annular;  when  the  thickness  is  uniform 
and  small  in  proportion  to  the  diameter,  the  square  of  the  diameter  varies 
as  the  distance  from  the  end,  or  the  diameter  varies  as  the  square  root  of 
the  distance,  and  the  semi-beam  is  formed  by  the  revolution  of  a  parabola 
on  its  axis. 

If  the  thickness  varies  with  the  diameter,  the  diameter  varies  as  the  cube 
root  of  the  distance  from  the  end,  and  the  semi-beam  is  cubic-parabolic, 
like  Fig.  154. 

When  the  section  of  the  semi-beam  is  elliptical,  the  sections  being  similar  at 
all  points  of  the  length,  the  cube  of  the 
depth  varies  as  the  distance  from  the  end, 
or  the  depth  varies  as  the  cube  root  of  the 
distance,  and  the  elevation  of  the  beam  is 
cubic-parabolic,  like  Fig.  154. 

When  the  section  of  the  semi-beam  is  hollow- 
elliptical,  the  beam  being  of  similar  sections 
throughout.  When  the  thickness  is  uniform, 
and  is  small  in  proportion  to  the  depth,  the 

Square  Of  the   depth  varies    as   the  distance       Fig.  154.— Semi-beam  loaded  at  one  end. 

from  the  end,  or  the  depth  varies  as  the 

square   root   of   the   distance,   and  the   side   elevation   of  the  beam  is 

parabolic. 


520 


THE   STRENGTH   OF    MATERIALS. 


If  the  thickness  varies  with  the  depth,  the  depth  varies  as  the  cube  root 
of  the  distance  from  the  end,  and  the  beam  is  cubic-parabolic  in  side 
elevation,  like  Fig.  154. 

SEMI-BEAMS  OF  UNIFORM  STRENGTH  UNIFORMLY  LOADED. 

The  moment  of  stress  due  to  the  weight  when  uniformly  distributed 
increases  as  the  square  of  the  distance  from  the  end  of  the  beam,  as  will  be 
shown  in  the  following  case: — 

When  the  semi-beam  is  rectangular  in  section,  and  its  breadth  is  constant. 
Suppose  the  load  equally  divided  and  distributed  as  a  great  number  of 
weights,  W,  W",  W",  &c.,  Fig.  155;  and  suppose  the  beam  to  be  divided 
into  an  equal  number  of  corresponding  sections  at  c' ,  c",  c'" ,  &c.  The  loads 
supported  by  the  successive  sections  are  W,  2  W,  3  W3  &c. ;  the  distances 
of  the  centres  of  gravity  of  these  loads,  from  the  respective  sections,  are  as 


W~  W     W 


Figs.  155,  156. —Semi-beams  uniformly  loaded. 

i,  2,  3,  «Sz:c.  Therefore,  the  moments  of  stress  at  the  successive  intersec- 
tions, c',  c",  c",  &c.,  are  as  i2,  22,  32,  &c.,  or  as  the  square  of  the  distance 
from  the  end.  But  the  moments  of  resistance  at  the  intersections  are  as 
the  squares  of  the  depths  at  c',c",c"',  &c.;  and  so  the  square  of  the  depth 
is  as  the  square  of  the  distance,  or  the  depth  is  as  the  distance  from  the 
end.  The  beam  is  therefore  triangular  in  elevation. 

When  the  semi-beam  is  rectangtdar  in  section,  and  has  the  depth  constant, 
Fig.  156.  As  the  depth  is  constant,  the  breadth  must  increase  as  the  square 
of  the  distance;  and  it  may  be,  in  outline,  of  the  form  of  two  parabolas 
b  <:,  b  c',  back  to  back,  touching  each  other  at  their  vertices  at  b ;  the  axes 
being  perpendicular  to  the  length. 

When  the  section  of  the  semi-beam  is  hollow-rectangular,  or  is  double-flanged; 
and  the  breadth  is  constant.  Calculating  the  strength  of  the  upper  and  lower 
members,  or  flanges,  only,  and  supposing  the  thickness  to  be  uniform,  the 
moment  of  resistance  is  as  the  depth;  the  depth  is,  therefore,  as  the 
square  of  the  distance  from  the  end,  and  is  of  the  form  of  a  parabola, 
Fig.  157,  of  which  the  vertex  is  at  b,  and  the  axis  is  perpendicular  to 
the  length. 

Calculating  the  strength  of  the  vertical  webs  or  rib  only,  the  beam  would 
be  triangular  in  side  elevation. 

Combining  the  webs  and  the  flanges  in  the  calculation,  the  form  of  the 
beam  would  be  intermediate  between  the  parabolic  and  the  triangular. 

2d.  When  the  depth  is  constant.  Calculating  for  the  flanges  only,  the 
thickness  being  uniform ;  the  breadth  of  the  flanges  is  as  the  square  of  the 
distance  from  the  end,  Fig.  158,  the  same  as  in  Fig.  156. 


BEAMS  SUPPORTED  AT  BOTH  ENDS. 


521 


If  the  vertical  web  or  rib,  of  uniform  thickness,  be  included  in  the  calcula- 
tion, it  does  not  materially  modify  the  form  of  the  flange. 

When  the  section  of  the  semi-beam  is  circular.  The  moment  of  resistance 
is  as  the  cube  of  the  diameter,  and  the  moment  of  stress  is  as  the  square 
of  the  length;  therefore  the  cube  of  the  diameter  is  as  the  square  of  the 


Figs.  157,  158.— Semi-beams  uniformly  loaded. 

length,  or  the  diameter  is  as  the  cube  root  of  the  square  of  the  length,  or 
as  the  ^  power,  or  .666  power  of  the  length.  The  solid  is  formed  by  the 
revolution  of  a  semi-cubic  parabola  on  its  axis. 

When  the  section  of  the  semi-beam  is  annular,  the  thickness  being  uniform 
and  small  compared  to  the  diameter.  The  moment  of  resistance  of  any 
section  is  as  the  square  of  the  diameter.  The  square  of  the  diameter  is, 
therefore,  as  the  square  of  the  length,  or  the  diameter  is  as  the  length,  and 
the  semi-beam  is  triangular  or  conical  in  elevation,  Fig.  159. 

When  the  thickness  diminishes  with  the  diameter,  the  moments  of  resist- 
ance of  sections  are  as  the  cubes  of  the  dia- 
meters, and  the  diameter  varies  as  the  ^ 
power,  or  .666  power  of  the  length,  as  with  a 
solid  circular  section,  and  the  form  is  derived 
from  the  revolution  of  a  semi-cubic  parabola 
on  its  axis. 

When  the  section  of  the  semi-beam  is  elliptical. 
The  moment  of  resistance  of  a  section  is  as 
the  cube  of  the  depth,  and  the  form  is  the 
same  as  that  of  a  circular  beam. 

When  the  section  of  the  semi-beam  is  hollow- 
elliptical.  The  form  is  the  same  as  that  of  a  beam  of  annular  section. 

BEAMS  OF  UNIFORM  STRENGTH  SUPPORTED  AT  BOTH  ENDS. 

The  forms  of  beams  supported  at  both  ends,  and  loaded  at  the  middle, 
are  simply  doubles  of  the  forms  of  semi-beams,  or  such  as  are  fixed  at  one 
end  and  unsupported  at  the  other  end.  In  the  beam  of  rectangular  section, 
for  example,  A  B,  Fig.  160,  the  diagonal  lines,  ca  and  cb,  from  the  top  at  the 
middle  to  the  supports  at  each  end,  are  simply  doubles  of  the  diagonal  b  d, 
in  the  semi-beam,  Fig.  149,  and  represent  the  graduated  moment  of  bending 
stress  from  the  middle,  where  it  is  a  maximum,  to  the  ends,  where  it  vanishes; 
and  the  parabolic  curves  ca  and  cb,  meeting  base  to  base  at  the  middle  cd, 
form  the  outline  of  the  rectangular  beam  of  uniform  strength,  when  the 
breadth  is  constant. 

The  beam,  rectangular  in  section,  of  uniform  strength,  loaded  at  the  middle, 


Fig.  159. — Annular  Semi-beam 
uniformly  loaded. 


522 


THE   STRENGTH   OF   MATERIALS. 


and  having  breadth  constant,  may  therefore  be  moulded  according  to  any  of 
the  parabolic  forms,  Figs.  161,  162,  163,  having  the  axes  horizontal,  and 
the  vertices  at  the  points  of  support. 


Fig.  160. — Stress  in  rectangular  beam 
supported  at  both  ends. 


Fig.  161. — Beam  loaded  at  the  middle. 


When  a  rectangular  beam,  with  a  constant  breadth,  is  loaded  uniformly; 
referring  to  formula  (5),  page  508.  When  the  weight  is  constant,  together 
with  the  breadth  b,  and  the  length  /,  the  square  of  the  depth,  d^,  varies  as 


Figs.  162,  163. — Beams  loaded  at  the  middle. 


the  products,  m  n,  of  the  segments,  m  and  «,  of  the  length  of  the  span  at 
any  point  of  the  length.  Or,  the  depth  varies  as  the  square  root  of  the 
product  of  the  segments,  and  the  form  of  the  beam,  Fig.  164,  is  a  semi- 
ellipse.  It  may  be  a  complete  ellipse,  Fig.  165. 


Figs.  164,  165. — Beams  uniformly  loaded. 

For  a  rectangular  beam,  with  a  constant  depth,  and  loaded  at  the  middle, 
the  form  of  the  breadth,  Fig.  166,  is  a  double  of  Fig.  151,  page  519;  con- 
sisting of  two  triangles,  in  plan,  united  at  their  base. 


Fig.  166. — Beam  loaded  at  the  middle. 


Fig.  167. — Beam  uniformly  loaded. 


When  a  rectangular  beam,  with  a  constant  depth,  is  uniformly  loaded; 
referring  to  the  formula  (  5  )  above  noticed,  the  variables  are  the  breadth  b 
and  the  product  mn,  and  the  breadth  varies  as  the  product  of  the 


BEAMS   SUPPORTED  AT   BOTH   ENDS. 


523 


segments  of  the  length  of  the  span  at  any  point  of  the  length.  The  form  of 
the  breadth  in  plan  is  therefore  that  of  two  parabolas  having  their  vertices 
at  the  middle  and  meeting  at  the  points  of  support,  Fig.  167. 

A  hollow-rectangular  or  double-flanged  beam  with  a  constant  breadth,  and 
loaded  at  the  middle,  consists  of  the  double  of  Fig.  152,  page  519;  being  two 
triangles  united  at  their  base,  at  the  middle,  Figs.  168,  169.  In  this  and 
the  three  following  cases,  the  resistance  of  the  flanges  only  is  calculated; 
and  the  flanges  are  supposed  to  be  of  uniform  thickness. 


Figs.  168,  169. — Beams  loaded  at  the  middle. 

When  a  hollow-rectangular  or  double-flanged  beam,  with  a  constant  breadth? 
is  uniformly  loaded;  the  depth  varies  as  the  product  of  the  segments  of  the 
beam  at  any  point  in  the  span;  and  the  side  of  the  beam,  Fig.  170,  is  of  the 
form  of  a  parabola,  having  its  axis  at  the  middle.  The  resistance  of  the 
flanges  only  is  here  calculated. 


Fig.  170. — Beam  uniformly  loaded. 


Fig.  171. — Beam  loaded  at  the  middle. 


A  hollow-rectangular  or  a  double-flanged  beam,  with  a  constant  depth,  and 
loaded  at  the  middle,  consists  of  the  double  of  Fig.  153,  page  519;  the  flanges 
being  of  uniform  thickness,  and  forming  two  triangles  in  plan,  joined  at 
their  base  at  the  middle  of  the  beam.  Their  form,  Fig.  171,  is  the  same 
as  that  of  a  rectangular  beam,  with  a  constant  depth,  Fig.  166. 

When  a  hollow-rectangular  or  a  double-flanged  beam,  with  a  constant  depth, 
is  uniformly  loaded,  the  form  of  the  flanges  in  plan  is  the  same  as  that  of  a 
rectangular  beam  (Fig.  167),  consisting  of  two  parabolas,  having  their  vertices 
at  the  middle  of  the  beam,  Fig.  172. 


Fig.  172. — Beam  uniformly  loaded. 


Fig.  173. — Beam  loaded  at  the  middle. 


When  the  section  of  the  beam  is  circular,  and  the  load  is  at  the  middle,  the 
form  is  the  double  of  Fig.  154,  page  519,  consisting  of  the  revolutions  of 
two  cubic  parabolas,  base  to  base,  at  the  middle,  Fig.  173. 


524  THE   STRENGTH   OF   MATERIALS. 

When  a  beam  of  circular  section  is  uniformly  loaded,  the  cube  of  the 
diameter  varies  as  the  product  of  the  segments  of  the  length  of  the  span  at 
any  point  in  the  length;  and  the  radius  varies  as  the  cube  root  of  the 
product  of  the  segments. 

When  the  section  of  the  beam  is  annular,  and  the  load  applied  at  the  middle, 
the  form  is  that  produced  by  the  revolution  of  two  parabolas,  base  to  base, 
with  their  vertices  at  the  ends  of  the  beam.  The  thickness  is  supposed  to 
be  inconsiderable. 

When  the  annular  beam  is  uniformly  loaded,  the  square  of  the  diameter 
varies  as  the  product  of  the  segments  of  the  length  of  the  span  at  any  point 
in  the  length;  and  the  radius  varies  as  the  square  root  of  the  product  of 
the  segments.  The  form  of  the  beam  is  that  produced  by  the  revolution 
of  an  ellipse  on  one  of  its  axes. 

When  the  section  of  the  beam  is  elliptical^  and  the  load  applied  at  the  middle, 
the  form  is  that  of  two  cubic  parabolas  joined  base  to  base. 

When  the  beam  of  elliptical  section  is  uniformly  loaded,  the  form  is  that  of 
an  ellipse. 

When  the  section  of  the  beam  is  hollow-elliptical,  and  the  load  applied  at  the 
middle,  and  the  thickness  is  uniform,  and  is  small  in  proportion  to  the  depth, 
the  form  of  the  beam  is  that  of  two  parabolas,  united  base  to  base,  having 
their  vertices  at  the  points  of  support. 

If  the  thickness  varies  with  the  depth,  the  forms  are  cubic  parabolas. 

When  the  hollow  beam  of  elliptical  section  is  uniformly  loaded,  the  form  of 
the  beam  is  elliptical. 

BEAMS  OF  UNIFORM  STRENGTH  UNDER  A  CONCENTRATED  ROLLING  LOAD. 

Reverting  to  formula  (5),  page  508,  it  signifies  that  the  breaking  weight 
varies  inversely  as  m  x  n,  or  the  product  of  the  segments  of  the  length  of 
the  span,  at  any  point  of  the  length;  but  if  the  weight  be  constant,  the 
moment  of  stress  at  any  point  is  as  the  product  m  x  n,  and  therefore,  also, 
the  moment  of  resistance  of  a  beam  of  uniform  strength  varies  as  m  x  n,  at 
all  points  of  its  length. 

Hollow-rectangular,  or  flanged  beam,  with  a  constant  breadth,  under  a  con- 
centrated rolling  load.  Calculating  the  resistance  of  the  booms  or  flanges 
only,  the  depth  varies  as  m  x  n,  and  is  according  to  the  form  of  a  parabola, 
of  which  the  axis  is  vertical,  when  the  upper  or  lower  side  is  horizontal, 
Figs.  174,  175;  or  of  two  parabolas,  on  the  same  axis,  meeting  at  the 
points  of  support. 

Under  these  conditions,  the  sectional  area  of  the  flanges  is  constant. 


Figs.  174,  175.— Flanged  Beams  under  a  Concentrated  Rolling  Load. 

Hollow -rectangular  or  flanged  beam,   with  a  constant  depth,  or  parallel 

flanges,  tinder  a  concentrated  rolling  load.     The  breadth  varies  as  m  x  n,  and 

the  flanges,  supposed  to  be  of  uniform  depth,  are  of  the  form  of  two 

parabolas  on  the  same  axis  passing  through  the  middle,  Fig.  172,  page  523. 


SHEARING   STRESS   IN   BEAMS  AND   PLATE-GIRDERS.         525 


Stress  in  the  curved  flange,  Figs.  174,  175.  Mr.  Stoney  gives  a  simple 
means  of  finding  the  stress  diagramatically.  Let 
A  B,  Fig.  176,  represent  the  horizontal  stress, 
which  is  uniform  throughout  the  length.  Draw 
A  c  parallel  to  the  tangent  of  the  curve  at  the 
given  point,  and  B  c  perpendicular  to  A  B.  Then 
A  c  is  the  maximum  longitudinal  stress  at  the 
given  point,  and  A  B  and  B  c  are  its  horizontal 
and  vertical  components.  It  follows  that  the  sec- 
tion of  the  curved  flange  should  increase  as  it  approaches  the  points  of 
support  in  proportion  to  A  c,  or  the  secant  of  the  angle  A. 


SHEARING  STRESS  IN  BEAMS  AND  PLATE-GIRDERS. 

Shearing  stress  in  beams  is  caused  by  the  vertical  pressure  of  the  load. 
A  conception  of  this  stress  is  easily  formed  on  reflecting  that  the  weight  of 
the  beam  and  its  load  tends  to  force  it  downwards  at  the  abutment,  whilst 
the  abutment,  by  its  upward  pressure,  tends  to  force  upwards  the  part  of 
the  beam  which  rests  on  it.  The  stress  thus  caused  tends  to  a  vertical 
rupture,  or  slicing  off  of  the  loaded  end  of  the  beam,  called  shearing  stress. 
The  same  kind  of  stress  acts  with  various  intensity  in  the  portion  of  the 
beam  between  the  abutments,  and  it  is  the  duty  of  the  web  to  resist  the 
shearing  stress. 

In  a  beam  supported  at  one  end,  and  loaded  at  the  other  end,  the  vertical 
shearing  stress  is  equal  to  the  weight,  at  every  point  of  the  length. 

In  the  same  beam,  uniformly  loaded,  the  shearing  stress  increases 
uniformly  from  the  end,  where  it  is  nothing,  to  the  abutment,  where  it  is 
equal  to  the  weight.  A  diagram  indicating  the  gradations  of  stress  would 
have  the  form  of  a  triangle. 

In  the  same  beam,  uniformly  loaded,  and  also  weighted  at  the  end,  the 
shearing  stress  is  represented  by  a  compound  diagram,  Fig.  177,  in  which 
the  triangle  a  b  c  represents  the  graduated  shearing  stress  due  to  a  uniform 
load,  in  a  beam  of  the  length  ab;  and  the  rectangle  abde,  the  uniform 
shearing  stress  due  to  a  weight  at  the  end.  The  whole  depth  dc  at  the 
abutment  represents  a  total  shearing  stress  equal  to  the  sum  of  the  dis- 
tributed and  end  loads;  and  the  total  stress  at  intermediate  points  is  repre- 
sented by  the  corresponding  ordinates. 


Fig.  177.— Shearing  Stress. 


Fig.  178. — Shearing  Stress. 


In  a  beam  supported  at  both  ends,  and  loaded  at  any  point,  the  shearing 
stress  in  each  segment  is  equal  to  the  pressure  on  its  abutment.     The 


THE   STRENGTH   OF   MATERIALS. 


pressures  at  a  and  b,  Fig.  178,  are  as  the  segments  m  and  n,  and  the  shearing 
stress  in  the  segments  ad  and  db  are  equal  to  W  x  —  and  W  x  —;  repre- 
sented by  the  graduated  rectangles  on  a  d  and  db. 

When  a  concentrated  load  is  moved  over  the  beam,  the  shearing  stress 
in  each  segment  varies  as  the  length  of  the  other  segment: — from  o  to  W, 
the  weight;  represented  by  the  two  graduated  triangles,  ab  c,  ab  d,  Fig.  179, 
in  which  the  verticals  a  c  and  b  d,  at  the  ends,  represent  the  weight. 
c 


Fig.  179.— Shearing  Stress.  Fig.  180.— Shearing  Stress. 

When  a  number  of  weights  are  placed  irregularly  on  a  beam,  the  shearing 
stress  of  some  is  neutralized  more  or  less  by  that  of  others;  and,  referring 
to  any  given  section  of  the  beam,  the  shearing  stress  is  equal  to  the  differ- 
ence of  the  sum  of  those  portions  of  the  weights  placed  on  one  side  of  the 
section  which  are  conveyed  to  the  abutment  on  the  other  side,  and  the 
sum  of  those  portions  of  the  weights  on  the  other  side  which  are  conveyed 
to  the  abutment  on  the  first  side. 

When  a  beam,  supported  at  both  ends,  is  loaded  uniformly,  the  shearing 
stress  is  o  at  the  centre,  as  in  Fig.  180,  and  increases  uniformly  towards 
the  abutments,  where  it  is  equal  to  half  the  weight. 

When  a  load  of  uniform  density,  as  a  railway  train,  traverses  a  girder, 
the  shearing  stress  at  the  front  of  the  train  increases  as  the  square  of  the 
length  of  the  loaded  segment.  Suppose  that  the  train  advances  from  b  to  a, 
Fig.  1 8 1,  covering  the  whole  length,  the  curve  of  increasing  shearing  stress, 
.b  c,  is  parabolic,  having  its  apex  at  b.  When  the  girder  is  wholly  covered, 
the  shearing  stress  follows  the  triangular  gradations  shown  by  dot-lines. 


Fig.  181. — Shearing  Stress.  Fig.  182. — Shearing  Stress. 

When  a  fixed  uniform  load  and  a  rolling  load  are  combined,  the  maximum 
shearing  stress  to  which  the  girder  is  liable  at  different  points  of  its  length 
is  shown  by  the  combined  ordinates  in  Fig.  182. 

Sectional  area  of  a  continuous  web  calculated  from  the  shearing  stress. — 
"When  the  flanges  are  parallel,"  says  Mr.  Stoney,1  "the  theoretic  area  of  a 
continuous  web  may  be  calculated  from  the  shearing  stress  by  the  following 
rule : — 

Sectional  area  of  web  =  shearing  stress 

unit-stress 

in  which  the  unit-stress  is  the  safe  unit-stress  for  shearing.     This  gives  the 
1  The  Theory  of  Strains  in  Girders  and  Similar  Structures. 


DEFLECTION   OF  BEAMS  AND   GIRDERS. 


527 


minimum  thickness,  which,  however,  is  often  much  less  than  a  due  regard 
for  durability  requires." 

"  When  a  girder,  with  parallel  flanges  and  a  continuous  web,  is  loaded 
in  the  manner  described  below,  where  /  =  the  length,  and  /  =  the  safe 
unit-strain  for  shearing  force,  the  theoretic  quantity  of  material  in  the  web 
would  be  as  follows:" — 


Theoretic 

KIND  OF  LOAD. 

Quantity  of 
Material  in  a 
Continuous 

Proportional 
numbers. 

Web. 

Fixed  central  load.     .     .      —  \V 

W/ 

12 

2/ 

Concentrated  rolling  load..  —  W 

3W/ 

18 

4/ 

Uniformly  distributed  load  =  W 

W/ 

6 

Distributed  rolling  load       —  "W 

fwi 

7 

24/ 

DEFLECTION   OF   BEAMS  AND   GIRDERS. 

Compressive  strain  is  taken  as  equal  to  tensile  strain,  per  ton  of  direct 
stress  on  the  fibres,  and  the  strain  is  directly  proportional  to  the  stress, 
within  the  elastic  limits.     When  a 
beam  is  deflected  under  a  load,  the 
lower  side   is   lengthened   and   the 
upper  side  is  shortened  in  propor- 
tion to  the  direct  stress  per  unit  of 
section  of  the  fibres. 

In  a  beam  of  uniform  strength, 
the  fibres  at  the  surface,  on  the  up- 
per and  lower  sides,  are,  by  the  de- 
finition, equally  stressed  and  equally 
strained  throughout  the  length  of  the 
beam;  and  the  form  assumed  by  a 
straight  parallel  beam,  when  deflected 
under  its  proper  load,  is  that  of  a  cir- 
cular arc. 

Let  a  b  c'  c',  Fig.  183,  be  a  parallel 
beam,  rectangular  in  section,  having 
a  constant  depth,  and  of  uniform 
strength,  when  loaded  at  the  middle. 
Let  its  lower  side  assume,  by  deflec- 
tion, the  form  of  the  circular  arc 


c' 


Fig.  183. — Deflection  of  a  Beam. 


ad' b,  the  ends  ac',  be',  which  were  upright,  in  their  normal  position,  are 
now  convergent  in  the  positions  ad',  be" ;  and  when  produced,  they  meet  at 
the  centre  of  the  arc,  O,  in  the  vertical  radius  d'O.  The  deflection  at  the 
centre,  dd ' ,  is  the  versed  sine  of  the  arc.  Let, 


528  THE   STRENGTH    OF   MATERIALS. 

R  =  the  radius  Od1, 

/    =  the  length  of  the  beam,  or  the  chord,  a  b, 

b   =  the  breadth  of  the  beam  at  the  middle, 

d  =  the  depth  of  the  beam  at  the  middle,  c  d. 

a   —  the  sectional  area  of  the  beam, 

D  =  the  deflection  of  the  beam,  dd'9 

I'  =  the  difference  of  length  of  the  upper  and  lower  sides, 

E  =  the  coefficient  of  elasticity,  or  the  denominator  of  the  fraction  of  the 
length,  by  which  the  beam  is  extended  or  compressed,  per  ton  of  direct 
stress  per  square  inch  of  section, 

s'  =  the  direct  tensile  stress  on  the  extreme  outer  fibres,  in  tons  per 
square  inch, 

/  =  the  direct  compressive  stress  on  the  extreme  outer  fibres,  in  tons 
per  square  inch, 

W  =  the  weight  in  tons. 

Note.  —  The  dimensions  are  to  be  all  in  inches,  or  all  in  feet. 

By  the  properties  of  the  circle,  the  square  of  half  the  chord  is  equal  to 
the  product  of  the  versed  sine,  or  deflection,  by  the  diameter  minus  the 
deflection;  or 


(  -  }  =  D  x  (2  R  -  D);  or  sensibly  (—  )   -  &  x  2  R;  and 


(a) 


Again,  by  similar  triangles,  in  Fig.  183,  Qc"  :  Qa  :  :  c'  c"  :  ab\  or,  in 
symbols,  R  :  d  ::/:/',  substantially;  whence 


Substituting  this  value  of  R  in  equation  (a), 
~      /2       /'      //' 


Now,  /'  =  'J  +  s  '  .     When  the  two  stresses,  s'  and  s",  are  equal  to  each 

Jt_j 

other,  let  them  be  represented  by  s.  When  they  are  not  equal  to  each 
other,  the  deflection  is  nevertheless  the  same  as  if  they  were  so,  and  that 
the  direct  tensile  and  compressive  stress  were  each  equal  to  the  mean  of 

/  and  /',  or  to  L±f_.      Putting  i±L  =  S)  then  /'  =  -^i-;  and,  substituting 

22  h, 

2  S/2 

this  value  of  /'  in  equation  (<:),  D  =  —  -  --  ;  or, 

o  a\Li 


/     x 
and,  s  =  *     2     ............................  (2) 


DEFLECTION   OF   BEAMS   OF   RECTANGULAR   SECTION.         529 

DEFLECTION  OF  BEAMS  OF  RECTANGULAR  SECTION. 

No.  i.  Rectangular  beam,  of  constant  depth,  of  uniform  strength,  loaded  at 
the  middle,  Fig.  166,  page  522. — This  beam  is  double-triangular  in  plan. 
The  value  of  s,  the  direct  stress  on  the  fibres  at  the  upper  and  lower 
surfaces,  in  terms  of  the  weight,  is,  by  formula  2,  page  507, 

.(3) 


Equating  this  value  of  s  and  the  above  value  (2), 
t       _W/ 

whence 


w_  4.62  £</3  ED  (     , 

/3 

....  (6) 


These  equations  express  the  relations  of  the  weight,  the  coefficient  of 
elasticity,  and  the  deflection. 

The  formula  (4),  for  the  value  of  the  deflection,  signifies  that  the  deflec- 
tion varies  directly  as  the  weight,  and  as  the  cube  of  the  span ;  and  that  it 
varies  inversely  as  the  breadth,  the  cube  of  the  depth,  and  the  coefficient 
of  elasticity. 

No.  2.  Rectangular  beam,  of  constant  breadth,  of  uniform  strength,  loaded  at 
the  middle,  Figs.  161,  162,  163,  page  522. — The  form  of  the  beam  in  side 
elevation  is  parabolic.  The  average  depth  is,  by  the  properties  of  the  para- 
bola, two-thirds  of  the  depth  at  the  middle,  or  of  the  depth  of  the  circum- 
scribed rectangle;  and  the  beam  may  be  treated,  for  finding  the  deflection, 
as  a  parallel  beam,  or  beam  of  constant  depth,  having  two-thirds  of  the 
depth  of  No.  i  beam,  and  under  the  same  stress  on  the  extreme  fibres. 
As  the  strain,  and  the  difference  of  length  of  the  upper  and  lower  sides,  of 
the  suppositious  beam,  are  approximately  the  same  as  those  of  the  Original 
beam ;  and  as  the  deflection  is  inversely  as  the  depth  (see  equation  ( c ), 
page  528),  or  as  two  to  three,  in  No.  i  and  No.  2  beams;  then,  modifying 

formula  (4),  above,  D  =  -^-  x — _.  •  or, — 

2      4.62<£</3E 


3.08  £  </3 


No.  3.  Rectangular  beam,  of  uniform  section,  loaded  at  the  middle,  Fig. 
160,  page  522.  —  Compared  with  No.  i  beam,  Fig.  166,  No.  3  beam  is 
rectangular  in  plan,  and  contains  twice  the  surface  of  No.  i,  which  is 
triangular.  Under  a  given  weight,  therefore,  the  stress  on  the  extreme,  or 
surface,  fibres  of  No.  3,  averages  only  half  the  stress  on  those  of  No.  i. 


530  THE   STRENGTH   OF   MATERIALS. 

The  deflection  would  also,  if  it  followed  the  same  proportion,  be  just  half 
that  of  No.  i.  But  it  must  be  more  than  half,  since  it  is  not  according  to 
a  circular  outline,  but  follows  an  outline  like  that  of  a  hyperbolic  section  — 
the  curvature  being  localized  mostly  at  the  middle.  It  appears  from  ex- 
perimental results  that  the  deflection  may  be  approximately  taken  as  equal 
to  that  of  a  beam  of  No.  i  form,  and  the  formula  for  No.  i  is,  therefore, 
provisionally  adopted  for  No.  3,  until  more  complete  data  are  established  :  — 


No.  4.  Rectangular  beam,  of  constant  depth,  of  uniform  strength,  uniformly 
loaded,  Fig.  167,  page  522.  The  stress  in  the  upper  and  lower  surface  fibres 
of  No.  4  is  only  half  the  stress  in  those  of  No.  i,  under  equal  loads;  and 
therefore  also  the  deflection  is  only  half.  Doubling,  accordingly,  the  nu- 
merical coefficient  of  formula  (  4  ),  — 

W/3 


No.  5.  Rectangular  beam,  of  constant  breadth,  of  uniform  strength,  uniformly 
loaded,  Figs.  164,  165,  page  522.  This  beam  is  elliptic  in  elevation,  and 
the  area,  and,  therefore,  the  average  depth,  are  four-fifths  of  those  of  the 
circumscribed  rectangle,  or  of  No.  4  beam.  Reasoning  on  this  beam,  as 
on  No.  2  beam,  the  deflection  is  five-fourths  of  that  of  No.  4  beam,  and 
the  numerical  coefficient  is  four-fifths  ;  or, 

W/3 
= 


No.  6.  Rectangular  beam,  of  uniform  section,  uniformly  loaded,  —  The  deflec- 
tion under  a  uniform  load  is  found,  by  experiments  with  timber,  to  be 
about  five-eighths  of  that  of  the  same  beam  loaded  with  an  equal  weight 
at  the  middle.  Increase,  therefore,  the  numerical  coefficient  for  No.  3  to 

o 

eight-fifths,  or  (4.62  x  -  =)  7.40:  — 

W/3 
=  -- 


DEFLECTION  OF  DOUBLE-FLANGED,  OR  HOLLOW-RECTANGULAR  BEAMS. 

No.  7.  Double-flanged  beam,  of  constant  depth,  of  uniform  strength,  loaded 
at  the  middle,  Fig.  171,  page  523. 

\st.  When  the  strength  of  both  the  flanges  and  the  web  is  calculated.  The 
value  of  s,  the  direct  stress  on  the  fibres  at  the  upper  and  lower  surfaces, 
is,  by  inversion  of  formula  (  19  ),  page  511, 

W/ 

5=-- .-      (  12  ) 

a  (40+  1.155  a  ) 

in  which  d"  is  the  distance  apart  between  the  centres  of  the  flanges;  a  is 
the  sectional  area  of  one  flange;  and  a"  the  sectional  area  of  the  web, 


DEFLECTION   OF   DOUBLE-FLANGED   BEAMS.  531 

taking  the  height  of  the  web  =  d" '.     Equating  this  value  of  s  to  the  value  (2), 
page  528,  in  terms  of  the  deflection, — 

W/ 


a  ^40+1.1550  ;         t* 
tiere,  d  is  taken  as  equal  to  d" ' ;  whence, 

W/3 


From  this  equation  it  may  be  inferred  that  the  deflection  varies  inversely 
as  a  power  of  the  depth  greater  than  the  square,  and  less  than  the  cube. 

2d.  If  the  strength  of  the  flange  alone  be  calculated.     By  inversion  of 
formula  (20),  page  511, 

W/  , 


Equating  this  value  to  the  value  (2),  page  528, 

JW/      4-TED        d 

^ad  /2 

whence, 


In  this  equation,  it  is  seen  that  the  deflection  varies  inversely  as  the  square 
of  the  depth. 

No.  8.  Double-flanged  beam,  of  constant  breadth,  of  uniform  strength,  loaded 
at  the  middle,  Figs.  168,  169,  page  523.  The  side  of  the  beam  is  triangular, 
and  the  average  depth  is  half  the  maximum  depth.  Reasoning  on  this  beam, 
as  on  No.  2  beam,  page  529,  the  deflection  is  found  to  be  twice  that  of  No.  7 
beam.  The  numerical  coefficients  for  No.  7  are  therefore  halved,  and, 

ist.    When  the  strength  of  both  the  web  and  the  flanges  is  calculated:  — 

' 


2d.    When  the  strength  of  the  flanges  only  is  calculated: 


No.  9.  Double-flanged  beam,  of  uniform  section,  loaded  at  the  middle.  The 
superficies  of  the  flanges  is  double  that  of  the  triangular  flanges  of  No.  7  ; 
and  the  average  stress  is  a  half.  Reasoning  on  this,  as  on  No.  3  beam, 
the  deflection  is  taken  as  equal  to  that  of  No.  7  beam,  and  is  determined 
by  the  formulas  (13)  and  (15). 

No.  10.  Double-flanged  beam,  of  constant  depth,  of  uniform  strength,  uni- 
formly loaded,  Fig.  172,  page  523.  The  deflection  is  half  of  that  of  No.  7, 
with  equal  loads.  Doubling  the  numerical  coefficients  of  the  formulas 
(13)  and  (15),— 

i  st.    When  the  strength  of  both  the  flanges  and  the  web  is  calculated:  — 

D_  W/3 

2"' 


532 


THE   STRENGTH   OF   MATERIALS. 


2d.    When  the  strength  of  the  flanges  only  is  calculated: — 

D  =  -^^E (,9j 

No.  n.  Double-flanged  beam,  of  constant  breadth,  of  uniform  strength,  uni- 
formly loaded,  Fig.  170,  page  523. — This  beam  is  parabolic  in  elevation, 
and  has  an  average  depth  two-thirds  of  the  maximum.  The  deflection  is 
three-halves  of  that  of  No.  10,  and  two-thirds  of  the  numerical  coefficients 
of  formulas  (18)  and  (19)  are  to  be  taken. 

ist.    When  the  strength  of  both  the  flanges  and  the  web  is  calculated: — 

W/3 

II55a») (2°) 

2d.    When  the  strength  of  the  flanges  only  is  calculated: — 

W/3 
°».33*^E 

No.  12.  Double-flanged  beam,  of  uniform  section,  uniformly  loaded. — Increase 
the  numerical  coefficient  for  No.  9  to  eight-fifths,  as  was  done  correspond- 

O  Q 

ingly  for  No.  6;  or  to  (4  x  —  — )  6.4,  and  (16  x  —  =  )  25.6. 

ist.    When  the  strength  of  both  the  flanges  and  the  web  is  calculated: — 

W/3 

D  =  7 —    -     jp- (22) 

6.4^  2E(4#  +  1.1550") 

2d.    When  the  strength  of  the  flanges  only  is  calculated: — 

D-       W/3 

~  25.6  d"2  E*  ' 

Note. — As  to  double- flanged,  or  hollow-rectangular  beams,  Nos.  7  to  12.  It  has  been 
supposed,  for  convenience  of  investigation,  that  the  flanges  are  uniformly  thick  ;  and  that 
the  variation  in  their  section  takes  place  entirely  in  the  breadth. 

Relative  Deflections  of  the  six  forms  of  beams,  both  solid-rectangular,  and 
double-flanged. — The  deflections  are  inversely  as  the  numerical  coefficients 
in  the  respective  formulas,  and  are  as  follows,  table  No.  169: — 

Table  No.   169. — RELATIVE    DEFLECTION    OF    BEAMS,   VARIOUSLY   PRO- 
PORTIONED AND  LOADED. 


LOADED  AT  THE  MIDDLE. 

1.  Constant  depth,  uniform  strength, 

2.  Constant  breadth,          do. 

3.  Uniform  section, 

UNIFORMLY  LOADED. 

4.  Constant  depth,  uniform  strength, 

5.  Constant  breadth,          do. 

6.  Uniform  section... 


Rectangular. 


ratio. 

i.o  or  i 
1.5  or  i 
i.o  or  i 


.5      or 

.625  or 
.625  or 


Double-flanged. 


ratio. 

i.o  or  i 
2.0  or  2 
i.o  or  i 


•5      or    % 

•75    or    # 
.625  or    % 


UNIFORM   BEAMS   SUPPORTED   AT   POINTS.  533 


No.   13.   Deflection  of  a  Cylindrical  Beam  of  Uniform  Diameter.  —  By 

rmula  (2),  page  528,  s  =  ^  —  —  —  ;  and  by  i 

W  / 

5  10,  s  -  -  -j-  .     Equating  these  values  of  s, 
.7854  a3 


formula  (2),  page  528,  s  =   —  —  —  ;  and  by  inverting  formula  (15),  page 
W  / 


Deflection  of  Semi-Beams  and  Semi-Girders.  —  The  deflection  of  a  semi- 
beam  or  semi-girder,  loaded  at  one  end,  is  double  that  of  a  beam  of  twice 
the  length,  loaded  with  twice  the  weight,  at  the  middle  :  —  comparing  beams 
and  semi-beams  of  the  same  principle  of  uniform  strength,  or  of  uniform 
section.  Therefore,  the  deflection  of  a  beam  loaded  at  one  end  is 
(2X2X23  =  )32  times  that  of  the  same  beam  supported  at  both  ends  and 
loaded  at  the  middle. 

To  find  the  deflection  of  semi-beams  or  semi-girders  uniformly  loaded; 
ascertain,  first,  the  deflection  as  found  by  the  ratio  just  stated,  applicable 
when  the  load  is  applied  at  one  point;  secondly,  multiply  the  deflection 
thus  ascertained  by  the  respective  multipliers  subjoined.  The  product  is 
the  deflection  for  a  uniform  load  :  — 

MULTIPLIERS  FOR  UNIFORM  LOADS. 
Rectangular    Double-flanged 
section.  section. 

(Fig.  156)  For  constant  depth,  uniform  strength,...  .5  .5 

(Fig.  155)  For  constant  breadth,         do.  ...  .67  .76 

(Fig.  149)  For  uniform  section,  .........................  625  .625 

These  multipliers  have  been  deduced  by  the  consideration  of  average 
stress  combined  with  average  depth,  already  employed  for  beams  and  girders 
of  constant  depth,  and  of  constant  breadth. 

UNIFORM   BEAMS   SUPPORTED   AT   THREE   OR   MORE 

POINTS. 

The  distribution  of  weight  of  a  continuous  beam  uniformly  loaded  on 
three  or  more  points  of  support,  at  equal  spans,  is  deducible  from  the  laws 
that  regulate  the  deflection  of  such  a  beam  between  the  supports.  Let  the 
load  per  unit  of  length  =  wt  and  the  length  of  the  span  =  //  then  the  total 
load  for  one  span  =  w  I. 

1.  Beam  of  two  equal  spans,  on  three  supports:  — 

Weight  resting  on  ist  and  3d  supports,  =  f  w  I. 
Do.  do.  zd  do.  =  V°  wl- 

2.  Beam  of  ttiree  equal  spans,  on  four  supports  :  — 

Weight  resting  on  ist  and  4th  supports,  =^wl. 
Do.  do.  2d  and  3d  do.  =  -j-J  w  I. 

3.  Beam  of  four  equal  spans,  on  five  supports:  — 

Weight  resting  on  ist  and  5th  supports,  =^wl. 
Do.  do.  2  d  and  4th  do.  =  ff«//. 
Do.  do.  3d  do.  =ffw/. 


534  THE   STRENGTH   OF   MATERIALS. 

Deflection  of  Continuous  Beams  or  Girders. — When  a  continuous  girder, 
uniformly  loaded,  is  supported  at  three  points,  by  two  equal  spans,  the 
middle  portion  is  deflected  downwards  over  the  middle  pier,  and  it  sustains, 
by  suspension,  the  extreme  portions,  which  also  have  a  bearing  on  the  outer 
supports.  The  middle  portion  is,  by  deflection,  convex  upwards,  and  the 
outer  portions  are  concave  upwards;  and  there  is  a  point  of  "contrary 
flexure,"  where  the  curvature  is  reversed,  being  at  the  junction  of  the 
convex  and  concave  curves,  at  each  side  of  the  middle  support.  This 
point  is  distant  from  the  middle  pier,  on  each  side,  one-fourth  of  the  span. 
Of  the  remaining  three-fourths  of  each  span,  a  half  is  carried  by  suspension 
by  the  middle  portion,  and  a  half  is  supported  by  the  abutment.  Hence, 
the  distribution  of  the  load  on  the  supports  is  easily  computed,  as  given 
above.  The  deflection  of  each  span  is  to  that  of  an  independent  beam 
of  the  same  length  of  span,  as  2  to  5. 

In  a  beam  of  three  equal  spans,  the  deflection  at  the  middle  of  either  of 
the  side  spans  is  to  that  of  an  independent  beam,  as  13  to  25. 

In  a  long  continuous  beam,  supported  at  regular  intervals,  the  deflection 
of  each  span  is  to  that  of  an  independent  beam  of  one  span,  as  i  to  5. 

TORSIONAL  STRENGTH  OF  SHAFTS. 

Solid  Round  Shaft. — When  a  solid  round  shaft  is  subjected  to  torsional 
stress,  the  centre  is  a  neutral  axis,  about  which  the  intensity  and  the 
leverage  of  the  resistance  each  increase  as  the  radius;  and  the  two  in 
combination,  or  the  moment  of  resistance  per  square  inch,  increases  as  the 
square  of  the  radius.  Again,  the  ring,  or  annular  area  of  surface,  exposed 
to  stress,  increases  as  the  radius;  therefore,  the  moment  of  resistance  for 
each  ring  is  as  the  cube  of  the  radius;  and  the  total  moment  of  resistance 
for  shafts  of  different  diameters,  is  as  the  cube  of  the  radius,  or  of  the 
diameter. 

The  radius  of  the  resultant  ring  of  resistance  is  the  radius  of  gyration  of 
the  section,  being  the  same  as  that  of  a  circular  plate  revolving  on  its  axis, 
namely,  .7071  r,  the  radius  being  equal  to  r.  (See  page  289.)  By  reasoning 
analogous  to  that  which  was  applied  to  the  transverse  resistance  of  beams, 
it  is  deducible  that,  whilst  the  resultant  radius  is  .7071  r,  the  intensity  of 
resistance  over  the  whole  sectional  area  of  the  shaft  may  be  taken  as 
equivalent  to  that  of  the  resistance  at  the  circumference.  The  ultimate 
moment  of  resistance  is,  then,  expressed  by  the  product  of  the  sectional 
area  of  the  shaft  by  the  ultimate  shearing  resistance  per  square  inch,  and 
by  the  radius,  and  by  .7071;  that  is  to  say,  by 

.7854^2  x  -  x  h  x  0.7071; 

or  by  .2 78^3/z, % (a) 

in  which  d  =  the  diameter  in  inches,  and  h  =  the  ultimate  shearing  resistance 
per  square  inch. 

The  moment  of  the  load  W  is  the  product  of  the  load  by  the  radius  R 
through  which  it  is  applied,  or  W  R;  and,  W  R  =  .278  d3  h\  or, 

, (0 


TORSIONAL   STRENGTH   OF   SHAFTS.  535 

that  is  to  say,  the  breaking  force  is  equal  to  the  product  of  the  cube  of  the 
diameter  by  the  ultimate  shearing  strength  per  square  inch,  and  by  .278, 
divided  by  the  radius  of  the  force.  Also, 


WR  ,     , 

(3) 


^8175 

Hollow  Round  Shafts.  —  The  diagonal  resistance  to  torsion  is  diminished 
by  the  hollowing  of  the  shaft;  and  in  the  absence  of  experimental  evidence, 
it  will  be  provisionally  assumed  that  the  torsional  strength  is  equal  to  that 
of  a  solid  shaft  of  the  same  diameter,  minus  the  resistance  contributed 
by  the  imaginary  core;  though  this  assumption  can  afford  but  a  rough  ap- 
proximation for  a  rule.  The  stress  h'  at  the  circumference  of  the  core  is  less 
than  the  stress  h  at  the  outer  circumference  in  proportion  to  the  diameter,  or 

h  :  h'  :  :  d  :  a",  and  h'  =  h  —  .     The  resistance,  W,  of  the  imaginary  core,  adapt- 
a 

ing  formula  (  i  ),  is 


W  =  -  and,  by  substitution  for  /;',  W  *  '  .  .  (  £  ) 

R  R  d 

The  strength  of  the  hollow  shaft  is,  therefore,  by  deduction, 


R  d 


That  is  to  say  :  Multiply  the  difference  of  the  4th  powers  of  the  outer  and 
inner  diameters  by  the  ultimate  shearing  strength  per  square  inch,  and  by 
.278,  and  divide  by  the  product  of  the  outer  diameter  and  the  radius  of  the 
force.  The  quotient  is  the  ultimate  torsional  strength  of  the  hollow  shaft. 

2d  Method.  —  When  the  section  is  comparatively  thin,  the  material  may 
be  conceived  to  be  collected  at  the  circumference,  for  which  the  radius  of 
gyration  is  equal  to  the  radius  of  the  shaft.  Let  t  =  the  thickness,  then  the 

sectional  area  =  3.  14  d  x  t,  and  W==3.i4</x/x  —  x^-i-R;  or, 


(5) 


Square  Shafts.  —  The  calculable  moment  of  torsional  resistance  of  a  square 
shaft  is  greater  than  that  of  a  round  shaft  having  the  same  sectional  area, 
since  the  corners  of  the  square  project  farther  from  the  centre  than  any 
portion  of  the  circle.  On  the  contrary,  the  material  is  less  favourably  dis- 
posed for  resisting  torsional  stress,  as  the  corners  are  comparatively  unsup- 
ported. It  may,  therefore,  be  assumed  that  practically  the  torsional 
strength  of  a  square  shaft  is  equal  to  that  of  a  round  shaft  having  the  same 
sectional  area.  The  side  of  the  square  section  is  to  the  diameter  of  the 


536  THE   STRENGTH   OF   MATERIALS. 

round  section  as  i  to  1.128;  and  putting  £  =  the  breadth  of  the  side,  and 
^=the  diameter  of  the  equivalent  round  shaft,  d=  1.128  b.     Substitute  this 

,17.     .278  x  (1.128  b}3  x  h 

value  of  a  in  formula  (  i  )  ;  then,  W  =  —  '-  —  •  or, 

R 


(6) 


Inversely,  the  breadth  of  a  square  shaft  having  the  ultimate  torsional  stress 
W  R  is,  after  reduction, 


That  is  to  say,  the  breaking  force  (  6  )  is  equal  to  the  product  of  the  cube 
of  the  breadth  of  the  shaft  by  the  ultimate  shearing  strength  per  square 
inch,  and  by  .4;  divided  by  the  radius  of  the  force. 

Also,  the  breadth  of  the  shaft  is  equal  to  the  cube  root  of  the  quotient 
obtained  by  dividing  the  product  of  the  force  and  its  radius  by  the  shearing 
strength,  multiplied  by  1.36. 

TORSIONAL  DEFLECTION. 

When  a  round  shaft  is  twisted  by  torsional  stress,  the  angular  deflection 
within  the  elastic  limit  is  approximately  proportional  to  the  twisting  force, 
and  to  the  length  of  the  shaft.  Let 

d—  the  diameter  of  the  shaft, 

/=the  length  of  the  shaft  subjected  to  torsion, 

h  -  the  shearing  stress  at  the  circumference  in  tons  per  square  inch 
within  the  elastic  limits, 

D  =  the  total  angular  deflection  of  the  shaft  subjected  to  torsion, 
expressed  in  parts  of  one  revolution, 

E'  -  the  coefficient  of  elasticity,  being  the  denominator  of  the  fraction 
of  the  length  by  which  the  circumference  of  the  shaft  is 
deflected,  or  the  ratio  of  the  length  to  the  circumferential  arc 
of  deflection  per  ton  of  shearing  stress  per  square  inch  at  the 
circumference. 

R  =  the  radius  of  the  force. 

W  =  the  twisting  force  in  tons. 

The  total  circumferential  deflection  is  equal  to 

,     i      j      Ih 

l*W*k=W>   ........................  <'> 

and  if  the  circumferential  arc  of  deflection  be  divided  by  the  circumference, 
equal  to  3.1416  d,  the  quotient  is  the  angular  deflection,  or 

D=-  (8) 

3.i4i6</E'   ' 
and 


,     x 
......................  (9) 

Equating  this  value  of  h  and  the  previous  value  of  ft,  (3  ). 


STRENGTH  OF  TIMBER.  537 

D      WR 


/  .278  a3 

whence 


R  /= 


WR/  ,       . 


These  equations  express  the  relations  of  the  weight  or  force,  the  coeffi- 
cient of  elasticity,  and  the  deflection  within  the  elastic  limits.  The  formula 
(10)  for  the  deflection  signifies  that  the  deflection  varies  directly  as  the 
force,  and  as  the  radius  of  the  force,  or,  jointly,  as  the  moment  of  the 
force;  and  as  the  length  of  the  shaft;  and  that  it  varies  inversely  as  the 
4th  power  of  the  diameter,  and  as  the  coefficient  of  torsional  elasticity. 

The  torsional  deflection  of  a  square  shaft  may  be  found  by  means  of  the 
same  formula,  substituting,  for  calculation,  a  round  shaft  of  equivalent 
strength. 

The  torsional  deflection  of  round  and  square  shafts  varies  with  the  diameter 
in  the  same  ratio  as  the  transverse  deflection;  namely,  as  the  4th  power. 

Hollow  Round  Shafts.  —  Equating  the  value  of  h,  obtained  by  inversion  of 
formula  (4),  and  the  value  in  equation  (9):  — 


.278  (d*- 
whence 


STRENGTH   OF   TIMBER. 

A  number  of  delicate  experiments,  described  by  M.  Morin,1  were  made 
by  various  experimentalists,  with  specimens  of  wood  of  different  kinds, 
uniform  in  texture,  of  very  small  scantling,  and  on  very  wide  spans,  loaded 
at  the  middle.  It  was  satisfactorily  proved  by  the  results  of  these  experi- 
ments: ist,  that  the  deflection  was  sensibly  proportional  to  the  load;  2d, 
that  the  compression  and  extension  were  nearly  the  same,  though  the  com- 
pression was  slightly  the  less;  3d,  that,  to  produce  equal  deflections,  the 
load  when  placed  on  the  middle,  was  to  the  load  when  uniformly  dis- 
tributed, as  .638  to  i,  or  as  5  to  7.84;  4th,  that  the  deflections  under  equal 
loads  were  inversely  as  the  breadths,  inversely  as  the  cubes  of  the  depths, 
and  directly  as  the  cubes  of  the  spans. 

Thus,  the  correctness  of  the  principles  of  the  deflection  of  beams  under 
transverse  stress  is  established  by  the  results  of  most  carefully  conducted 
experiments  ;  though  in  ordinary  practice,  no  doubt,  there  are,  in  individual 
instances,  considerable  degrees  of  divergence  from  those  laws  of  deflection 
in  the  behaviour  of  timber,  which  are  attributable  to  the  want  of  uniformity 
of  structure. 

1  Resistance  des  Materiaux. 


538  THE   STRENGTH   OF   MATERIALS. 

MM.  Chevandier  &  Wertheim,  who  have  made  many  experiments  on  the 
strength  of  timber,  arrived  at  the  following  general  conclusions: — ist.  That 
the  density  of  wood  varies  very  little  with  the  age.  2d.  That  the  coeffi- 
cient of  elasticity  diminishes  after  a  certain  age;  and  that  it  depends  also 
on  the  dryness  and  the  aspect  of  the  ground  where  the  wood  is  grown. 
Woods  from  a  northerly  aspect,  on  dry  ground,  have  always  a  high  co- 
efficient, whilst  woods  from  swampy  districts  have  the  lowest  coefficients. 
3d.  That  the  cohesive  strength  is  influenced  by  the  age  and  the  aspect. 
4th.  The  coefficients  of  elasticity  of  trees  cut  down  in  full  vigour,  and  of 
those  cut  down  before  they  arrive  at  this  condition,  do  not  present  any 
sensible  difference.  5th.  That  there  is  no  limit  of  elasticity,  properly 
so  called,  in  wood.  There  is  a  permanent  set  for  every  elastic  extension. 

A  condensed  table  of  the  results  of  their  experiments  on  the  elastic 
and  absolute  strength  of  timbers,  is  given  in  the  section  on  the  elastic 
strength  of  timber.  It  may  be  added  that  the  same  woods  were  tested 
for  tensile  strength,  in  directions  at  right  angles  to  the  length  of  the  trees, 
in  a  radial  line,  and  in  a  line  tangential  to  the  annular  layers.  The  average 
ultimate  strengths  were  as  follows : — 

Parallel  to  the  axis  of  the  tree,. .  .3.08    tons  per  square  inch,  or  as  i 

Radially 305     „  „  „       */IO 

Tangentially 323     „  „  „       '/I0.5 

MR.  LASLETT'S  EXPERIMENTS. 

The  recently  published  results  of  Mr.  Laslett's  experiments  on  the  strength 
of  timber,  afford  valuable  data  for  the  ultimate  strength  of  timbers.1  The 
specimens  tested  for  tensile  and  transverse  strength  were  2  inches  square. 
For  transverse  strength  they  were  7  feet  long,  on  a  6-foot  span,  with  the  load 
applied  at  the  middle;  and  for  tensile  strength  they  had  usually  a  clear 
length  of  30  inches.  The  specimens  tested  for  crushing,  or  compressive 
strength,  consisted  of  cubes  of  from  i  to  4  inches,  and  of  pieces  2  inches 
square  and  upwards,  of  various  lengths. 

English  Oak. — Twelve  specimens  were  cut  side  by  side,  in  a  line  dia- 
metrically across  one  tree.  Six  on  one  side  of  the  centre  came  out  with  a 
long  clean  straight  grain;  six  on  the  other  side  had  a  wavy  and  twisted 
grain  with  a  short  fibre.  They  were  tested  for  transverse  strength;  the 
breaking  weight  varied  from  390  Ibs.  to  740  Ibs.,  and  the  ultimate  deflec- 
tion from  3.5  to  7  inches. 

Straight  Grain.     Wavy  Grain.         Together. 

Average  breaking  weight 562  Ibs.       40 7  Ibs.       484  Ibs. 

Average  ultimate  deflection 5.10  in.       3.95  in.       4.52  in. 

Average  specific  gravity 858  .867  .862 

The  stronger  half-dozen  specimens  were  afterwards  tested  for  tensile 
strength.  The  results  are  given  in  table  No.  170,  together  with  a  selection 
of  results  of  specimens  from  two  trees  of  average  quality,  fairly  seasoned. 
The  tensile  strength  is  shown  by  the  table  to  increase  with  the  specific 
gravity;  and  it  ranges  from  i  ton  to  4  tons  per  square  inch.  The  trans- 

1  Timber  and  Timber  Trees,  Native  and  Foreign.  By  Thomas  Laslett,  Timber  Inspector 
to  the  Admiralty.  1875.  The  data  extracted  from  Mr.  Laslett's  work  are  here  published 
by  permission  of  the  proprietors  of  the  work. 


STRENGTH   OF   ENGLISH   OAK. 


539 


verse  strength  varies  proportionally  as  from  i  to  2.27;  and  the  deflection 
under  390  Ibs.,  from  1.5  to  4  inches,  or  as  i  to  2.63. 

Table  No.  170. — TRANSVERSE  AND  TENSILE  STRENGTH  OF  ENGLISH  OAK. 

(Reduced  from  Mr.  Laslett's  Experiments.) 
First — Specimens  cut  from  one  side  of  a  tree. 


No.  of  Specimen, 
2  inches  square. 
Span,  6  feet. 

Specific 
Gravity. 

Transverse  Strength. 

Tensile  Strength 
per  Square  inch. 

Deflec- 
tion under 
390  Ibs. 

Set  for 
390  Ibs. 

Ultimate 
Deflec- 
tion. 

Breaking 
Weight. 

i  (next  centre)  

.900 
.900 
.854 
.864 
•838 
.79I 

inches. 
2.0O 
2.00 
2.25 

3-5° 
3-75 
4.00 

inches. 

inches. 
7.00 
4.50 
5.00 
4.50 
5.00 
4.50 

Ibs. 
740 
630 
620 
470 
480 
430 

Ibs.             tons. 

5,320  or  2.375 
4,400  „  1.964 
4,200  „  1.875 
4,340  „  1.938 
2,520  „  1.125 
2,240  „  i.  ooo 

2             

•2 

A 

5  

6  (outside)      

Averages  

.858 

2.916 

— 

5.10 

562 

3,837  or  1.713 

Second  —  From 
Averages  

two  trees, 

1.003 
1.005 

1.002 

.905 
.720 
.725 

good  av< 

i.75 
1.625 
1.50 
3-5o 
3-50 
3-25 

irage  qu< 
.000 

.125 

.000 

.200 

.250 
.125 

ility,  mo 

9.25 
9.50 

8.75 
5.25 
7.00 
6.50 

ierately  s 
882 

977 
827 
590 
804 
797 

easoned. 

8,890  or  3.969 
7,840    ,  3.500 
8,400    ,  3.750 
8,260    ,  3.687 
6,160    ,  2.750 
5,880    ,  2.625 

.893 

2.524 

.117 

7.71 

813 

7,571  or  3.380 

Total  Averages. 

.876 

2.720 

— 

6.40 

688 

5,704  or  2.546 

Dantzic  Fir. — In  a  series  of  six  experiments  for  transverse  strength — 

The  specific  gravity  varied  from  .478  to  .673;  average,  .582 
The  deflection  under  3 90 Ibs.  „     1.25  to  2.25;        „         1.63  inches. 
The  set  under  390  Ibs.           „     .000  to  .100;        „        .066      „ 
Ultimate  deflection                 „     4.50  to  6.15;        „        5.14      „ 
Breaking  weight                      „     700  to  970;         „        877  Ibs. 

In  experiments  for  tensile  strength — 

The  specific  gravity  varied  from        .512  to  .673;  average  .603 
The  breaking  weight  per  square  inch  i.o    to  2.0;          „        1.5  tons. 

The  resistance  to  crushing  of  i-inch,  2-inch,  3-inch,  and  4-inch  cubes  of 
various  woods  was  practically  the  same  per  square  inch  of  surface  for  the 
different  sizes  of  cube;  though  there  was  in  general  a  slight  difference  in 
favour  of  the  smaller  cubes. 

The  table  No.  171,  compiled  from  the  results  of  Mr.  Laslett's  experi- 
ments, shows  the  average  transverse  strength  and  tensile  strength  of  various 
woods,  hard  and  soft;  and  table  No.  172  shows  their  compressive  strength, 
or  the  resistance  of  cubes  to  crushing : — 


540 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  171. — TRANSVERSE  AND  TENSILE  STRENGTH  OF  TIMBER. 
(Reduced  from  Mr.  Laslett's  data.) 

The  specimens  for  transverse  strength  were  supported  at  both  ends  and  loaded 
at  the  middle. 


Name  of  Timber. 

Transverse  Strength. 

Tensile  Strength. 

Aver- 

cagefi 
bpecmc 
Grav- 
ity. 

Deflection. 
Span,  6  feet. 

Break- 
ing 
Weight. 

Aver- 
age 
Specific 
Grav- 

fry. 

Breaking 
Weight 
per  square 
inch. 

Specimens,  2  inches  square. 

Load, 

390  Ibs. 

Set. 

Ulti- 
mate 
Deflec- 
tion. 

Oak:— 
-r-,     r  ,    (  one  side  of  tree 
English  jothersideoftree 

Do 

.858 
.867 

.893 
.976 

1.040 
.990 

.836 
1.042 
.983 

•747 
•993 
.776 
.809 

1.176 
1.116 
1.150 
.917 
.769 

•659 
.678 

1.169 

1.  010 

1.142 

1.029 
•736 
.480 

.558 
.748 

.582 

•541 
.484 
.646 

•439 
•554 
•435 
•55i 
.552 
•659 
.710 

•538 
•550 

inches. 
2.92 
3-25 
2.52 
1.48 
I.58 
3.76 
2.6l 

5.00 
4.03 
1.92 
1.47 
2.25 
1.65 
1.94 

.96 
.92 

2-'5 
.96 
1.208 
1.916 
1.125 

1.27 
3.21 
•94 
1.26 
1.62 
2-75 

4.90 
1-75 

1.63 

1.29 
1.23 
!-57 

2.27 

1.67 

2.12 
I.7I 
2.09 
1.  12 
1.24 
1.42 

i-39 

inches. 

.117 
.041 
•125 

•"3 
.125 

.240 
.250 
.208 
.191 
.050 
•083 
.083 

•033 
.025 
.066 

•033 
.025 
•  083 
.058 

.108 
•133 

.000 

.100 
.050 

•  125 

1.300 
.290 

.066 

.092 

.055 
•  175 

.258 

.133 
1.833 
.714 
.706 
.075 
.063 

.104 

.125 

inches. 
5.10 

3-95 
7.71 
6.00 
7.58 
7.66 
6.50 

6.46 
6.62 
8.83 
7.13 
5-14 
3.38 
6.49 

4-25 
2.83 
4.62 
3-75 
3-45 
4.06 

3-92 

4-75 
4.71 
3-8i 
4.21 
8.63 
7-37 

5-29 
8.79 

5i4 
3-63 

5-19 
4-33 

4-37 
4-63 
4.66 

3-39 
3-45 
4-79 
4.67 
4.42 
4.00 

Ibs. 
562 
407 
813 
877 
831 
758 
758 

474 
562 
804 

723 
1,108 

913 

843 

i,273 
975 
i,333 
1,293 

856 
802 
783 

1,029 
686 
1,407 
712 
862 
638 

393 
920 

877 
600 
670 
626 

560 

?53 
627 

483 
3°4 
1,049 

93° 
744 
719 

.858 

•893 
.976 

.838 

.969 
.742 
.971 
•777 

1.176 

I-I34 
1.141 

.917 
.765 
•659 
•655 

1.169 
.996 
1.150 
1.049 

•750 
.588 

•705 
.642 
.748 
.819 
.603 

•553 
.484 
.649 

.469 
•553 

•551 

•552 
•659 

•544 

Ibs. 
3,837 

7,571 
8,102 

4,217 

7,021 

3,832 
7,052 
3,301 

9,656 

7,199 
8,820 

5,558 
3,791 
2,998 

3,427 

10,284 

2,940 

8,377 
6,048 
3,78o 
5,495 
4,853 
5,460 
9,182 
6,405 
3,231 
4,051 
3,934 
4,203 

2,870 
2,705 

2,759 
2,259 
4,666 

4,040 

tons. 
I.7I3 

3.380 
3.617 

1.882 
3-143 

J-443 
3.148 
1.474 

4-3H 
3.214 

3-937 
2.481 
1.692 
1.338 
1.530 

4.591 
1.312 

3-740 
2.700 
1.687 

2-453 
2.166 

2-437 
4.100 
2.860 
1.442 
i.  808 
1.756 
1.876 

1.281 

1.207 

1.231 

1.008 
2.083 

1.803 

French  
Do  

Tuscan                    

Sardinian  

Dantzic  

Spanish 

American  White  

Do.      Baltimore  
African  (or  teak)  
Teak,  Moulmein  

Do  

Iron  Wood,  Burmah  
Chow,  Borneo  
Greenheart,  Guiana  
Sabicu,  Cuba  
Mahogany   Spanish 

Honduras  
Mexican  .. 

Eucalyptus,  Australia  :  — 
Tewart 

Mahogany  ... 

Iron-Bark 

Blue  Gum 

Ash,  English  

Canadian 

Beech  

Elm,  English  
Rock  Elm,  Canada  

Hornbeam,  England  
Fir   Dantzic 

Ri^a 

Spruce,  Canada  
Larch,  Russia  

Cedar,  Cuba  

Red  Pine,  Canada  
Yellow  Pine,  Canada  
Do.               do  
Do.               do  
Pitch  Pine,  American  
Do.               do  

Do.               do. 

Kauri  Pine,  New  Zealand.... 

CRUSHING  RESISTANCE   OF   TIMBER. 


541 


Table  No.  172. — CRUSHING  RESISTANCE  OR  COMPRESSIVE  STRENGTH 

OF  TIMBER. 

(Reduced  from  Mr.  Laslett's  data.) 


Name  of  Timber. 

Average 

Name  of  Timber. 

Average 

Specimens  —  i-inch,  2-inch,  3-inch, 
and  4-inch  cubes. 

per  square 
inch. 

Specimens  —  i-inch,  2-inch,  3-inch, 
and  4-inch  cubes. 

per  square 
inch. 

Oak,  English  (unseasoned) 
Do.      (seasoned)... 

tons. 
2.194 
3-337 

Eucalyptus,  Mahogany  
Iron-Bark  

tons. 
3.198 

4.601 

French. 

•2  C/T7 

Blue  Gum 

0  078 

Tuscan  

OO^f/ 
2.4^7 

Ash,  English  

o-w° 

•2  JOG 

Sardinian  

s 
2.6O4. 

Canadian 

2  AZ1 

Dantzic  

T..  -34.4. 

Elm,  English..  .    . 

*"*r5J 

2  S83 

American,  White  

2.709 

Rock  

3.832 

Do.         Baltimore 

2.63O 

Hornbeam 

•2  7T  T 

Teak,  Moulmein  

2.51JQ 

Fir,  Dantzic  

J-/  A  L 

3  IO2 

Iron  Wood  

5.208 

Riora 

2  1A.2 

Chow 

5  621 

Spruce 

^•j'r* 

2  166 

Greenheart  

6.438 

Larch  

2  so6 

Sabicu  

•2.776 

Cedar 

2  OOO 

Mahogany,  Spanish  

j-i  /v 

2.863 

Red  Pine.    . 

2  ?^7 

Honduras...  . 

2  8S3 

Yellow  Pine 

-^OO/ 
I  877 

Mexican  

2  so3 

Pitch  Pine 

2  885 

Eucalyptus,  Tewart  

A.  174. 

Kauri  

2  867 

CRUSHING  RESISTANCE  OF  COLUMNS  OF  WOOD. 
English  Oak,  3  inches  square: — 

per  square  inch. 

Unseasoned,  9  specimens,      8  to  16  inch,  high,  spec.  grav.  .922,  1.68  tons. 

Seasoned,       2        do.,        17  and  18     „         „  „          .778,  2.52     „ 

Average  of  4  specimens,  6  inches  square,  1 2  to  36  inches  high,  3.68     „ 

Do.       4        do.,       9      „          „        12  to  21      „          „    '  2.85     „ 


One  specimen,  9  x  10  inches, 


24 
1 8  and  21 


3-24 
„  2.72,  2.91 


Two  specimens, lox  n 

Indian  Teak: — 

6  inches  square,  specific  gravity  .795 4.38     „ 

9  »  »  »  -838 3.81     „ 

Dantzic  Fir,  under  30  inches  high,  average  results: — 

6  inches  square,  specific  gravity  .600 3-897  „ 

9x10  inches,  do.  .608 2.562,, 

10  inches  square,  do.  .660 1.812,, 

10      „  „  do.  .563 2.446  „ 

English    Oak  and  Fir  of   considerable   length   in   proportion   to   the 
scantling : — 


542 


THE   STRENGTH   OF   MATERIALS. 


English  Oak, 

English  Oak, 

Dantzic  Fir, 

Riga  Fir, 

T    .--.,.', 

2  inches  square. 

4  inches  square. 

2  inches  square. 

2  inches  square. 

L-engtn. 
of 
Specimens. 

Specific 
Gravity. 

Crushing 
Weight 
per  square 
inch. 

Specific 
Gravity. 

Crushing 
Weight 
per  square 
inch. 

Specific 
Gravity. 

Crushing 
Weight 
per  square 
inch 

Specific 
Gravity. 

Crushing 
Weight 
per  square 
inch. 

inches. 

tons. 

tons. 

tons. 

tons. 

I 

.740 

3-37 

— 

— 

.756 

2.72 

— 

247 

2 

341 

— 

— 

•756 

3-17 

— 

2.  1  1 

3 

„ 

3-47 

— 

— 

.720 

2-97 

— 

2.88 

4 

?J 

3.50 

— 

— 

.756 

3-44 

— 

2.25 

5 

n 

3-94 

— 

— 

.669 

3-44 

— 

2.63 

6 

„ 

3.72 

— 

— 

.648 

3-25 

— 

2.81 

7 

n 

3.69 

— 

— 

.617 

3-19 

— 

2.78 

8 

„ 

3.63 

— 

— 

.621 

3-03 

— 

2.75 

9 

5J 

3-75 

— 

— 

.720 

3-03 

— 

2.50 

10 

J) 

slipped. 

— 

— 

.669 

3-13 

— 

2.00 

ii 

J) 

3-69 

— 

— 

.726 

2.91 

— 

2.44 

12 

.720 

3-44 

— 

— 

•774 

3.00 

— 

2.78 

15 



— 

.958 

1.  60 

— 

— 

— 

— 

16 



— 

.972 

I.58 

— 

— 

— 

— 

17 



— 

•934 

1.69 

— 

— 

— 

— 

18 

.720 

2.75 

•930 

1.72 

.636 

2.88 

— 

2.47 

19 



•932 

I.76 

— 

— 

— 

20 

— 

— 

.972 

I.76 

— 

— 

— 

— 

21 



— 

.946 

i-75 

— 

— 

— 

— 

22 



— 

•932 

1.63 

— 

— 

— 

— 

23 



— 

.921 

1.47 

— 

— 

— 

— 

24 

.720 

2.63 

1.003 

1.88 

.684 

2.72 

— 

1.72 

30 

•734 

2.44 

— 

— 

.662 

2.63 

— 

1.84 

MR.  FINCHAM'S  EXPERIMENTS  ON  THE  TRANSVERSE  STRENGTH  OF 

SOFT  WOODS.  l 

Mr.  Fincham  made  many  experiments  on  3-inch  square  scantlings,  at 
spans  of  4  feet,  of  wood  of  three  degrees  of  seasoning: — "green"  wood, 
"  dry"  wood,  and  "  very  dry  and  particularly  good"  wood.  Table  No.  173 
contains  results  of  his  experiments  on  very  dry  wood,  to  which  are  added 
the  average  results  for  the  same  woods  "green"  and  "  dry." 

These  results  show  that  the  ultimate  strength  of  the  woods  is  the  same 
whether  green  or  dry,  but  that  the  stiffness  is  materially  increased  by 
thorough  drying.  It  seems  from  the  experiments  that  the  elastic  limit  of 
strength  of  dry  wood  is  about  a  half  of  the  breaking  strength,  and  that  the 
deflection  is  about  y2  inch  for  a  load  of  .75  ton,  or  .64  inch  per  ton. 

TRANSVERSE  STRENGTH  OF  BEAMS  OF  LARGE  SCANTLING. 

Mr.  H.  H.  Maclure  made  experiments  on  the  transverse  strength  of 
Memel  fir,  supported  at  both  ends,  and  loaded  at  the  middle;  for  the 
results  of  which  see  table  No.  174. 


Papers  on  Naval  Architecture,  vol.  i. 


TRANSVERSE   STRENGTH   OF  SOFT   WOODS. 


543 


Table  No.   173. — TRANSVERSE  STRENGTH  OF  SOFT  WOODS  (VERY  DRY). 

(Reduced  from  Mr.  Fincham's  tables.) 
Specimens  3  inches  square,  4  feet  span  ;  loaded  at  the  middle. 


Description 
of 
Timber. 

Specific 
Gravity. 

Load  1680  Ibs., 
or  .75  ton. 

Load  2520  Ibs., 
or  1.125  tons- 

Load  2520  Ibs., 
or  1.125  tons. 
After  i  hour's 
pressure. 

Breaking 
Weight. 

Defl'tion. 

Set. 

Defl'tion. 

Set. 

Defl'tion. 

Set. 

Riga  Fir  
Red  Pine  
Yellow  Pine... 
Norway  Fir.... 
Scotch  Pine  .  .  . 
Kauri  

.6lO 

•544 
•439 
•517 

•453 
•579 

inch. 

•37 

!62 

.29 

inch. 
.00 
.01 
.07 
.01 
.02 
.00 

inch. 

'.78 
.61 

•93 
.46 

inch. 
.04 
.06 
.06 
.01 

•03 
.02 

inch. 
.40 
.86 
1.  00 

.86 
•50 

inch. 
.07 
.08 
.18 
•23 

•05 

Ibs.            tons. 
4530,  or  2.022 
3780,  or  1.688 
2756,  or  1.230 
3292,  or  1.470 
2520,  or  1.125 
4110,  or  1.835 

Average  i 

Green,  Top.... 
Do.,    Butt.... 
Dry,  Top  
Do.,  Butt  
Very  dry  

^esultsfo 

.704 

•645 
.466 

•541 
•  524 

r  the  ab 

•75 
•54 
.58 
.48 
•37 

ove  S 

.08 
.06 
.04 

.02 
.02 

ix  Wooa 

•94 
•75 
.92 
.70 
•64 

?s,  itn 

•13 
.04 

•03 
.10 
.04 

der  diffe 
1.09 

i-35 
1.04 

•95 
.72 

rent  c 

.22 
.70 
.09 
.16 
.12 

onditions. 

3431,  or  1.532 
3746,  or  1.672 
3050,  or  1.361 
2945,  or  1.315 
3498,  or  1.561 

Total  averages 

•576 

•54 

.04  1    -79 

.07 

1.03 

.26 

3334,  or  1.488 

Table  No.  174. — TRANSVERSE  STRENGTH  OF  MEMEL  FIR,  1849. 

(Reduced  from  Mr.  H.  H.  Maclure's  data.) 


Calculated  Tensile 

Breadth  and  Depth. 

Span. 

Breaking  Weight. 

Ultimate 
Deflection. 

Strength  per  square 
inch,  by  formula 

(  2  ),  page  507. 

inches.      inches. 

inches. 

pounds.        tons. 

inches. 

tons. 

I         X          I 

16 

483,  or    .215 

•75 

2.978 

I          X          I 

16 

450,  or    .201 

•75 

2.784 

2X2 

32 

1910,  or   .853 

1.  00 

2-953 

2X2 

32 

1311,  or    .584 

1.125 

2.023 

feet. 

3      x      3 

9 

1  104,  or   .493 

3-5 

1.707 

3><3 

9 

1482,  or    .661 

4-5 

2.289 

6       X     12 

12 

15-5 

2.0 

2.222 

9     x    12 

12 

—        17.0 

2.5 

I-635 

12        X     12 

12 

-       27.5 

3-25 

1.992 

Mr.  Edwin  Clark  tested  the  transverse  strength  of  red  pine  of  large 
scantling  selected  from  the  scaffolding  employed  in  constructing  the 
Britannia  Bridge : — two  whole  balks  1 7  feet  long,  and  a  piece  cut  from  the 
centre  of  a  balk. 

Mr.  C.  Graham  Smith  gives  the  results  of  tests  for  transverse  strength  of 


544 


THE   STRENGTH   OF   MATERIALS. 


pine  timber  of  large  scantling  at  Liverpool.     The  pieces  were  selected  as 
average  samples  from  cargoes.1 

The  table  No.  175  contains  the  leading  results  of  the  experiments  of  Mr. 
E.  Clark  and  Mr.  C.  G.  Smith. 

Table  No.  175. — TRANSVERSE  STRENGTH  OF  PINE  AND  FIR. 

(Reduced  and  arranged  from  the  experiments  of  Mr.  Edwin  Clark, 
and  of  Mr.  C.  Graham  Smith. ) 

(Mr.  Edwin  Clark.) 


Breadth  and  Depth. 

Span. 

Application 
of  the 
Load. 

Elastic 
Strength. 

Elastic 
Deflec- 
tion. 

Breaking 
Weight. 

Ultimate 
Deflec- 
tion. 

Ratio  of 
Elastic  to 
Breaking 
Weight. 

inches. 

feet. 

tons. 

inches. 

tons. 

inches. 

per  cent. 

American  Red  Pine 

I.         12      X     12 

15 

Centre. 

9.0 

1.  00 

14.82 

4.OO 

61 

(Sp.  gr.,  .509) 

2.         12      X     12 

15 

Do. 

9.0 

1.25 

13.24 

3.10 

68 

(Sp.  gr.,  .543) 

3.        6x6 

7-5 

Do. 

2.0 

.62 

3-29 

1.68 

61 

(Mr.  C.  G.  Smith.) 

Memel  Fir. 

4-       13-5  x    13-5 
(from  the  butt) 

10.5 

Distributed. 

38.0 

•37 

61.00 

— 

62 

5-       13-5  x    13-5 
(from  the  top) 

10.5 

Do. 

38.0 

•5i 

61.00 

— 

62 

Baltic  Fir. 

6.        6    x    12 

12.25 

Centre. 

6.0 

.66 

8.50 

1.  11  + 

75 

7.        6     x    12 

12.25 

Do. 

6.0 

.72 

10.50 

I-93  + 

57 

Pitch  Pine. 

8.        6    x    12 

12.25 

Do. 

5.0 

.28 

10.2 

1.31 

5° 

9.        6     x    12 

12.25 

Do. 

8.0 

•97 

10.5 

1.31  + 

76 

10.      14    x    15 

10.5 

Do. 

40.0 

•49 

60.0 

1.14 

67 

ii.      14    x    15 

10.5 

Do. 

35-o 

•49 

59-2 

59 

Red  Pine. 

12.           6      X     12 

12.25 

Do. 

5.0 

.70 

7-5 

— 

67 

13.        6     x    12 

12.25 

Do. 

5>o 

.70 

8-5 

1.94  + 

59 

Quebec  Yellow  Pine. 

14-       14     x    15 

10.5 

Distributed. 

35-° 

•39 

61.0 

— 

58 

15.       14     x    15 

10.5 

Do. 

35-° 

•39 

61.0 

— 

58 

16.      14     x    15 

10.5 

Centre. 

30.0 

.56 

38.3 

— 

78 

17.      14     x    15 

10.5 

Do. 

— 

34-0 

— 

Three  beams  of  oak,  mentioned  by  Mr.  Baker,2  appear  to  have  been 
broken  transversely  by  the  following  loads  at  the  middle : — 

1.  i  inch  square  x  2  feet  span 212  tons  breaking- weight. 

2.  8^  inches  square  x  u  feet  9  inches  span 14.365  „  „ 

3.  io2/3  in.  wide  x  12%  in.  deep,  24  ft.  6  in.  span..    8.780  „  „ 

1  See  Mr.  Smith's  paper  on  Pine  Timber,  read  before  the  students  of  the  Institution  of 
Civil  Engineers  in  1875,  and  published  in  Engineering,  vol.  xix.  page  392. 

2  On  the  Strengths  of  Beams,  Columns,  and  Arches.     1870. 


TRANSVERSE  STRENGTH   OF  FIR  AND   OAK. 


545 


MM.  Chevandier  and  Wertheim  tested  the  transverse  strength  of  rectan- 
gular beams  of  fir  and  oak  from  the  Vosges.1 

Table  No.  176. — TRANSVERSE  STRENGTH  OF  FIR  AND  OAK  FROM 

THE  VOSGES. 

(Reduced  from  MM.  Chevandier  and  Wertheim's  data.) 


VOSGES  TIMBER. 
Specific  Gravity. 

Breadth  and  Depth. 

Span. 

Breaking  Weight  at 
the  middle. 

FIR. 

inches.        inches. 

feet. 

pounds.           tons. 

•530 

11.4     x    12.8 

42.64 

14,120,  or  6.30 

.506 

IO.O       X     1  1.2 

36.08 

11,867,  or  5.30 

.548 

8.8     x     9.6 

29.52 

7,584,  or  3.38 

.525 

6.7     x     7.7 

29.52 

4,580,  or  2.04 

.481 

3.65    x     4.85 

29.52 

1,137,  or    -508 

•493 

9.7     x     2.16 

9.91 

2,017,  or    -900 

479 

9.5      x      i.  ii 

9.91 

581,  or     .260 

OAK. 

1.008 

9.2      x    10.9 

18.04 

17,356,  or  7.75 

.958 

8.6     x     9.3 

18.04 

15,816,  or  7.06 

.922 

7.6     x     8.6 

18.04 

11,495,  or  5.23 

.928 

6.3      x      74 

18.04 

12,155,  or  543 

.985 

54     x     6.3 

18.04 

4,895,  or  2.19 

.636 

3.26   x      3.20 

9.84 

1,188,  or    .530 

•759 

3.07    x      3.16 

8.20 

1,617,  or    .722 

.685 

11.5      x      2.15 

18.04 

957,  or    .427 

.824 

5.64   x      1.66 

9.84 

825,  or    .368 

.712 

9.5      x      i.i  i 

9.84 

715,  or    .319 

ELASTIC  STRENGTH  AND  DEFLECTION  OF  TIMBER. 

Reverting  to  the  conclusions  of  MM.  Chevandier  and  Wertheim,  on  the 
strength  and  elasticity  of  timber,  page  538,  these  experimentalists  found 
that  there  was  no  limit  of  elasticity,  properly  so  called,  in  wood;  though 
there  was  a  permanent  set  for  every  elastic  extension.  They,  nevertheless, 
adopted  empirically,  as  the  limit  of  elasticity  for  tensile  strength,  the  point 
at  which  a  set  of  Va^oooth  of  the  length  is  acquired.  This  is  a  fanciful 
distinction,  for  a  set  of  i  in  20,000  parts  may  be  simply  the  effect  of  a 
straightening  of  the  fibres.  With  this  explanation,  the  following  table, 
No.  177,  of  the  tensile  strength  of  timbers,  condensed  from  their  tables,  is 
of  some  value;  although  the  fractions  of  extension  in  the  second  last 
column  are  scarcely  consistent  with  the  results  of  the  scanty  experiments 
of  others. 


1  Morin's  Resistance  des  Materiaux. 


546 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  177. — TENSILE  STRENGTH  OF  TIMBER. 

(Reduced  from  the  tables  of  MM.  Chevandier  and  Wertheim.) 


ist  SERIES.  —  Comparative  Elastic 
Strength  per  ton  per  square  inch, 
taken  when  the  set  is  i  in  20,000. 

2d  SERIES.  —  Elastic  and  Ultimate  Strength. 

Specific 
Gravity. 

Elastic  Strength,  when 
set  is  i  in  20,000. 

Ultimate 
Strength  per 
square  inch. 

Green 

Wood. 

Wood  Dried. 

Total  per 
square  inch. 

Extension 
per  ton  per 
square  inch 
in  parts  of 
the  length. 

In  closed 
premises. 

In  the  air 
and  the 
sun. 

Acacia  
Fir  

tons. 

.814 

.483 

.627 
1.046 

1.096 
.920 
1.462 

tons. 
2.016 
I.OI4 

I.28I 

1.229 
.883 

.762 

tons. 
2.O24 
1.367 

.027 

.471 

.491 
•037 
.170 
.462 

.288 
.149 

•957 
.724 
.942 

.717 

•493 
•756 
.812 
.822 

.808 
.872 
•559 
•723 
.692 

.697 
.601 
.602 
.674 
•477 

tons. 
2.O24 
1.367 
.814 
.027 
.471 

.491 

•037 
.170 

•723 

.728 
.712 

•657 
.678 

•639 

V8oi 
V707 
V69o 
*/632 
*/6»3 
1/621 
V585 
J/356 
x/740 
z/739 

'/7I2 
1/704 
1/683 
1/648 
'/328 

tons. 
4.978 
2.654 
1.899 

1.369 
2.267 

4-I2I 

3-594 
J-575 
4-439 
3.912 

4.305 
2.883 
4-572 
2.273 
1.240 

Hornbeam... 
Birch 

Beech  

Oak  
Do  
Pine  
Elm  

Sycamore.  ... 

Ash     .     . 

Alder  
Aspen  
Maple  

Poplar  

The  following  are  the  results  of  experiments  by  Mr.  Laslett  on  the 
elongation  of  hard  woods  under  tensile  stress.  The  specimens  were  2 
inches  square;  length,  36  inches.  The  elastic  limit  reached  up  to,  or 
nearly  to,  the  breaking  point : — 


Elastic 

Breaking 

ELASTIC  EXTENSION. 

WOOD. 

per  square 
inch. 

per  square 
inch. 

Total. 

Per  ton  per 
square  inch. 

Fraction  of 
length. 

tons. 

tons. 

inch. 

inch. 

English  oak  

...     2.75      ... 

— 

...       .25       ... 

...     .091      ... 

'/396 

Dantzic  oak  .... 

...    2.66    ... 

...    2.66    ... 

...     .094     ... 

J/383 

Indian  teak  

...    1.75    •.• 

...    1.92    ... 

...       .18      ... 

...     .108     ... 

X/333 

The  following  are  the  chief  results  of  tests  by  Mr.  Kirkaldy  of  the  com- 
pressive  resistance  of  two  balks  of  fir — White  Riga  and  Red  Dantzic,  about 
13  inches  square,  and  20  feet  long,  with  square  ends,  in  a  horizontal 
position.  They  were  "not  very  dry."  The  limits  of  the  elastic  strengths 
are  taken  at  the  points  where  the  rate  of  compression  for  equal  increments 
of  pressure  became  accelerated. 


TRANSVERSE   STRENGTH   OF   TIMBER. 


547 


COMPRESSION.  WHITE  RIGA. 

Elastic  strength 133-9  tons. 

Do.  •         per  square  inch . 792  „ 

Breaking  strength *47«9      » 

Do.               per  square  inch -875,, 

Ratio  of  elastic  to  breaking  strength 90  per  cent. 

Elastic  compression .523  inch. 

Do.  per  ton  per  square  inch, 

in  parts  of  the  length r/364 

Final  compression  before  rupture .642  inch. 

Set  under  total  elastic  stress ...  .022 


RED  DANTZIC. 

1 1 1. 6  tons. 
.627  „ 
138-0      „ 

•775  » 
8 1  per  cent. 
.41 4  inch. 

'/364   . 

.548  inch. 
.02 


MR.  BARLOW'S  EXPERIMENTS  ON  TRANSVERSE  STRENGTH,  I837.1 

Mr.  Barlow  made  a  number  of  experiments  to  test  the  transverse  deflec- 
tion and  strength  of  timber  of  average  quality,  taken  as  seasoned,  from  the 
stores  in  Woolwich  dockyard.  The  specimens  were  2  inches  square, 
and  tested  on  a  span  of  7  feet,  except  a  few  which  were  tested  on  a  span 
of  6  feet.  The  average  ratio  of  the  elastic  to  the  breaking  strength  is,  from 
the  table,  31  per  cent.;  but  Mr.  Barlow  has  not  stated  the  conditions  of  the 
elastic  limit  prescribed  by  him. 

Table  No.  178. — ELASTIC  TRANSVERSE  STRENGTH  OF  TIMBER. 

(Condensed  and  adapted  from  Mr.  Barlow's  experiments.) 
Specimens  2  inches  square,  7  feet  span ;  loaded  at  the  middle. 


Name  of  Timber. 

Specific 
Gravity. 

Elastic  Strength. 

Breaking  Weight. 

Ratio  of 

Elastic  to 
Breaking 
Strength. 

Weight. 

Deflection. 

Weight. 

Deflection. 

Teak 

•745 
•579 
•969 
•934 
.872 
.756 

•993 
.760 
.696 

•553 
.660 
.657 

•553 

•753 
•738 
.696 

•693 
•703 

•531 

.522 
.556 
•  560 
•577 

pounds. 
300 
150 
150 
2OO 
225 
200 

150 
225 

150 
125 
150 
150 

150 
125 
150 
125 
150 
150 

125 
125 

^o 

150 

2OO 

inches. 
I.I5I 

.822 

.080 
•590 

•430 
.266 
.026 
.685 
•134 

•755 

•931 
.870 
•883 
1.442 
i.  006 
1.006 

1.885 
.812 
•831 
•831 
.800 

pounds. 

938 
846 

450 
637 
673 
560 

526 

772 

593 
386 
622 
5ii 
420 
422 
467 
436 
561 
56i 

325 
370 
5oi 
510 
655 

inches. 
4.32 
5-92 
5-92 

8.10 
6.00 
4.86 

5-73 
8.92 

5-73 
6-93 
6.00 

5-83 

4.66 
6.00 
6.00 
6.00 
6.42 
6.42 

8.58 
5-00 
5.00 
5.00 
4.00 

per  cent. 

32 
17.7 

33 
3i-4 
33-4 
36 

28.5 
29 
25 
32.4 
24 
29 

36 
30 
32 
29 
27 

27 

38 
34 
30 
30 
30 

Poon  

English  oak      . 

Do  

Canadian  oak 

Dantzic  oak  

Adriatic  oak 

Ash  

Beech 

Elm  

Pitch  pine     .             . 

Red  pine  

New  England  fir  

Riga  fir 

Do.     (span  6  feet)  
Mar  Forest  fir 

Do.         (span  6  feet) 
Do. 

Larch  

Do.  (span  6  feet)  
Do.             „          
Do.            „          

Norway  spar  (span  6  feet) 

1  On  the  Strength  of  Materials;  edition  of  1845. 


548 


THE   STRENGTH   OF   MATERIALS. 


RULES   FOR  THE   STRENGTH   AND   DEFLECTION 
OF   TIMBER. 

The  results  of  Mr.  Laslett's  experiments,  tables  Nos.  171  and  172,  throw 
some  light  on  the  relations  of  the  tensile,  compressive,  and  transverse 
strength  of  timber.  Employing  formula  (  2  ),  page  507,  namely, 


s  = 


W/ 


1.155  bd2'  ' 


to  calculate  the  direct  tensile  strength  of  the  specimens,  the  results  may  be 
classified  as  follows: — 

Calculated  Tensile  Strength  WITHIN  5  PER  CENT,  of  the  Experimental 

Strength. 


Six  HARD  WOODS. 

Transverse 
Breaking 
Weight. 

Tensile 
Strength, 
calculated. 

Tensile 
Strength, 
experimental. 

Compressive 
Strength, 
experimental. 

English  Oak  (mean) 

Ibs. 
68? 

tons. 
2  3Q2 

tons. 
2  C4.6 

tons. 

-3    -5  -2  7 

Iron  Wood                  

1.271 

•«••  jy 
4.4.28 

A,  "3  I  I 

5  208 

Chow  

07  c 

•3.^02 

3.214. 

c  621 

Iron  Bark 

I  4O7 

A  OOO 

-j  740 

•>  ^ 

4  DO  I 

Blue  Gum                       ..  . 

712 

2  4.77 

2  7OO 

-3  078 

Canadian  Ash  

638 

2.2IQ 

2.4.C3 

2  ACT. 

Averages  (hard  woods)  .  .  . 

949 

3-I5I 

3.l6l 

3.615 

Calculated  Tensile  Strength  MUCH  GREATER  than  Experimental  Strength. 

EIGHT  HARD  WOODS. — Baltimore  oak,  African  teak,  Moulmein  teak, 
greenheart,  sabicu,  average  of  American  mahoganies,  Eucalyptus  mahogany, 
English  ash : — 

Averages,  967  3.354  2.120  3.493 

NINE  SOFT  WOODS. — Dantzic  fir,  Riga  fir,  spruce  fir,  larch,  cedar,  red 
pine,  yellow  pine,  pitch  pine,  Kauri  pine : — 

683  2.375  1.597  2.486 

Calculated  Tensile  Strength  MUCH  LESS  than  Experimental  Strength. 

Six  HARD  WOODS. — French  oak,  Dantzic  oak,  American  white  oak, 
Eucalyptus  Tewart,  English  elm,  Rock  elm : — 

750  2.607  3'295  3-365 

Averages  of  the  Twenty  Hard  Woods  preceding: — 

896  3.069  2.785  3.490 

Averages  of  the  Nine  Soft  Woods  preceding : — 

683  2.375  1.597  2.486 

Averages  of  Twenty-nine  Woods,  Hard  and  Soft: — 

830  2.853  2-4!6  3.168 


STRENGTH   OF  TIMBER  OF  LARGE  SCANTLING.  549 

This  analysis  shows  that  for  only  six  out  of  twenty-nine  woods  does  the 
formula  (  i  )  give  the  experimental  tensile  strength  in  terms  of  the  trans- 
verse strength;  and  these  are  all  hard  woods.  For  the  remainder  of  the 
woods,  comprising  the  soft  woods,  the  formula  (  i  )  shows  a  tensile  strength 
varying  extremely,  both  by  excess  and  by  deficiency,  from  the  experimental 
strength ;  and  for  all  the  soft  woods  the  calculated  tensile  strength  is  far  in 
excess  of  the  experimental  tensile  strength.  In  every  instance  the  experi- 
mental compressive  strength  is  greater  than  the  experimental  tensile  strength 
— for  the  soft  woods  much  greater; — and  the  calculated  tensile  strength 
excepting  for  six  hard  woods,  lies  between  these  values.  It  is,  therefore,  to 
be  inferred  that  the  transverse  strength  is  a  function  of  the  compressive 
strength  as  well  as  of  the  tensile  strength  •  and  that  it  would  be  safe  to 
calculate  the  transverse  strength  in  terms  of  the  mean  of  the  tensile  and 
compressive  strengths,  supposing  that  these  values  can  be  truly  averaged 
for  large  scantlings. 

Calculating  likewise  the  tensile  strength  of  the  pieces  of  soft  woods 
tested  for  transverse  strength  by  Mr.  Fincham,  table  No.  173,  page  543, 
they  are  as  follows: — 

Calculated 
Tensile  Strength. 

Soft  woods,  six  specimens,  green,  top 2.358  tons. 

Do.  do.  green,  butt 2-573    » 

Do.  do.  dry,  top 2.095    » 

Do.  do.  dry,  butt 2.024    » 

Do.  do  very  dry 2.403    „ 

Average, 2.290    „ 

Average  from  Mr.  Laslett's  experiments  on  soft  woods,  2.375    » 

showing  a  fair  accord  between  the  two  calculated  tensile  strengths;  though 
Mr.  Fincham's  3-inch  square  specimens  give  a  lower  value  than  Mr.  Laslett's 
2-inch  square  specimens. 

CALCULATED  TENSILE  STRENGTH  OF  TIMBER  OF  LARGE  SCANTLING. 

Selecting  the  experimental  results  for  the  transverse  strength  of  beams 
of  larger  scantling,  from  six  inches  square  upwards,  the  calculated  tensile 
strengths,  by  formula  (  i  ),  averaged  for  each  set  of  specimens,  are  as 
follows : — 

Calculated 
Tensile  Strength. 

Maclure,  last  3  pieces,  table  No.  174,  page  543,  Memel  fir 1.950  tons. 

Smith,  2      „  „         175,     „     544)     Do.     „  1.334    „ 

Smith>  2      »  „         175,     „     544,  Baltic    „  1.400    „ 

Chevandier,  4      „  „         176,    „     545,  Vosges  „  1.483    „ 

Average  for  Fir, 1.542     „ 

E.  Clark,        3  pieces,  table  No.  175,  page  544,  Red  pine 1.240  tons. 

Smith,  2      „  „         175,     „     544,      Do 1.163     » 

Average  for  Red  Pine, 1.202    „ 

Smith,  4  pieces,  table  No.  175,  page  544,  Quebec  yellow  pine  1.200  tons. 

.Smith,  2      „  „         175,     „     544,  Pitch  pine 1.834    „ 

Baker,  2      „  —       „     544,  English  oak 1.416     „ 

Chevandier,  5      „  „         176,    „     545,  Vosges  oak 1.943    „ 


550  THE   STRENGTH   OF   MATERIALS. 

FORMULAS  FOR  THE  TRANSVERSE  STRENGTH  OF  TIMBER  OF 
LARGE  SCANTLING. 

Adopting  the  foregoing  data  as  the  proper  values  of  J,  the  tensile  strength 
in  tons  per  square  inch,  in  the  general  formula  (  i  ),  page  507,  as  applied 
to  find  the  breaking  weight  of  timber  beams  of  considerable  scantling,  the 
numerical  constant,  1.155  s>  f°r  eacn  is  obtained:  — 

Ultimate  Transverse  Strength  of  Timber  of  Large  Scantling,  loaded  at 

the  middle. 

Fir  ............................  W 


..................  ...... 

Red  pine  .....................  W^1'39^  ........................  (3) 

Quebec  yellow  pine  ........  w 


Pitch  pine  ...................  W=2'12^2  ........................   (5) 


English  oak w=i^— - (6) 

T-  i  i  -ITT       2.2A.bd2  i 

French  oak W  = — (?) 

W  =  the  breaking  weight,  in  tons;  b  the  breadth,  d  the  depth,  and  /  the 
span,  all  in  inches. 

For  other  timbers,  in  the  absence  of  direct  experimental  data,  formulas 
may  be  deduced  for  transverse  strength  by  substituting  for  s,  in  the  general 
formula,  the  mean  of  the  tensile  and  crushing  resistances  of  a  given  wood, 
reduced  in  the  proportion  by  which  the  strength  of  large  scantlings  is  less 
than  that  of  small  scantlings ;  which  may  be  taken  at  two-thirds. 

Meantime,  Mr.  Laslett's  data,  table  No.  171,  may  be  utilized  by  fixing 
the  value  of  the  coefficient,  1.155  st  directly  from  the  transverse  breaking 
weights  of  the  timbers,  taken  at  two-thirds  of  the  observed  values.  Invert- 
ing the  general  formula  (  i  ),  page  507, 

W/ 


By  means  of  this  formula,  the  values  of  the  numerical  coefficients  to  be 
substituted  for  the  coefficient  in  any  of  the  formulas  (  2  )  to  (  7  ),  for  the 
ultimate  transverse  strength  of  other  timbers,  are  found  to  be  as  follows  in 
table  No.  179: — 

FORMULAS  FOR  THE  TRANSVERSE  DEFLECTION  OF  TIMBER  BEAMS 
OF  UNIFORM  RECTANGULAR  SECTION. 

The  deflection  of  beams  of  small  scantling  may  aid  as  a  basis  for  calcu- 
lating the  deflection  of  large  beams,  by  means  of  the  general  formula  ( 4 ), 
page  529,  in  which  the  value  of  E,  the  coefficient  of  elasticity,  may  be 
calculated  from  the  various  data  already  given  for  such  timber  by  means  of 
the  inverted  general  formula  (  8  ),  preceding. 


DEFLECTION   OF   TIMBER   BEAMS. 


551 


Table  No.  179. — VALUES  OF  1.155  J>  NUMERICAL  COEFFICIENT  FOR  THE 
TRANSVERSE  STRENGTH  OF  TIMBER  BEAMS;  TO  BE  USED  IN  ANY  OF 
THE  FORMULAS  (  2  )  TO  (  7  ),  page  550.  (From  Mr.  Laslett's  data.) 


Description  of  Timber. 

Values  of 

1.155*. 

Description  of  Timber. 

Values  of 
1.155  *• 

Oak,  English  (average) 

1.63 

Iron  Bark,  Australia 

3.87 

French  

2.4.1 

Blue  Gum,       do  

>°/ 

I.  q6 

Do 

2  28 

English  fish 

2  -37 

Tuscan 

2.08 

Canadian  ash 

2.  ^O 

Sardinian  

2.08 

Beech  (estimated)  

2.4.O 

Dantzic 

I.3O 

English  elm 

I  08 

Spanish  . 

l.CA 

Rock  elm    Canada  . 

2.  C7 

American  White  

2.21 

Hornbeam,  England  

2.C3 

Baltimore     . 

I  QQ 

Dantzic  fir 

2  4.T 

African  (or  teak)  

3.OC 

Riga  fir            .    . 

1.65 

Moulmein  teak  

2.  SI 

Spruce  fir  

<vo 
1.84. 

Do 

2  32 

Larch   Russia 

I  62 

Iron  \Vood,  Burmah  

3.  CO 

Cedar  Cuba 

I  54. 

Chow  Borneo 

268 

Red  pine   Canada 

180 

Greenheart  Guiana 

367 

Yellow  pine   Canada 

I  71 

Sabicu,  Cuba  

3.r6 

Do             do 

I  33 

Mahogany,  Spanish  

2.3C 

Do.             do  

*4 

Honduras 

2  21 

Pitch  pine  American 

288 

Mexican  

2  K 

Do               do 

2  ;6 

Tewart,  Australia  

2.8^ 

Do.              do  

2.05 

Mahogany    do 

«"VJ 

I  80 

Kauri  pine  New  Zealand 

i  08 

L.<_>y 

Transverse  Deflection  of  Rectangular  Timber  Beams  of  uniform  section : — 

W/3 


4.62 


(9) 


D  =  the  deflection,  /  the  span,  b  the  breadth,  and  d  the  depth,  all  in  inches; 
W  the  load  at  the  middle  in  tons,  E  the  coefficient  of  elasticity. 

The  values  of  E  and  4.62  E  are  given  in  the  annexed  table  No.  iSo.1 

SHEARING  STRENGTH  OF  TIMBER. 

Oak  treenails,  firmly  held,  of  from  i  inch  to  i^  inches -in  diameter, 
were  found  by  Mr.  Parsons  to  have  a  shearing  strength  of  about  2  tons  per 
square  inch  of  section.  For  the  development  of  so  much  resistance,  Pro- 
fessor Rankine  deduces  that  the  planks  connected  by  the  treenails  should 
have  a  thickness  of  at  least  three  times  their  diameter.  Treenails  of  i^ 
inches,  in  3-inch  planks,  bore  only  1.43  tons  per  square  inch;  and  in  6-inch 
planks,  1.73  tons. 

1  It  may  here  be  stated,  that,  whilst  the  value  of  E  possesses  importance  as  an  element 
in  a  scientific  theory  of  deflection,  it  is  not  necessary,  for  the  purposes  of  calculation  for 
the  deflection  of  beams,  that  the  value  of  E  should  be  exactly  ascertained,  since,  in  its 
employment  in  the  formula  for  deflection,  it  is  merged  in  the  compound  coefficient  4.62  E, 
the  value  of  which  can  be  determined,  independently,  from  practical  data. 


552 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  180. — VALUES  OF  E  AND  4.62  E  IN  FORMULA  ( 9  ),  PAGE  551, 
FOR  THE  TRANSVERSE  DEFLECTION  OF  TIMBER  BEAMS. 


Description  of  Timber. 

By  Laslett's 
Data. 

By  Barlow's 
Data. 

Various  Data. 

Averages. 

E. 

4.62  E. 

E. 

4.62  E. 

E. 

4.62  E. 

E. 

4.62  E. 

English  oak  

348 
576 
234 
338 
450 
458 

598 

390 
494 

916 

956 
410 
916 
726 
460 

780 
692 
948 
698 
542 
320 

502 

538 
680 

7H 
578 
3f 
560 

526 

704 
414 

672 

1611 

2656 
1080 

1555 
2080 
2114 

2761 

1804 
2276 

4228 
4412 
1888 
4228 
336o 
2118 

3608 
3196 
4378 
2559 
2506 
1476 

450 

746 
376 

934 
654 

2072 

3445 
1735 

43" 
3018 

2939 

2418 
1227 

2072 

1615 

3286 
2187 

2602 

— 

— 

400 
576 
234 
338 
450 
458 

598 
746 
376 
390 
7'4 
654 

9l6 
956 
4IO 
916 

726 
460 

780 
692 
948 
698 
588 
320 

524 
266 
502 
776 

538 
616 

7H 
578 
344 
456 

|S28 

622 

J45> 

534 
624 

1848 
2656 
1080 

1555 
2080 
2114 

2761 
3445 
1735 
1804 

3293 
3018 

4228 
4412 
1888 
4228 
336o 
2118 

3608 
3196 
4378 

2559 
2722 
1476 

2418 
1227 
2319 
3630 
2490 

2920 

3300 
2669 
1583 

2100 

2434 
2968 
2084 

2465 
2884 

French  do. 

Tuscan  do  

Sardinian  do 

Dantzic  do  

American  white  do. 

Baltimore  do  
Canadian  do  
Adriatic  do. 

African  do.  (or  teak) 
Moulmein  teak 

— 

Poon  

Iron  wood 

Chow  

Greenheart 

Sabica  . 

— 

Spanish  mahogany.. 
Honduras  do  

Mexican  do  
Tewart 

Iron  Bark  
Blue  Gum        . 

English  ash  

636 

524 
266 

450 

350 
712 

474 
564 

Canadian  do.      . 

— 

Beech  

Elm 

Rock-elm  

2319 
2490 

3H7 

33°o 
2669 
I5H 

2585 

2430 

3257 
1915 

2920 

i&  776 

F.  756 

S.578 

F^358 

E.G.  460 
S.  474 
F.  464 
8.690 
F.  410 
S-530 
F.  504 
F.  616 

Memel  fir  
Dantzic  do    . 

3630 

3491 
2669 

Spruce  do. 

New  England  do.  .  .  . 
Scotch  do 

1652 

2124 
2188 
2141 
346l 
1891 

2445 
2328 

2847 

Larch    

Red  pine 

Pitch  do 

Yellow  do  

Norway  spar  
Kauri  pine 

1  E.  C.,— E,  Clark;  S.,— G.  G.  Smith;  F.,— Fincham. 


STRENGTH   OF   CAST   IRON. 


553 


STRENGTH   OF   CAST   IRON. 
TENSILE  STRENGTH  AND  COMPRESSIVE  STRENGTH. 

Mr.  Hodgkinson's  experiments  and  investigations  form  the  basis  of  most 
of  what  is  known  on  the  strength  of  cast  iron.  To  ascertain  the  relative 
strength  of  cast  iron  according  to  the  form  of  the  cross  section,  he  tested 
specimens  of  cruciform,  rectangular,  and  circular  sections — the  first  melting 
of  the  pigs.  The  area 

for  each  section,  Figs.      f*-- "V^S *  INS 

184,  185,  and  186,  was  f-~  23 ->, 

intended  to  be  four 
square  inches,  but  the 
castings  were  accurately 
measured,  and  the  exact 
area  of  each  was  ascer- 
tained. The  following 
were  the  average  breaking  weights  or  absolute  tensile  strengths  per  square 
inch  for  the  different  sections : — 


Figs' l84' l85' l86— Trial  Sectlons  for  Cast  Iron- 


Bowling  iron,  No.  2 

Brymbo  iron,  No.  3 

Blaenavon  iron,  No.  2  ... 


Section.  Tensile  strength  per  square  inch. 

(  Cruciform 6.784  tons. 

(Rectangular 6.267     „ 

j  Cruciform 6.661     „ 

(  Rectangular 6. 1 1 5     „ 

I  Cruciform 6.253     » 

I  Circular 6.614     » 


Total  average 6.450     ,, 

From  these  results  it  appears,  that,  taking  the  strength  of  the  cruciform 
section  as  i,  the  strengths  of  the  other  sections  were  relatively  as  follows: — 

Cruciform. 

Bowling  iron,  No.  2, as  i  to    .924  rectangular. 

Brymbo  iron,  No.  3, as  i  to    .918  rectangular. 

Blaenavon  iron,  No.  2, ...as  i  to  1.054  circular. 

The  section  of  the  specimens  tested  by  Mr.  Hodgkinson 
for  tensile  strength  was  cruciform,  and  the  specimens  were 
V  /111  /\  of  the  form  Fig.  187;  having  a  uniform  section 

\  (III  fffrs  A  for  one  foot  of  length.     For  compression,  they 

were  cylindrical,  ^  inch  in  diameter,  and  were 
made  to  two  heights,  respectively  equal  to  i 
diameter  and  2  diameters,  and  they  were  placed 
for  testing  within  a  cylinder  under  a  loaded 
plug,  as  shown  in  Fig.  188.  He  tested  the 
strength  of  16  denominations  of  cast  iron,  51 
specimens  of  which  were  tested  for  tension 
and  8 1  for  compression.  The  results  of  the 
tests  are  condensed  from  the  Commissioner's 
Figs.  187, 188.— Specimens  for  Report  on  the  Application  of  Iron  to  Railway 


Testing  Tensile  Strength  and 
Compressive  Strength. 


Structures,  in  table  No.  181. 


554 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  181. — TENSILE  AND  COMPRESSIVE  STRENGTHS  OF  CAST  IRONS 
AND  STIRLING'S  IRON. 

(Mr.   Hodgkinson.) 


Iron. 

Mean 
Specific 
Gravity. 

Mean 
Tensile 
Strength 
per  Square 
Inch. 

Mean  Compressive 
Strength  per 
Square  Inch. 

Ratio  of  Tensile  to 
Compressive  Strength. 

Height 
of  Speci- 
men. 

Strength. 

Lowmoor,  No.  I  ... 
Lowmoor,  No.  2  ... 
Clyde,  No.  I  

7.074 

7.043 
7.051 

7.093 
7.IOI 
7.042 

7.H3 
7.051 
7.025 
7.024 
7.071 
7.037 
6.989 
7.II9 

7.034 
7.013 

tons. 
5.667 
6.901 
7.198 

7-949 
10.477 
6.222 
7466 
6.380 
6.131 
6.820 
6.440 
6.923 
6.032 
6.478 
6.228 
5-959 

inch. 

X 

1/2 

X 

1/2 

X 
I# 
X 

i# 

X 

1% 
X 

1/2 

X 
I# 
X 

I'A 

X 

1% 
X 

1/2 

X 

1/2 

X 

1/2 

£ 
*# 

X 

1% 
X 
I* 

X 

1/2 

tons. 
28.809 
25.198 
44430 
41.219 
41.459 
39.6l6 
49.103 

45-549 
47.855 
46.821 
40.562 

35.964 
52.502 

45-7I7 
30.606 

30-594 
32.229 

33-921 

44.723 
45.460 

33-390 
33784 
33.988 
34.356 
33.987 
33.028 
44.610 
42.660 
37.281 
35.H5 
34-43° 
33-646 

:  5.084 
:  4.446 
:  6.438 
:  5-973  ( 
:  5-759 
:  5'5°3 
:  6.177  I 
:  5.729 
:  4-568 
:  4.469 
:  6.519 
:  5.780 
:  7.032  ) 
:  6.123  } 
:  4-797 
:  4-795 
:  5.256 

:  5-532 
:  6.557 
:  6.665 
.-5.186 
:  5.246 
:  4.909 
:  4-963 
:  5.635  I 
:  5.476  } 
:  6.886 
:  6.585 
:  5.985 
:  5-638 
:  5778 
:  5.646  \ 

mean. 

i  :  4.765 
i  :  6.205 
i  :  5.631 

1  •  5-953 
i  14.518 
i  :  6.149 
i  :  6.577 
i  :  4-796 
i  :  5-394 
i  :  6.611 
i  :  5.216 
i  :  4.936 
i  :  5-555 
i  :  6.735 
i  :  5.811 
i  :  5.712 

Clyde  No  2  ...    . 

Clyde,  No.  3  

Blaenavon,  No.  i... 

Blaenavon,  No.  2,  ) 
ist  sample  ( 

Blaenavon,  No.  2,  ) 
2d  sample     .       ( 

Calder  No   i 

Coltness,  No.  3  
Brymbo,  No.  I  
Brymbo,  No.  3  

Bowling   No  2  . 

Ystalifera  anthra- 
cite, No  2  

Yniscedwyn     an- 
thracite, No.  i.. 
Yniscedwyn      an- 
thracite, No.  2.. 

Averages   of  cast  ) 
irons  ...           .     ) 

7.055 

6.830 

38.525 

i  :  5.641 

Stirling's  iron,  2d 
aualitv 

7.165 
7.108 

11.502 
10.474 

X 

1/2 

X 

1/2 

55-952 

53.329 
70.827 
57.980 

i  :  4.865   • 
i  :  4-637 
i  :  6.762 
i  :  5-536 

i  14.751 
I  :  6.149 

Stirling's  iron,  3d 
Quality 

Average   of    Stir-  ) 
ling's  iron             ( 

7.136 

10.988 

59.522 

I  :  5.417 

STRENGTH   AS  AFFECTED  BY  THE   MASS  OF   METAL.        555 

It  appears  from  the  table  that  the  tensile  strength  of  cast  iron  varied 
from  5.667  to  10.477  tons,  and  averaged  6.830  tons  per  square  inch. 

That  the  compressive  strength  varied  from  25.198  tons  to  52.502  tons, 
averaging  38.525  tons  per  square  inch. 

That  the  compressive  strength  was  from  4.518  to  6.735  times  the  tensile 
strength;  average  ratio  of  tensile  to  compressive  strength,  i  to  5.641. 

That  the  specific  gravity  varied  from  6.989  to  7.113,  and  averaged  7.055, 
and  that,  generally,  the  strength  increased  with  the  specific  gravity,  though 
there  were  many  exceptions  to  such  relation. 

That  the  tensile  strength  of  Stirling's  metal  (a  mixture  of  cast  and  wrought 
iron)  averaged  10.988  tons  per  square  inch,  and  the  compressive  strength 
59.522  tons  per  square  inch;  ratio,  i  to  5.417. 

The  average  compressive  resistances  of  the  pieces  one  and  two  diameters 
high  were  respectively  as  100  to  95.6. 

Dr.  Anderson  tested,  at  Woolwich  Arsenal,  850  specimens  of  cast  iron. 
The  ultimate  tensile  strength  of  selected  specimens  varied  from  4.90  tons 
to  14.5  tons  per  square  inch,  averaging  9.45  tons,  and  of  all  the  850  speci- 
mens, from  4.20  tons  to  15.30  tons.  He  found  that  the  average  tensile 
strength  of  ordinary  irons  of  commerce  was  6  tons  per  square  inch.  It  is 
probable  that  the  higher  strengths  were  those  of  bars  of  20!  or  3d  meltings. 

STRENGTH  AS  AFFECTED  BY  THE  MASS  OF  METAL. 

Mr.  Hodgkinson,  comparing  the  tensile  strength  of  bars  of  cast  iron, 
i  inch,  2  inches,  and  3  inches  square,  found  that  the  relative  strengths  were 
approximately  as  100,  80,  77. 

Captain  James  found  that  the  tensile  strengths  of  i-inch,  2-inch,  and 
3-inch  bars  were  as  100,  66,  60;  and  that  the  tensile  strength  of  24 -inch 
bars  cut  out  of  2 -inch  and  3-inch  bars  had  only  half  the  strength  of  the  bar 
cast  i  inch  square. 

The  ascertained  inferiority  in  strength  of  massive  castings  as  compared 
with  thinner  castings  is  attributable  to  the  greater  proportion  of  surface  or 
"skin"  on  the  thinner  castings.  It  is  known  that  the  skin  is  harder  and 
stronger  than  the  interior  of  a  casting.  Besides,  the  interior  of  massive 
castings  becomes  more  spongy  in  texture  as  the  thickness  is  increased. 

STRENGTH  OF  CAST  IRON  AS  AFFECTED  BY  COLD  BLAST 
AND  HOT  BLAST. 

Mr.  Hodgkinson  tested  several  cast  irons,  made  by  cold  blast  and  hot 
blast,  with  the  following  results,  table  No.  182;  showing  an  average 
tensile  strength,  of  all  irons,  7.36  tons  per  square  inch,  and  average 
compressive  strength,  47.0  tons;  ratio,  i  to  6.11.  At  the  same  time,  it 
is  shown  that  the  hot-blast  irons  had  9.17  per  cent,  less  tensile  strength, 
but  that  they  had  3.39  per  cent,  more  compressive  strength,  than  the 
cold-blast  irons. 

Mr.  Robert  Stephenson  concluded  from  experiments  of  more  recent 
date,  conducted  by  him,  that  the  average  strength  of  hot-blast  iron  was  not 
much  less  than  that  of  cold-blast  iron;  but  that  cold-blast  irons,  or  mixtures 
of  cold-blast  irons,  were  more  certain  and  regular,  and  that  mixtures  of  cold- 
blast  and  hot-blast  irons  were  better  than  either  separately  mixed. 


556 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  182. — STRENGTH  OF  COLD-BLAST  AND  HOT-BLAST  IRON. 

(Mr.  Hodgkinson.) 


Description  of  Iron. 

Tensile  Strength 
per  Square  Inch. 

Compressive  Strength 
per  Square  Inch. 

Cold  Blast. 

Hot  Blast. 

Cold  Blast. 

Hot  Blast. 

Carron  iron   No   2 

tons. 

7-45 
6-43 

7.80 
8.42 
6.49 

7.32 

tons. 
6.03 
7.84 
9.68 

6.00 

7-45 

tons. 
47-5° 

S'-S0 

41.65 
36.50 

tons. 
48.50 

59-50 
64.9 

38.50 
36.90 

Carron  iron   No   3 

Devon  iron  No.  3          

Buffery  iron   No  i    

Coed-Talon  iron,  No.  3  

Lowmoor  iron,  No.  3  

Total  averages         ..    .        .... 

7.40 

44.3° 

49.70 

Comparative  averages  of  cold  ) 
and  hot  blast                          f 

7-. 
7-52 

36 
6.83 

47 
44-3° 

.O 
45.80 

Sir  William  Fairbairn,  writing  in  1870,  maintained  that  the  quality  of  iron 
had  been  greatly  improved  since  the  introduction  of  the  hot  blast,  and  that 
nothing,  at  the  time  of  writing,  was  said  of  the  difference  between  hot-blast 
and  cold-blast  irons.  Dr.  Siemens,  on  the  same  occasion,  stated  that  the 
ironmasters  had  seen  the  advantage  of  raising  the  temperature  of  the  blast, 
and  that,  in  using  the  Siemens-Cowper  regenerative  hot-blast  stoves,  the  tem- 
perature had  been  raised  as  high  as  1400°  F.,  without  any  deterioration  of 
the  quality  of  the  metal  having  been  observed.1 

STRENGTH  OF  CAST  IRON  INCREASED  BY  REMELTING. 

The  strength,  as  well  as  the  density,  of  cast  iron  are  increased  by  repeated 
remeltings.  The  increase  of  strength  and  density  appears  to  be  the  conse- 
quence of  the  gradual  abstraction  of  the  constituent  carbon  of  the  iron,  and 
the  approximation  of  the  metal  in  composition  to  wrought  iron. 

Mr.  Bramwell  proved  the  increase  of  the  tensile  strength  of  Acadian 
cold-blast  iron  by  remelting  it.  The  tensile  strengths  of  successive  samples 
were  as  follows : — 

ACADIAN  IRON. 

Tensile  strength  per 

square  inch. 
Sample  bars.  tons. 

i  st  samples 7.5 

2d      do.     after  2  hours  longer  fusion 8.3 

3d      do.     after  i^       „          „           10.8 

4th     do.     remelted,  with  fresh  pigs n.o 

5th     do.     after  4  hours  longer  fusion 18.5 

Maximum  of  5th  samples 19.6 

1  Proceedings  of  the  Institution  of  Civil  Engineers,  "Regenerative  Hot  Blast  Stoves," 
by  Mr.  E.  A.  Cowper,  vol.  xxx.  p.  321. 


STRENGTH  OF  CAST  IRON  INCREASED  BY  REMELTING.      $$? 


Showing  that  the  tensile  strength  was  increased  150  per  cent,  by  8  hours  of 
continued  fusion,  and  by  remelting.  The  compressive  strength  averaged 
3^  times  the  tensile  strength.1 

Sir  William  Fairbairn  tested  for  compressive  strength,  samples  of  Eglinton 
No.  3  hot-blast  iron  of  from  i  to  18  meltings  —  the  resistance  was  doubled 
by  1  8  meltings;  but  the  maximum  resistance  was  attained  at  the  i4th 
melting,  and  amounted  to  2.2  times  the  first  resistance.  The  following 
are  the  results  of  these  tests  :  — 


EGLINTON  No.  3  HOT-BLAST  IRON. 


Melting. 


Compressive 

strength. 

tons. 


44.0 

43-6 
41.1 

40.7 
41.1 
41.1 
40.9 
41.1 


Melting. 

IO. 
II. 
12. 

14. 
15- 

16. 
18. 


Compressive 

strength. 

tons. 

••  57-7 
..  69.8 

••   73-1 

..  66.0  (defective) 

. .  O  C.O 
..  76.7 
..  70.5 

.   88.0 


Remelting,  or  continued  fusion,  of  cast  iron  is  practised  in  the  United 
States.  The  pig  iron  generally  used  has,  in  the  state  of  pig,  a  tensile 
strength  of  from  5  to  6^  tons  per  square  inch.  When  melted,  it  is  kept 
for  some  time  in  a  state  of  fusion,  and  the  first  castings  have  a  tensile 
strength  of  about  9  tons  per  square  inch.  For  guns,  the  metal  is  melted 
three  or  four  times  in  an  air-furnace,  and  at  each  melting  is  retained  in 
fusion  for  from  one  to  three  hours  before  being  poured;  and,  according  to 
the  experiments  of  Major  Wade,  the  strength  of  iron  so  treated  was  succes- 
sively increased.  The  following  are  some  of  the  results  obtained  by  Major 
Wade:— 

AMERICAN   IRON.  TENSILE  STRENGTH. 

tons  per  square  inch. 

Pigs-.... 5  to  6^ 

ist  melting 9.3 2 

2d      do ii. 06 

3d      do 11.96 

4th     do 12.45 

Maximum  strength  observed 20.5 

Samples  from  100  gun-heads 14.9 

Proof  bars  (in  other  trials) 16.23 

38  samples  from  a  Rodman  gun 15.3  to  19.8 

Do.        average 16.88 

A  lot  of  pig  iron,  in  the  crude  state 5.66 

27  guns  cast  from  this  pig  iron,  3d  melting 15. 75 

The  specific  gravity  of  the  metal  was  increased  by  successive  meltings 
and  protracted  fusion,  from  6.90,  in  some  instances,  to  7.40. 

The  compressive  strength  of  the  irons  tested  by  Major  Wade,  varied 
from  37.7  tons  to  78  tons  per  square  inch.  The  specimens  were  y2  inch 

1  The  above  particulars  are  reduced  from  the  Proceedings  of  the  Institution  of  Civil 
Engineers,  vol.  xxii.  page  559. 


558 


THE   STRENGTH   OF   MATERIALS. 


in  diameter,  and   ij^  inches  high.     Some  of  the  mean  results  were  as 
follows :  2d  Melting  3d  Melting> 

No.  i  cast  iron,  44.5  tons  62.5  tons. 

Mixtures  of  Nos.  i,  2,  3,  69.4    „  74.6    „ 

It  may  be  inferred  that  the  ratio  of  tensile  to  compressive  strength  of  the 
American  irons  above  tested,  was  about  i  to  4. 

ELASTIC  STRENGTH  OF  CAST  IRON. 

Mr.  Hodgkinson  made  experiments  with  round  cast-iron  bars  of  one 
square  inch  sectional  area,  and  50  feet  long,  suspended  in  a  lofty  building, 
to  find  the  extension  and  permanent  sets.  These  experiments  were  made 

C 


2  3  4.  5  G      G*    7  TONS 

LocuL   per    square   inch. 

Fig.  189. — Diagram  to  show  Rate  of  Extension  and  Set  of  ic-feet  bars  of  cast  iron.     Table  No.  183. 

with  the  object  of  insuring  exceptional  accuracy  of  results.  But,  by  much 
the  greater  number  of  Mr.  Hodgkinson's  experiments,  both  for  extension 
and  for  compression,  were  made  with  bars  limited  to  10  feet  in  length, 
i  inch  square.  The  results  of  the  observations  on  the  extension  and  com- 


ELASTIC   STRENGTH   OF   CAST   IRON. 


559 


pression  of  10  feet  bars,  are  plotted  in  Figs.  189  and  190,  in  which  the  base- 
line A  B  represents  the  loads,  and  the  verticals  in  light  lines  are  the  observed 
extensions,  compressions,  and  sets,  for  the  given  loads.  The  curves  A  c  and 
A  D  are  traced  through  the  ends  of  these  verticals,  and  the  vertical  black 
lines  show  the  extensions  and  sets,  for  integral  tons  of  load. 


A 

6       I      2      3     4     5      6     7      8     9     10      It      12     13      14     15     16     (7TONS 

Loads  per  square  inch'. 

Fig.  190. — Diagram  to  show  Rate  of  Compression  and  Set  of  xo-feet  bars  of  cast  iron.     Table  No.  184. 

The  following  table,  No.  183,  is  constructed  from  the  diagram  of  exten- 
sion and  set,  Fig.  189,  and  it  shows  the  mean  extension  and  set  for  given 
stresses  in  integral  tons  up  to  6^  tons  on  a  i-inch  square  bar  of  average 
quality  10  feet  long. 

The  table  No.  184  is  likewise  constructed  from  the  diagram  of  compres- 
sion and  set,  Fig.  190,  and  it  shows  the  mean  compression  and  set  for  given 
stresses  in  integral  tons  up  to  17  tons  on  a  i-inch  square  bar  of  average 
quality  10  feet  long. 


560 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  183. — MEAN  REDUCED  EXTENSION  AND  SET  FOR  GIVEN 
STRESSES,  OF  A  BAR  OF  CAST  IRON  OF  AVERAGE  QUALITY,  i  INCH 
SQUARE,  10  FEET  LONG. 

Deduced  from  diagram,  Fig.  189,  page  558. 


EXTENSION. 

SET. 

Total  Extension. 

STRESS. 

Increment 
of  Exten- 

Fraction of  Length. 

Increment 
of  Set  for 

Total  Set  in 

Ratio  of 
Total  Set  to 

sion  for 

each  Ton 

each  Ton. 

In  Inches. 

x  ension. 

Total. 

Per  Ton  per 

Square  Inch 

tons. 

inches. 

inches. 

ratio. 

length  =  i. 

inches. 

inches. 

ratio. 

ratio. 

I 

.0196 

.0196 

I 

I/6l22 

.000163 

.00058 

.00058 

I 

to  34 

2 

.O222 

.0418 

2.n 

V287i 

.000174 

.00139 

.00197 

3-4 

to  21 

3 

.0238 

.0656 

3-35 

Vi829 

.000183 

.00228 

.00425 

7-3 

to  15.4 

4 

.0276 

.0932 

4-75 

1/1287 

.000194 

.00325 

.0075 

13 

to  12.4 

5 

.0300 

.1232 

6.29 

V974 

.OOO2O5 

.00463 

.01213 

21 

to  10.2 

6 

.0378 

.l6lO 

8.21 

*/745 

.000223 

.00720 

•01933 

33 

to    8.3 

6.5 

• 

.0210 

.1820 

9.29 

J/6S9 

.000257 

.00407 

.0234 

40 

to    7.8 

Table  No.  184. — MEAN  REDUCED  COMPRESSION  AND  SET  FOR  GIVEN 
STRESSES,  OF  A  BAR  OF  CAST  IRON  OF  AVERAGE  QUALITY,  i  INCH 
SQUARE,  10  FEET  LONG. 

Deduced  from  diagram,  Fig.  190,  page  559. 


COMPRESSION. 

SET. 

Total  Compression. 

STRESS. 

Increment 
of  Com- 
pression 
for  each 

In  Inches. 

Fraction  of  Length. 

Increment 
of  Set  for 
each  Ton. 

Total  Set. 

Ratio 
of  Total  Set 
to  Com- 
pression. 

| 

Ion. 

Total. 

Per  Ton  per 

Square  Inch 

tons. 

inches. 

inches. 

ratio. 

fraction. 

length  =  i. 

inches. 

inches. 

ratio. 

ratio. 

I 

.02 

.02 

I 

1/6*00 

.000167 

.000567 

.000567 

I 

to  35-3 

2 

,O2l6 

.0416 

2.08 

1/2884 

.000173 

.00198 

.00255 

4-5 

to  17.6 

3 

.O224 

.064 

3-2 

XA*75 

.000178 

.OO2I2 

.00467 

8.2 

to  13.7 

4 

.022 

.086 

4-3 

Vi395 

.OOOI79 

.00230 

.00697 

12.2 

to  12.3 

5 

.O222 

.1082 

/nog 

.000180 

.OO26l 

.00958 

17 

to  11.3 

6 

.0228 

.1310 

£55 

J/gi6 

.OOOl82 

.OO3O2 

.0126 

22 

to  10.4 

7 

.0234 

•1544 

7-72 

J/777 

.000184 

.00315 

•01575 

28 

to  9.8 

8 

.0236 

.178 

8.9 

1/674 

.OOOl85 

•00355 

.0193 

34 

to  9.2 

9 

.0238 

.2018 

10.09 

J/595 

.000187 

.0040 

•0233 

41 

to  8.  7 

10 

.023 

.2248 

11.24 

z/534 

.000188 

.0043 

.0276 

48 

to  8.  i 

ii 

.0236 

.2484 

12.42 

.000188 

•0043 

.0319 

56 

to  7.8 

12 

.026 

.2744 

13.72 

V437 

.000191 

.0056 

•0375 

66 

to  7-3 

13 

•0258 

.3002 

15.01 

1/400 

.000193 

.0077 

.0442 

78 

to  6.8 

14 

.0306 

•3308 

16.54 

.000197 

.0087 

.0529 

93 

to  6.3 

15 

.0222 

•353 

17-65 

T/34<> 

.000196 

.008I 

.061 

1  08 

to  5.8 

16 

.04 

•393 

19-65 

T/3°5 

.000205 

.0193 

.0803 

142 

to  4-  9 

17 

•033 

.426 

21.3 

1/282 

.000209 

.0060 

.0863 

152 

to  4-9 

TRANSVERSE  STRENGTH  OF  CAST  IRON.         561 

It  is  clear  from  the  diagrams,  Figs.  189  and  190,  and  the  tables  Nos.  183 
and  184,  that  both  the  extension  and  the  compression  of  cast  iron,  with  the 
respective  sets,  begin  at  the  beginning  of  the  loading;  and,  strictly  inter- 
preted according  to  the  definition  of  elasticity,  the  evidence  is  to  the  effect 
that  there  is  no  such  thing  as  perfect  elasticity  in  ordinary  cast  iron.  The 
progression  of  extension,  compression,  and  set,  moreover,  is  regular,  and  it 
is  gradually  accelerated  whilst  the  stress  is  increased  in  arithmetical  propor- 
tion. There  is  no  sudden  change  in  the  rate  of  progression  anywhere,  no 
"yielding  point"  for  cast  iron,  and  no  indication  of  a  permanent  elastic 
limit  before  rupture  takes  place. 

In  this  respect  cast  iron  radically  differs  from  wrought  iron  and  steel,  for 
in  the  behaviour  of  these  metals  the  "yielding  point"  is  a  clearly  defined 
characteristic. 

SHEARING  STRENGTH  OF  CAST  IRON. 

Professor  Rankine  states  that  the  shearing  strength  of  cast  iron  is  12.37 
tons  per  square  inch.  But  Mr.  Stoney  found  by  experiment  that  it  was 
from  8  to  9  tons  per  square  inch.  Both  of  these  data  may  be  correct  :  it 
has  been  seen  that  cast  iron  varies  very  much  in  tensile  strength,  according 
to  the  character  of  the  specimens  operated  upon. 

It  is  very  probable  that  the  shearing  resistance  of  cast  iron  is,  by  reason 
of  its  comparative  incompressibility,  equal  to  its  direct  tensile  resistance. 

MALLEABLE  CAST  IRON. 

The  tensile  strength  of  annealed  malleable  cast  iron  is  guaranteed  by 
manufacturers  to  25  tons  per  square  inch.  It  is  capable  of  supporting  10 
tons  per  square  inch,  tensile  strength,  without  permanent  distortion. 

TRANSVERSE  STRENGTH  OF  CAST  IRON. 

Cast-  Iron  Bars  of  Rectangular  Section.  —  Mr.  Barlow  found,  by  experi- 
ment, that  for  i  -inch  square  bars  of  cast  iron,  the  breaking  weight  in  tons, 
applied  at  the  middle,  was  expressed  by  the  formula, 


in  which  b,  the  breadth,  d  the  depth,  and  /  the  span,  are  in  inches.  Mr. 
Robert  Stephenson  arrived,  by  experiment,  at  exactly  the  same  coefficient. 
If  the  coefficient  be  taken  as  only  12,  the  breaking  weight  of  a  i-inch 
square  bar,  at  12  inches  span,  is,  by  the  formula,  just  i  ton;  and  if  the  span, 
/,  be  expressed  in  feet,  the  formula  (  i  ),  with  a  coefficient  of  12,  is  re- 
solved into  the  form, 


If  cast-iron  bars  were  homogeneous,  and  of  uniform  density  for  all 
dimensions,  the  formulas  i  and  2  would  give  the  breaking  strengths  correctly 
for  all  sizes  of  bars  ;  but  so  great  is  the  diminution  of  strength  in  thicker 
castings,  due  to  the  comparatively  open  or  spongy  structure  of  cast  iron  in 
thick  masses,  that  3-inch  square  bars,  relatively,  have  scarcely  two-thirds 
of  the  transverse  strength  of  i-inch  bars,  and  the  proper  coefficient  for 

36 


562 


THE   STRENGTH   OF    MATERIALS. 


formula  (i),  is  only  8.6,  as  applicable  to  3-inch  bars.  It  is  obvious  that  no 
constant  coefficient  can  be  employed,  even  in  iron  of  the  same  denomina- 
tion, for  the  transverse  strength  of  cast-iron  bars,  when  the  thickness  is 
various. 

To  apply  the  general  formula  (i),  page  507,  for  the  transverse  strength 
of  rectangular  bars  of  cast  iron,  in  terms  of  the  tensile  strength;  namely, 

b  d2  s  ,       x 

W=i.i55— -j-\  (  3  ) 

in  which  <£,  d,  and  /  are  in  inches,  s,  the  tensile  strength,  in  tons  per  square 
inch,  and  W,  the  breaking  weight,  in  tons  at  the  middle;  the  mean  tensile 
strength  of  i-inch  square  bars  of  ten  different  irons  was  found  by  Mr. 
Hodgkinson  to  be  16,502  Ibs.,  or  7.36  tons,  and  their  transverse  strength, 
at  54  inches  of  span,  was  464  Ibs.  By  the  formula  (3),  taking  the  forces 
in  pounds,  the  transverse  strength  is, 

1.155*  i»  x  16,502  =  353lb&. 

54 

or  seven-ninths  of  the  actual  strength.  The  excess  of  actual  strength,  3 1  per 
cent,  results  from  the  distribution  of  the  stronger  portion  of  the  section  of 
the  bar  at  the  outside, — the  skin,  in  fact, — where  its  moment,  or  power 
of  resistance,  is,  by  reason  of  the  leverage  of  the  resistance,  much  more 
effective  than  that  of  the  interior  and  weaker  portions.  By  such  tubular 
distribution,  a  greater  total  strength  transversely  is  exerted  than  would 
have  been  exerted  if  the  material  had  been  of  uniform  tensile  strength 
throughout  the  section,  as  was  assumed  in  the  construction  of  the  formula. 
In  bars  of  greater  section,  the  influence  of  the  skin  on  the  strength  is 
comparatively  less.  Accordingly,  in  the  following  examples  selected  for 
comparison,  from  data  supplied  by  Mr.  Edwin  Clark,  the  excess  of  strength 
diminishes  generally  as  the  scantling  of  the  specimen  is  increased.  A 
tensile  strength  of  7  tons  per  square  inch  is  assumed  in  the  calculation  of  the 
strength,  by  the  formula  (3) — 


BARS. 


TRANSVERSE  STRENGTH. 


Width. 

Depth. 

Span.                Calculated. 

Actual.               Excess,  Actual. 

(i.) 

i 

inch 

X 

i  inch  x 

4.5    feet,        .158 

ton, 

.2^2  ton,       60  per  cent. 

(2-) 

i 

„ 

X 

3 

x 

1  8         „           .337 

„ 

.429 

27       „ 

(3-) 

3 

JJ 

x 

i 

x 

2.25 

„ 

I 

-376 

53 

(4-) 

2 

„ 

x 

2 

x 

13-5 

•399 

„ 

-475 

19       » 

(5-) 

2 

55 

X 

3 

x 

9 

1-347 

n 

I 

.800 

35 

(6.) 

3 

J? 

x 

2 

X 

4-5 

1.800 

?> 

2 

.410 

34       „ 

(7-) 

3 

„ 

x 

3 

x 

13-5 

1-347 

„ 

I 

.436    „           6.5     „ 

The  i -inch  square  bar  has  60  per  cent,  excess  of  strength.  The  2d  bar 
has  only  i  inch  of  bottom  skin  for  three  times  the  depth  of  the  ist,  and  so 
has  only  27  per  cent,  excess.  The  3d  bar,  of  the  same  section  as  the 
2d,  was  tried  on  its  side,  and  has  three  times  as  much  bottom  skin  as  the 
2d;  and  so  has  nearly  double  the  excess  of  the  2d,  but  not  so  much  as 
that  of  the  ist,  which  has  comparatively  more  side  skin.  The  4th  bar, 
2  inches  square,  has  less  skin  in  proportion  to  its  bulk,  and  has  only  19  per 
cent,  excess  of  strength;  whilst  the  5th  bar,  of  the  same  thickness,  but 
deeper,  has  a  greater  excess,  for  its  bottom  skin  has  more  leverage.  The 


TRANSVERSE   STRENGTH   OF   CAST   IRON.  563 

6th  bar  has  the  same  section  as  the  5th,  but  is  tested  on  its  side,  and  has 
the  same  excess  of  strength;  whilst  the  7th  bar,  3  inches  square,  has  only 
6^2  per  cent,  excess  of  strength  above  that  calculated  for  it  by  formula  (3). 
The  strength  of  the  yth  bar  is,  on  the  contrary,  34  per  cent,  less  than  what 
would  be  calculated  for  it  by  the  formula  (i). 

Diminishing  differences  with  increasing  sections,  are  also  exemplified 
by  experimental  observations  of  Mr.  Hodgkinson,  selected  from  one  of  his 
tables,1  with  bars  of  Carron  No.  2,  averaged  for  hot  and  cold  blast,  of  three 
sizes,  comprising  two  or  more  bars  of  each  size.  The  sizes  are  here  given 
in  round  numbers  :  — 

BARS.  TRANSVERSE  STRENGTH. 

Width.  Depth.  Span.  Calculated.  Actual.  Excess. 

(i.)     I  inch  x  i  inch  x  54  inches,         .158  ton,        .219  ton,       40  per  cent. 
(2.)     i    „      x  3     „      x  54         „          1.381    „         1.736    „          26       „ 
(3.)     i    „      x  4    „     x  54         „         3.794    „        4.600   „          21 

For  the  calculation  of  the  transverse  strength  of  cast-iron  bars  of  rectan- 
gular sections,  and  of  the  larger  scantlings,  even  of  the  commonest  quality, 
formula  (3)  may,  it  appears,  be  safely  employed,  allowing  a  wide  margin, 
with  a  minimum  factor  of  7  tons  per  square  inch  tensile  strength.  This 
gives  a  numerical  coefficient  of  (1.155  x  7  =  )  8.08;  say,  8,  in  formula  (  3  ). 

Transverse  Strength  of  Rectangular  Bars  of  ordinary  Cast  Iron,  of  the  first 
melting.     Tensile  strength,  7  tons  per  square  inch:  — 


Loaded  at  the  middle,  ......   W  =          ;  ......................  (  4  ) 


Loaded  at  one  end,    .........   W  =  -;  ......................  (  5  ) 

Round  Cast-Iron  Bars.  —  The  strength  of  round  cast-iron  bars,  taking  a 
tensile  strength  of  7  tons,  is  found  from  the  general  formula  (15),  page  510; 
in  which  .7854  x  ^  =  .7854  x  7  =  5.50. 

Transverse  Strength  of  Round  Bars  of  ordinary  Cast  Iron:  — 
Loaded  at  the  middle,  ......   W=5'5°^3;  ....................  (  6  ) 

Loaded  at  one  end,   .........   W^1'375^;  ..................  (  7  ) 

in  which  b,  d,  and  /  are  in  inches,  and  W  is  the  breaking  weight  in  tons. 
With  tensile  strengths  greater  than  7  tons,  the  constants  to  be  used  in  these 
formulas  are  as  follows:  — 


Tensile  strength  per             Constant  in            Constant  in             Constant  in             Constant  in 
square  inch.                    formula  (4).             formula  (5).             formula  (6).             formula  (7). 

8  tons,   

9-2   

2.3  

6-3 

1.6 

9     »       

10.4   

2.6     

7-i 

1.8 

TO     „       

n-5  

2.9  

7-9 

2.O 

ii     „       

12.7  

3-2   

8.6 

2.2 

12       „         

13.8  

3-4  

9-4 

2.4 

On  the  Strength  and  Properties  of  Cast  Iron,  1846,  pp.  398,  399. 


564  THE   STRENGTH    OF   MATERIALS. 

Test  Bars. — It  is  usual,  in  specifications  for  cast-iron  work,  to  require 
that  sample  bars  of  cast  iron,  say  i  inch  thick,  2  inches  deep,  and  on 
bearings  36  inches  apart,  shall  support  a  given  weight  applied  at  the  centre. 
Ten  years  ago,  a  weight  of  25  cwt.  was  considered  sufficient  as  a  test  load; 
but  the  load  has  since  been  increased  to  28  cwt.  and  30  cwt.  This  is  not 
very  severe,  for,  by  formula  (i),  based  on  the  strength  of  i-inch  square 
bars,  the  modern  test-bar,  of  ordinary  iron,  should  support  27.2  cwt.  before 
breaking. 

TRANSVERSE  DEFLECTION  AND  ELASTIC  STRENGTH  OF  CAST  IRON. 

Cast-Iron  Rectangular  Bars. — Mr.  Hodgkinson  gives  particulars  of  the 
deflection  of  rectangular  bars  under  various  loads.  The  deflections  under 
medium  loads  are  here  averaged  as  follows;  and  the  coefficients  of  elas- 
ticity E,  are  calculated  by  means  of  the  formula  (  8  ),  page  530:— 

SPAN.  LOAD.      DEFLECTION.         E. 

inches.  tons.  inches. 

Average  of  8  bars,  i  inch  square, 54          .100        .561        6076 

i  bar,  i  inch  x  3  inches  deep,  54         1.066         .216         6230 
i  bar,  i  inch  x  5  inches  deep,  54        3-348        .153         5966 


Average  value  of  coefficient  of  elasticity 6090 

This  value  of  E  agrees  with  that  which  was  found  for  direct  tensile  strength, 
table  No.  183,  page  560,  and  the  resulting  numerical  coefficient,  4.62  E  = 
4.62  x  6090  =  28,136,  say  28,000;  which  may  be  substituted  for  4.62  E  in 
the  formula  (  8  ),  page  530,  to  give  the  deflection  of  cast-iron  bars,  within 
ordinary  elastic  limits,  thus : — 

Deflection  of  Cast-Iron  Rectangular  Bars  of  Uniform  Section. 

w  73 

Loaded  at  the  middle,  D  = — ....  (8) 

28,000^ 

Loaded  at  one  end, ...  D  =  — — —    (9) 

875  b  a3 

D  =  the  deflection,  b  -  the  breadth,  d=  the  depth,  /=  the  span,  all  in  inches; 
W  =  the  load  in  tons. 

Cast-Iron  Round  Bars. — For  round  bars  of  uniform  diameter,  substitute 
the  above-found  average  value  of  E,  in  the  general  formula  (  26 ),  page  533. 
Then,  3.1416  x  E  =  3.1416  x  6090=  19,132,  say  19,000. 

Deflection  of  Cast-Iron  Round  Bars. 

Loaded  at  the  middle,  D-     W /3  ..  (  10  ) 

19,000  d* 

Loaded  at  one  end,...  D  =  - — (  n  ) 

594  ^ 


TORSIONAL   STRENGTH   OF   CAST   IRON. 


565 


TORSIONAL  STRENGTH  OF  CAST  IRON. 

The  only  direct  experiments  recorded,  worth  notice,  on  the  torsional 
resistance  of  cast-iron,  are  those  of  Mr.  Dunlop  at  Glasgow,  in  iSig.1 
They  were  made  to  ascertain  the  torsional  strength  of  shafts  as  usually  cast 
in  Glasgow  at  the  time.  Two  old  bars  of  cast  iron,  about  5  feet  long  each, 
one  of  them  3  inches  and  the  other  4  inches  square,  were  turned  down  in 
the  lathe  at  five  different  places,  to  ten  different  diameters,  of  from  2  to  4^ 
inches.  The  load  was  applied  at  the  end  of  a  lever  14  feet  2  inches  long. 
Particulars  of  the  experiments  are  given  in  table  No.  185;  the  values  of  h, 
the  shearing  resistance,  calculated  by  the  general  formula  (  3  ),  page  535, 
are  added. 

Table  No.  185. — TORSIONAL  STRENGTH  OF  CAST  IRON.     1819. 

(Reduced  from  Mr.  Dunlop's  data.) 


Diameters. 

Cubes 
of  Diameters. 

Ratio  of 
the  Cubes. 

Breaking 
Weight. 

Ratio  of 
the  Breaking 
Weights. 

Shearing  Stress 
per  square  inch. 

inches. 

tons. 

tons. 

2 

8 

I 

.IIl6 

I 

8-530 

2X 

11.4 

1.4 

.1714 

i-5 

9-201 

2^ 

15.6 

2.0 

.182 

1.6 

7.123 

2^ 

20.8 

2.6 

.312 

2.8 

8.761 

3 

Failed 

— 

— 

— 

3X 

34-3 

4-3 

.522 

4-7 

9.299 

3X 

42.9 

54 

•554 

5.0 

7.902 

3^ 

52.7 

6.6 

.742 

6.6 

8.604 

4 

64 

8.0 

.865 

7-7 

8.265 

4X 

76.8 

9.6 

•963 

8.6 

7.691 

Average 

8  T;; 

wo/  j 

It  seems  that  the  ultimate  torsional  strength  increased  very  nearly  as  the 
cube  of  the  diameter,  and  that  the  average  torsional  resistance  per  square 
inch  of  section  was  8.375  tons.  Assuming,  as  explained  at  page  561, 
that  the  shearing  resistance  of  cast  iron  is  equal  to  its  direct  tensile  resist- 
ance, the  general  formulas  for  torsional  strength  (  i  ),  page  534,  and  (  6  ), 
page  536,  become,  by  substitution, 


For  cast-iron  round  shafts,  W  = ' 2  ?         s  - 


R 


For  cast-iron  square  shafts,  W  =  '4° 


3.6  R 


R 


(12) 
(13) 


W  =  the  force,  in  tons. 
R  =  the  radius  of  the  force,  in  inches. 
WR  =the  moment  of  the  force,  in  statical  inch-tons. 
d  =  the  diameter  of  the  round  shaft,  in  inches. 
b   —  the  side  of  the  square  shaft,  in  inches. 
s    =  the  ultimate  tensile  strength,  in  tons  per  square  inch. 


Annals  of  Philosophy,  vol.  xiii.  1819. 


565  THE   STRENGTH   OF   MATERIALS. 

If  the  tensile  strength,  s,  be  taken  at  7.2  tons,  for  iron  of  average  quality, 
then,  by  substitution  and  reduction: — 

Ultimate  Torsional  Strength  and  Sizes  of  Cast-Iron  Shafts  of  average  quality. 

Round  shaft, W=  2          (J4) 

R 

d  = 


Square  shaft, W  =  ^  .   ( 16  ) 


TORSIONAL  DEFLECTION  OF  CAST-IRON  BARS. 

In  the  absence  of  direct  data  for  the  torsional  deflection  of  cast-iron  bars, 
it  is  assumed  that  it  is  i  ^  times  that  of  wrought-iron  shafts — the  same  pro- 
portion as  that  of  the  transverse  deflections  of  cast-iron  and  wrought-iron 
shafts,  as  indicated  by  a  comparison  of  the  formulas  (8),  page  564,  and 
(5)j  Page  59°-  Multiply,  therefore,  the  second  member  of  the  formula 
(14),  page  592,  by  1^3;  the  coefficient  1070,  or  exactly  1073,  becomes 
(1073x3/5  =  )  644:— 

Torsional  Deflection  of  Round  Cast-Iron  Bars. 

' (.8) 


STRENGTH    OF   WROUGHT   IRON. 


567 


STRENGTH   OF   WROUGHT   IRON. 


TENSILE  STRENGTH. 

Mr.  Telford  deduced  from  his  experiments,  an  average  tensile  strength  of 
29.25  tons  per  square  inch  for  wrought-iron  bars. 

Mr.  Barlow  deduced  from  the  results  of  eight  bars  of  wrought  iron — 
Swedish,  Russian,  and  Welsh,  from  i^  inches  square  to  2  inches  in 
diameter — an  average  tensile  strength  of  25  tons  per  square  inch. 

Mr.  Barlow  also  deduced  from  experiments  on  bars  of  from  i  inch  in 
diameter  to  2  inches  square,  that  the  elastic  tensile  strength  of  good 
medium  wrought  iron  was  10  tons  per  square  inch;  and  that  the  extension 
was  at  the  rate  of  Vio.oooth  part  of  the  length  per  ton  per  square  inch; 
and  that,  therefore,  the  elasticity  was  fully  excited  when  the  bar  was 
stretched  Vioooth  part  of  its  length. 

Sir  WTilliam  Fairbairn  published,  in  1861,  results  of  experiments  on  the 
tensile  strength  of  wrought  iron,  which  are  rendered,  slightly  adapted,  in 
tables  Nos.  186  and  187: — 

Table  No.  1 86.— TENSILE  STRENGTH  OF  WROUGHT  IRON.     1861. 
(Sir  William  Fairbairn.) 


DESCRIPTION  OF  IRON. 

Mean  breaking 
Weight 
per  square  inch. 

Ultimate 
Elongation. 

With 
Fibre. 

Across 
Fibre. 

Lowmoor  iron  (specific  gravity  7.6885).... 
Lancashire  boiler  plates  (9  specimens)  
Staffordshire    iron    (two    ^-inch    plates  ") 
rivetted  together)                                      j 

tons. 
28.66 
21.82 

21 
28.40 
2O.  IO 

22.30 
26.71 
27.36 
22.69 
26.80 
26.56 
26.65 

37-96 
21.25 
22.29 

tons. 

23.43 
20.  IO 

36 

18.49 

20.75 
24.47 
24.03 
23.58 

19.82 
19.62 

fraction  of  length. 

'/23  and  V36 

Vs 
V«,  and  «/aa 

*/2oand'/26 
'As  and  '/as 
Vis  and  '/a* 
V*,  and  Ya3 

•',: 

i 

'As  and  '/35 
Vas  and  Vas 

Charcoal  bar  iron  .  . 

Best   best    Staffordshire    charcoal    plate  ) 
(4  experiments)  J 

Best  best  Staffordshire  plates  (4  experiments) 
Best  best  Staffordshire  plate 

Best  Staffordshire  

Common  Staffordshire  

Lowmoor  rivet  iron  (2  experiments)  

Staffordshire  rivet  iron.  .    .  . 

Staffordshire  rivet  iron  

Bar  of  the  same,  cold-rolled  

Staffordshire  bridge  iron 

Yorkshire  bridge  iron  

568 


THE    STRENGTH    OF    MATERIALS. 


Table  No.  187. — TENSILE  STRENGTH  OF  IRON  AND  STEEL  PLATES  THAT 

HAD    BEEN    SUBJECTED    TO     EXPERIMENT    WITH    ORDNANCE    AT   SHOE- 
BURYNESS.        1 86 1. 

(Sir  William  Fairbairn.) 


DESCRIPTION. 

THICKNESS  OF  PLATES,  IN  INCHES. 

Averages 
of  plates 
of  one 
make. 

X 

% 

# 

1% 

2 

2^ 

3 

IRON. 
Lowmoor  

tons. 

24-34 
24.17 
p 

tons. 

25-75 
23.22 

tons. 

29-43 
26.47 

tons. 
24.16 
22.3O 
25.16 

tons. 

25.35 
23.66 
24.63 

tons. 
24.11 
23.92 
22.73 

tons. 
25.04 

23-54 
24.16 

tons. 
24.79 
24.32 
24.63 

Thames  Co.'s.... 
Beale  &  Co.'s.... 

Averages    of     j 
plates    of    the  > 
same  thickness  ) 

24.26 

24.49 

27-95 

23.87 

24-55 

23-59 

24.25 

— 

Average  speci-  ( 
fie  gravity  .  .  .  j 







7.70 

7.72 

7.72 

7.72 

— 

STEEL. 
Howell&  Co.'s  | 
homogeneous  > 
metal  j 
Specific  gravity... 

30.70 

33-69 

30.91 

26.2O 
7.89 

27.04 
7.91 

27-5I 
7.9T 

27-39 
7.91 

29.06 

Elongations  before  rupture  of  specimens,  in  part  of  the  length 
(2^4  to  3  inches). 

IRON. 
Lowmoor  
Thames  Co. 

per  cent. 
6.2 

3-o 

per  cent. 

7-6 
4.0 

percent. 

IO.O 
4.0 

per  cent. 
I7.6 
14.6 
19-3 

per  cent. 

30-5 
25-3 
17.9 

per  cent. 
28.8 
32.0 

16.0 

percent. 
32.0 
26.5 
23-3 

percent. 
20.5 
I6.5 

16.1 

Beale  &  Co  

Averages  

4.6 

5-8 

7.0 

17.2 

24.6 

25.6 

27-3 

— 

Homogeneous  ) 
metal     J 

25-6 

IO.O 

20.8 

19-3 

34-5 

29-5 

25-8 

— 

From  the  first  table,  No.  186,  it  appears  that  the  strength  of  iron  plates. 
in  the  direction  of  the  fibre,  varied  from  28.66  tons  for  Lowmoor  iron,  to 
20.10  tons  for  Staffordshire  charcoal-iron;  and  that  there  is  no  tensile 
strength  in  the  direction  of  the  fibre  so  low  as  20  tons  per  square  inch. 

Also,  that  the  averages  of  nine  irons  show,  for  the  breaking  weight,-— 

With  the  fibre 23.68  tons  per  square  inch. 

Across  the  fibre 2I-59         »  » 


Difference 2.09  tons,  or  about  9  per  cent. 


STRENGTH    OF    WROUGHT    IRON.  569 

From  the  second  table,  No.  187,  it  appears  that  the  thicker  plates  have 
less  tensile  strength  than  the  thinner  plates;  whilst  the  elongation  before 
rupture  is  greater;  thus: — 

X  to  %  inch  thick.  ij^  to  3  inches  thick. 

-,  ,. .  tons.  per  cent.          tons.  per  cent. 

Iron 25.57  elongation    5.8       24.06  elongation  23.7 

Homogeneous  metal... 3 1. 7 7          „          18.8       27.03          „          27.3 

Sir  William  Fairbairn  tested  the  resistance  of  iron  plates  to  a  bulging- 
stress.  He  stretched  two  ^-inch  plates  and  two  ^-inch  plates  over  a 
cast-iron  frame  12  inches  square  inside,  as  in  Fig.  191,  and  subjected 
them  to  pressure  from  an  iron  bolt  3  inches  in  diameter,  with  a  hemispher- 
ical end,  which  was  applied  to  the  plate. 


Fig.  191.  —  Specimen  Plate  to  resist  Bulging  Stress.  Fig.  192.  —  Effects  of  Bulging  Stress. 

Sir  W.  Fairbairn. 

The  ^-inch  plates  were  indented  ^  inch  and  ^  inch  respectively,  when 
they  commenced  to  fail  by  cracking  on  the  convex  side,  under  a  pressure 
of  4.7  tons  applied  at  the  centre.  Under  7  tons  of  pressure  they  were 
cracked  through. 

The  j^-inch  plates  were  indented  .33  inch  when  they  commenced  to 
crack,  under  a  pressure  of  9  tons  ;  and  they  were  cracked  through  under  a 
pressure  of  17  tons.  See  Fig.  192. 

The  resistance  to  bulging  in  these  experiments  was  in  proportion  to  the 
thickness  of  the  plates. 

To  test  the  effect  of  cold-rolling  on  iron  bars,  Sir  William  Fairbairn 
tested  three  bars,  and  obtained  the  following  results  :  — 


Black  bar  from  the  rolls  ........................   26.0  tons.         20  per  cent. 

„        turned  to  i  inch  in  diameter  .....   27.1     „  22        „ 

„        cold-rolled  to  i  inch  in  diameter  39.4     ,,  8        „ 

showing  that  the  tensile  strength  was  increased  by  one-half;  but  that  the 
elongation  was  reduced  to  less  than  a  half. 

With  respect  to  the  influence  of  temperature,  Sir  William  Fairbairn,  in 
1857,  found  that  the  strength  of  ordinary  Staffordshire  iron  plates,  either 
with  or  across  the  grain,  remained  the  same  for  temperatures  varying  from 
o°  F.  to  400°  F.  At  higher  temperatures  the  strength  declined,  until,  at  a  red 
heat,  it  fell  from  an  ordinary  average  of  20  tons  to  15^  tons  per  square  inch. 

Mr.  Thomas  Lloyd  tested  the  tensile  strength  of  Staffordshire  S.  C.  Crown 
bars,  i^  inches  in  diameter,  of  one  kind.  The  same  bars  were  broken 
four  times  in  succession,  and  the  successive  breaking  weights  were  for  the 

Tensile  Strength. 

ist  Breakage  (10  trials)  ............  23.94  tons  per  square  inch. 

2d         „         (io     „    )  ............  25.86         „  „ 

3d         ,,         (7       „     )  ............  27.06 

4th        „         (6       „     )  ............  29.20         „  „ 


5/O  THE   STRENGTH    OF    MATERIALS. 

showing  a  variation  in  the  same  bars  of  from  23.94  to  29.20  tons,  or  18 
per  cent,  of  the  maximum  tensile  strength. 

The  tensile  strength  of  i  ^-inch  round  Staffordshire  bars  of  various  lengths 
was  found  by  Mr.  Lloyd  as  follows:— 

Breaking  Weight. 

6  Bars,  10    feet  long 32.21  tons. 

6    Do.     3.5  „        „    32-12    » 

6    Do.     3      „        „    32.35    » 

6    Do.     2      „        „    32-00    » 

6    Do.   10  inches  long 32-29    » 

showing  that  the  strength  was  not  affected  by  the  length. 

Mr.  Edwin  Clark  found  the  average  tensile  strength  of  iron  per  square 
inch  as  follows : — 

Staffordshire  boiler  plates,  y^  to  II/I6  inch  thick,  with  fibre,   19.6    tons. 

•  i  ,.  •  i  f     with  fibre,   IQ.Q^ 

Do.        special  trials {  across  fibre!  16.82     '„ 

Difference,  i$j4  per  cent,  less 3.11      „ 

Best  scrap  rivet  iron,  ^  inch  diameter, 24.0       „ 

He  also  found  that  the  ultimate  resistance  to  compression  in  a  wrought- 
iron  bar  was  16  tons  per  square  inch,  at  which  pressure  the  metal  began  to 
ooze  away.  Under  1 2  tons  per  square  inch,  the  set  was  so  great  that  the 
form  began  to  change;  and  this  pressure  was  taken  as  the  average  ultimate 
resistance  to  compression  that  may  be  recognized  in  practice.  The  elastic 
limit  of  tensile  strength  was  also  taken  as  12  tons  per  inch;  and  Mr.  Edwin 
Clark  concluded  thus : — 

"  It  is  very  nearly  true,  and  very  convenient  in  practice,  to  assume  both 
the  extension  and  compression  to  take  place  at  the  rate  of  one  ten-thousandth 
Cro-Trw)  °f  the  length  for  every  ton  of  direct  strain  per  square  inch  of 
section  " — agreeing  in  this  respect  with  Mr.  Barlow. 

Mr.  Edwin  Clark  found  that  the  resistance  of  7/6-inch  rivet  iron  to 
shearing  was  as  follows : — • 

Tons  per  Square  Inch 
of  Section. 

Resistance  by  single  shearing, 24. 1 5 

Do.        by  double  shearing, 22.1 

Do.        in   two   ^-inch    plates   rivetted    together 

(one  section), 20.4 

Do.        in   three   ^/s-inch  plates  rivetted   together 

(two  sections), 22.3 

Tensile  strength, 24.0 

When  three  ^-inch  plates  were  rivetted  together  with  a  ^j-inch  rivet, 
the  frictional  resistance  to  displacement  of  the  middle  plate,  the  hole 
through  which  was  larger  than  the  rivet,  was  from  6  to  8  tons. 


STRENGTH   OF   WROUGHT   IRON. 


571 


EXPERIMENTS  ON  THE  TENSILE  STRENGTH  OF  WROUGHT  IRON  AND  STEEL. 
By  Mr.   Kirkaldy,   1858-61. 

Mr.  David  Kirkaldy  conducted,  for  Messrs.  Robert  Napier  &  Sons,  an 
extensive  series  of  trials  of  the  tensile  strength  of  iron  and  steel  bars  and 
plates,  the  results  of  which  threw  much  light  on  the  properties  of  these 
metals.1 

The  specimens  of  bars  were  formed  with  a  head  at  each  end,  united  to 
the  body  of  the  bar  by  taper  necks,  to  receive  the  shackles,  as  shown  in 
Figs.  193  and  194.  Screw  bolts  and  nuts  were  shackled  as  in  Fig.  195. 


o 


\ 


J    \ 
o 


9      9 

O 

@       9 


L\ 


O 


d> 


Figs.  193, 194,  195. — Mr.  Kirkaldy's  Test  Speci- 
mens of  Bars. 


Figs.  196  to  199. — Mr.  Kirkaldy's  Test  Specimens 
of  Plates. 


The  clear  length  of  bar  was  about  7  inches.  The  plate  specimens  were 
formed  as  in  Figs.  196  to  199,  the  ends  of  the  thinner  plates  being  forti- 
fied by  flitches  rivetted  on  both  sides.  The  increments  of  load  were 
applied  slowly  and  gradually. 

The  specimens  of  bar  iron  varied  from  ^  inch  to  i^  inch  in  diameter; 
but  they  were,  for  the  most  part,  from  3^  inch  to  i  inch  in  diameter.  There 
did  not  appear  to  exist  any  material  difference  of  strength  that  could  be 
ascribed  to  difference  of  diameter. 


Tensile  Strength  and  Elongation  of  Iron  Bars. 

The  average  ultimate   tensile  strengths  of  iron  bars,  and   their   total 
elongations  or  stretching  when  fractured,  were  as  follows : — 


1  The  results  of  Mr.  Kirkaldy's  important  investigations  are  published  in  his  work, 
Experiments  on  Wrought  Iron  and  Stee!, — a  mine  of  experiment  and  research.  The  data 
in  the  text  are  reduced  from  this  work. 

It  is  right  to  explain  that  Mr.  Kirkaldy  expresses  the  resistance,  in  all  cases,  in  pounds; 
and  that  they  are,  in  the  text,  converted  into  tons.  For,  though  the  pound-unit  commends 
itself  as  a  basic  unit  of  great  simplicity,  yet  engineers  are  accustomed  to  think  in  terms  of 
tons,  and  will  continue  to  do  so,  until  some  universal  decimal  system  is  adopted,  by  which 
the  ordinary  ton  may  be  superseded.  It  must  be  admitted  that  the  ton  of  2240  Ibs.  is  a 
barbarous  unit,  and  that  the  New  York  ton  of  2000  Ibs.  is  in  every  sense  superior  to  the 
old  British  relic. 


5/2  THE   STRENGTH    OF    MATERIALS. 


Yorkshire  rolled  bars,  ...................  27.39  tons.  25.2  per  cent. 

Staffordshire      do  ....................  25.90    „  23.5       „ 

Lanarkshire       do  ....................  26-55    »  T9-4       » 

Rivet  iron,         do  ....................  26.00    „  20.5       „ 


Averages, 26.46    „  22.2 


Hammered  scrap,  forged  down, 23.85  ,, 

Bushelled  iron  (turnings),  forged  down,  24.95  »  l6-8  » 

Crank-shaft,  scrap  iron,  with  fibre,...   20.37  „  21.8  ,, 

Do.             do.          across  fibre,     18.55  „  I2-5  » 

Armour  plate,  across  fibre, 16.92  „  9.0  ,, 

Averages, 20.93    »  I7-°       » 

The  lowest  tensile  strengths  of  the  better  kinds  were  not  more  than 
i  J/2  ton  below  the  average  for  each  brand. 

Contraction  of  the  Sectional  Area  in  Fracture. 

The  reduction,  in  size,  of  a  piece  subjected  to  tensile  strain  is  practically 
uniform  throughout  the  portion  that  is  strained,  except  near  to  the  point 

of    rupture,    where    the 

^^Ci"       piece    is    locally    much 

K      j  »        more  contracted  in  size, 

.„ ^--.rrf—        as  illustrated  by  Fig.  200, 

-"- -y which  shows  the  contrac- 
tion of  a  i -inch  round 
bar  of  soft  Bowling  iron, 

with  the  line  of  fracture.  The  diameter,  in  this  instance,  was  reduced 
3/32  inch,  and  the  sectional  area  was  reduced  at  the  fracture  to  43.56  per 
cent,  of  the  area  of  the  original  section.  In  the  following  selection  of 
examples  of  fractured  sectional  areas,  the  percentage  ranges  from  29.5  per 
cent,  to  87.8  per  cent. 

Iron  Bars — Fractured  Sectional  Areas. 

Per  cent,  of  original  area. 

Swedish  R.  F.  charcoal, 29.5 

Staffordshire,  charcoal, 38.4 

Yorkshire,  Lowmoor, 46.3 

Staffordshire  B.  B.  scrap, 47.6 

Do.  S.  C.  crown, 53.4 

Scotch  extra  best  best, 58.5 

Do.     best  best, 68.9 

Do.     common, 71.6 

Do.     common, 85.2 

Russian  C.  C.  N.  D.  (for  steel), 89.8 


Average, 59.0 


STRENGTH    OF   WROUGHT   IRON.  5/3 

Strength  as  the  Diameter  is  Reduced  by  Rolling. 

Four  pieces  were  cut  off  a  i^-inch  bar,  reheated  and  rolled  down  to 
different  sizes.     They  had  the  following  tensile  strengths:  — 


inch.  tons.  per  cent. 

1%  ...........................     22.38     .........     28.3 

I         ...........................     25.60     .........     26.7 

3A  ...........................   25-97   .........   25.2 

%  ...........................   26.65   .........   23.8 

showing  an  increase  of  tensile  strength,  and  less  elongation. 

Strength  as  Affected  by  Turning,  or  Removing  the  Skin. 

Rolled  bars  i%  inch  diameter  were  turned  down  to  i  inch,  and  tested. 
The  average  results  of  four  irons,  so  treated,  were  as  follows  :  — 

Tensile  strength.  Elongation. 

tons  per  square  inch.  per  cent. 

Rough  bars,  ..........................   24.38   ............    17.2 

Turned  bars,  .......................  ..   25.00   ............    19.3 

showing  that  the  turned  bars  were  at  least  as  strong  as  the  rough  bars. 

Strength  as  Affected  by  Forging. 

i^-inch  round  bars  of  four  kinds  of  iron  were  reduced  by  forging  to 
i  inch  and  ^  inch  in  diameter. 

Tons  per  inch.  Elongation. 

Rough  bars,  ................   25-I3   .........   24.5  per  cent. 

Forged  bars,  ................   26.10   .........   17.3       „ 

showing  an  increase  of  strength  equal   to   i  ton  per  square  inch  by  the 
forging  down,  and  a  reduction  of  elongation. 

Strength  as  Affected  by  Reheating  only. 

Five  different  irons,  i  inch  in  diameter,  of  which  the  collars  had  failed 
at  the  first  trial,  had  the  collars  replaced.  The  effects  of  the  reheating  to 
which  the  five  bolts  were  subjected  in  the  operation  are  thus  shown  :  — 

Tons  per  inch.  Elongation. 

1  st  Trial,  .................   25.86   ............    10.1  per  cent. 

2  d  Trial,  ...................   24.88   ............   32.6       „ 

showing  a  reduction  of  i  ton  per  square  inch  of  tensile  strength,  whilst  the 
elongation  was  trebled.   . 

On  the  contrary,  two  pieces  of  a  24  -inch  bar  of  good  iron  were  tested,  — 
one  in  its  ordinary  condition,  and  the  other  after  it  had  been  brought  to  a 
welding  heat,  and  cooled  slowly. 

Tons  per  inch.  Elongation. 

In  ordinary  condition,  .............   25.27   ......   22.3  per  Cent. 

Heated,  and  cooled  slowly,  .......   25.21    ......   17.7       „ 

Strength  as  Affected  by  Intense  Cold. 

Three  pieces  of  a  24  -inch  bar  were  tested,  one  at  64°  K;  the  others,  after 
having  been  exposed  over  night  to  intense  frost,  were  broken  at  23°  F.:  — 


574 


THE    STRENGTH   OF   MATERIALS. 


Tons  per  inch.  Elongation. 

At  64°   24.87   24.9  per  cent. 

At  23°   24.28   23.0       „ 

showing  that  at  the  lower  temperature  the  strength  was  a  little  less  by 
0.59  ton. 

Strength  as  Affected  by  Notching  the  Bar. 

The  two  ends  of  i  inch  round  specimen  bars  were  screwed,  and  the 
screw  at  one  end  was  divided  by  turning  out  a  square  notch  or  groove 
Y%  inch  wide,  as  at  Fig.  201,  leaving  a  diameter  of  .70  inch  at  the  bottom 

of  the  groove.  After  having  been 
broken  at  the  notch,  the  body  of  the 
bar  was  turned  down  to  the  same 
diameter  as  the  notch  had  originally, 
and  the  bar  broken  a  second  time, 


-Notched  Specimen  Bar. 


through  the  body.  The  following  are  selections  from  the  results  of  such 
tests  applied  to  bar  irons;  the  strength  of  the  rough  bar  being  added  for 
comparison.  The  elongations  are  not  recorded;  but  the  contracted  sec- 
tional areas  of  fracture  are  here  given : — 


Lowmoor  (hardest  bar),  .  .  . 
Bowling  (softest  bar),  
Govan  Diamond, 

Tensile  Strengths. 

Contracted  Sectional  Area. 

Notched. 

Turned 
Down. 

Rough 
Bar. 

Notched. 

Turned 
Down. 

Rough 
Bar. 

tons. 

40-95 
29.87 
30.96 
29.87 

tons. 

31-57 

25.01 
25.70 
28.17 

tons. 
29.10 
26.20 
25-7I 
23-I5 

per  cent. 
92.0 

72.2 

75-9 

IOO.O 

per  cent. 
50.8 
44.4 
46.4 
92.0 

percent. 
49.0 

43-6 
50.8 
96.1 

Dundyvan,  common,  

32.91 

27.61         26.04 

85.0 

58.4 

59-9 

showing  a  remarkable  excess  of  resistance,  more  than  5  tons,  at  the  notch, 
due  apparently  to  the  shortness  of  the  notched  portion,  which  was  partially 
sustained  by  the  thicker  parts  on  each  side,  whilst  the  contraction  of  area 
was  in  a  measure  prevented. 

Strength  of  Bars  or  Bolts  as  Affected  by  Screwing. 

Screws  with  rounded  threads  were  cut  in  three  modes — by  means  of  old 
blunt  dies,  new  sharp  dies,  and  chasers;  and  tested  for  tensile  strength. 
The  following  selection  of  results  has  been  made  for  the  purpose  of  fair 
comparison;  premising  that  the  diameter  at  the  base  of  the  thread  was  the 
same  for  all  the  screwed  bolts  of  one  diameter : — 

Not  Screwed.     Screwed.     Difference. 


tons.  tons. 

bolts,  screwed, 24.47  18.22 

,,         „  chased, 24.45  17.20 

i-inch  bolts,  screwed  (old  dies),  27.60  24.62 

„         „         screwed  (new  dies),  27.10  19.04 

„         chased,  27.95  20.02 

bolts,  screwed  (old  dies),  26.47  22.77 

„         screwed  (new  dies),  26.47  :947 

„         chased,  26.47  18.70 


6.25,  or  25  per  cent  less. 

7.25,  or  29 

2.98,  or  1 1 

8.06,  or  30 

7.93,  or  28 

3.70,  or  14 

7.00,  or  26 

8-77,  or  33 


STRENGTH   OF   WROUGHT   IRON. 


575 


A  decided  variation  of  strength  is  due  to  the  mode  of  cutting  the  screw. 
The  blunt  dies,  compressing  the  metal,  as  Mr.  Kirkaldy  argues,  harden  it, 
and  increase  its  tensile  resistance.  The  sharp  dies,  cutting  more  readily, 
do  not  compress  the  metal  so  much  as  the  blunt  dies.  The  chaser  does 
not  compress  it  at  all.  Hence,  the  reduction  of  strength  is  greater  in 
screwing  with  sharp  dies  than  with  blunt  dies,  and  is  greatest  when  the 
chaser  is  used. 

Strength  of  Bars  as  Affected  by    Welding. 

The  pieces  of  bar  iron  to  be  tested  were  cut  through  the  middle,  and 
scarfed  and  welded  in  the  ordinary  manner.  The  tensile  strength  at  the 
weld  was  in  all  cases  less  than  that  of  the  original  bar,  as  well  as  the  elonga- 
tion before  fracture.  The  following  are  the  averaged  results  for  each  class 
of  irons:  — 

Tensile  Strength.  Elongation. 

per  cent.  less.  per  cent. 

inch  diameter,  ............  17.6  .........  6.7 

„              ............  30.2  .........  7.3 

„              ............  14.5  .........  5.9 

„              ............  15.1  .........  10.5 


Glasgow  B.  Best,  i 
Farnley,  i 

Govan  B.  Best, 
Govan  Extra  B.  B. 


Average, 19.4 


7.6 


These  averages  conceal  the  excessive  fluctuations  of  strength,  which  varied 
from  2.6  to  43.8  per  cent,  below  the  normal  strength  of  the  bars. 

Strength  of  Bar  Iron  in  Resisting  Stress  Suddenly  Applied. 

In  a  special  series  of  trials,  when  the  steelyard  was  duly  loaded,  and  all 
taut,  it  was  suddenly  released  by  means  of  a  trigger;  so  that  the  stress  was 
delivered  upon  the  specimen  suddenly,  but  without  any  blow  or  jerk.  Mr. 
Kirkaldy  ascertained  the  value  of  the  rupturing  stress  for  each  iron  by 
taking  the  mean  between  the  lowest  stress  that  caused  rupture  and  the 
highest  that  did  not  do  so.  The  specimens  consisted  of  i-inch  round  bars. 
The  following  are  the  comparative  tensile  strengths : — 


Bradley  charcoal,  
Bradley  Crown  S.C., 
Lowmoor  .  . 

LOAD  APPLIED. 

DIFFERENCE. 

ELONGATION. 

Gradually. 

Suddenly. 

Gradually. 

Suddenly. 

tons. 
25.45 
27.82 
26.48 
26.81 
26.70 
26.70 

24.87 
19.54 

tons. 
22.05 
22.10 
21.37 
20.91 
20.45 
21.07 

22.50 
15.82 

tons,     percent,  less. 

3.40,  or    13.4 
5.72,  or    20.6 
5.11,  or    19.0 
5.90,  or    22.0 
6.25,  or    24.8 
5.63,  or    19.1 

2.37,  or     9.6 
3.72,  or    19.0 

per  cent. 
30.2 
25-3 

27.O 

24.9 
21.  1 

per  cent. 
40.1 
22.5 
25.0 
23.6 

25-3 
20.1 

17.3 
16.9 

Lowmoor,  

Glasgow  B.  Best,  
Glasgow  B.  Best,  
Glasgow    B.    Best  ) 
(X-inch  bars),      ) 
Crank-shaft            . 

25.54 

20.78 

4.76,  or    1  8.6 

24.6 

20.1 

(5  specimens.) 

5/6  THE   STRENGTH   OF    MATERIALS. 

These  results  show  that  the  tensile  resistance  to  fracture  by  suddenly- 
applied  stress  is  from  2  to  6  tons  per  square  inch,  or  from  10  to  25  per  cent, 
less  than  when  it  is  gradually  applied.  The  average  elongation  is  also  less, 
— decidedly  more  so,  if  the  exceptional  specimen,  Bradley  charcoal  bar,  be 
omitted  from  the  average. 

The  Influence  of  Frost,  at  23°  F.,  on  the  tensile  resistance  of  bar  iron  to 
strains  suddenly  applied  was  tried  on  seven  specimens  cut  from  a  ^-inch 
bar  of  Glasgow  B.  Best  iron.  The  average  results  showed  3.6  per  cent, 
diminution  of  resistance;  and  a  reduction  of  elongation  from  25  per  cent, 
to  20^  per  cent. 

Influence  of  Additional  Hammering  on  the  Iron  in  a  large  Crank-shaft. 

Three  pieces  i^  inches  square  were  cut  out,  and  forged  down  to  i^ 
inch,  and  turned  to  i  inch  in  diameter.  Compared  with  two  pieces  which 
were  simply  cut  out,  and  turned  down  to  i  inch,  the  results  were  as  follows : — 

Tons  per  inch.       Elongation. 

Cut  out  and  turned  down, 19.90  1 6. 8  per  cent. 

Cut  out,  forged  down,  and  turned,....   23.70   11.7        „ 

showing  20  per  cent,  increase  of  strength  with  reduced  elongation. 

Strength  of  Hammered  Iron  as  Affected  by  Removing  the  Skin. 

Two  i^-inch  square  bars  of  Govan  hammered  iron  were  turned  down 
to  i  inch  in  diameter.  Compared  with  i  inch  square  Govan  hammered 
bars,  in  their  skins,  they  gave  better  results,  thus : — 

Tons  per  inch.          Elongation. 

i -inch  square  bars, 28.60   20.6  per  cent. 

i j^-inch  square  bars  turned  down,....   30.35   23.5        „ 

Hardening  Iron  Bars. 

A  i^-inch  round  bar  of  Bowling  iron  was  cut  into  several  pieces,  which 
were  turned  and  forged  down,  and  hardened : — 

Diameter.  Tons  per  inch.         Elongation. 

Turned  to  i  inch, 27-I5  28.3  per  cent. 

Forged  to  .87  inch,  hardened  in  water,  32.79  19.6  „ 

Do.  .78  „  „  oil,...  28.85  J9-8  » 

Do.  .70  „  tar,...  28.06  22.4  „ 

showing  that  hardening  in  water  increases  the  strength  more  than  in  oil 
or  tar. 

The  tensile  strength  of  the  second  piece,  above  noted,  namely,  32.79 
tons  per  square  inch,  was  the  greatest  strength  of  iron  observed  by  Mr. 
Kirkaldy. 

Experiments  on  pieces  cut  from  the  large  crank-shaft  already  mentioned, 
and  from  an  armour-plate,  and  hardened,  show  that  there  was  no  increase 
of  tensile  strength  by  hardening,  and  that  the  elongation  was  reduced. 

Case-hardening  Iron  Bars. 

By  case-hardening  specimens  of  several  irons,  and  cooling  them  in  oil, 
or  in  water,  or  slowly,  the  loss  of  tensile  strength  averaged  2.21  tons  per 


STRENGTH   OF   WROUGHT   IRON. 


577 


square  inch;  whilst  three-fourths  of  the  elongation  was  gone.     The  averages 
may  be  placed  together  for  easy  comparison : — 


CASE-HARDENED. 

Tensile 
Strength. 

Elonga- 
tion. 

In  Ordinary  Con- 
dition. 

A    Forged  and  cooled  in  oil,  .... 

tons. 

25-39 

23.70 

25-36 
22.11 

per  cent. 
6.2 
2.9 

ii.  6 
4-9 

tons. 
26.50 
26.50 
27.07 
25-32 

per  cent. 
26.6 
26.6 
23-8 
20.7 

B    Forged  and  cooled  in  water,  

C    Forsred  and  cooled  slowly 

D.  Turned  down,  and  cooled  slowly,  
Averages   . 

24.14 

6.4 

26.35 

24.4 

Cold-rolled  Iron  Bars. 
Five  pieces  of  24 -inch  Blochairn  bar-iron  were  treated  as  follows : — 

Tons  per  inch.  Elongation. 

Cold-rolled  (2  pieces) 3J-86         n. 8  per  cent 

Cold-rolled  and  annealed  (2  pieces) 26.50         25.6         „ 

In  ordinary  condition  ( i  piece) 27.06         22.8         „ 

showing  that  cold-rolling  added  nearly  5  tons  to  the  strength,  which  was 
lost  when  the  bars  were  subsequently  annealed. 

Strength  of  Angle-iron,  Ship-strap,  and  Beam-iron. 

The  tensile  strength  of  angle-irons,  about  ^-inch  thick,  is  generally  less 
by  from  i  to  2  tons  per  square  inch  than  that  of  bar-iron.  The  tensile 
strength  of  ship-strap  and  beam-irons,  from  ^  to  i  inch  thick,  is  2  tons  less 
than  that  of  angle-irons.  The  elongations,  correspondingly,  are  also  less. 

Tensile  Strength  of  Iron  Plates. 

Iron  plates,  of  thicknesses  varying  for  the  most  part  from  |^  inch  to 
24  inch  thick,  cut  into  specimens  i  y2  and  2  inches  wide,  were  tested : — 


PLATES. 

TONS  PER  INCH. 

ELONGATION. 

With  Fibre. 

Across  Fibre. 

With  Fibre. 

Across  Fibre. 

Yorkshire 

tons. 

24-75 
23.01 
22.89 

23-37 
21.96 

tons. 
22.64 
21.40 
21.39 
19.22 
19.56 

per  cent. 
13-4 

9-3 
9-5 
9.6 

7.0 

per  cent. 

8.0 

5-3 
5-2 

2.8 

3-2 

Staffordshire      

Durham  

Shropshire 

Lanarkshire 

General  averages  

23.20 

20.84 

9.8 

4-9 

The  greatest  difference  of  the  lowest  tensile  strength  in  any  group  was 
3  tons  per  inch  below  the  average  of  the  group.  In  the  Yorkshire  plates 
it  did  not  exceed  2  tons. 

The  tensile  strength  across  the  fibre  is  from  i  ^  tons  to  4  tons  per  inch 
less  than  that  with  the  fibre.     The  average  difference  is  10  per  cent. 

37 


578 


THE   STRENGTH   OF   MATERIALS. 


Fractured  Sectional  Area  of  Iron  Plates. 

With  Fibre.  Across  Fibre. 

Yorkshire 63.5  percent.     79.7  per  cent,  of  original  area. 

„       76.5  83.7 


Staffordshire  crown  S.  C.  78.5 
„  Bradley....  84.3 

Scotch  best  boiler 87.3 

Staffordshire  best  best...  90.9 

Scotch  ship 95.4 

Scotch  common 94.4 


89.9 
92.0 
93-6 
94.6 

97-5 
98.5 


Cold-rolled  Iron  Plates. 

Pieces  of  Blochairn  plate  .345  inch  thick  were  reduced  by  cold-rolling 
to  .238  inch  thick,  or  to  two-thirds: — 


TONS  PER  INCH. 

ELONGATION. 

With  Fibre. 

Across  Fibre. 

With  Fibre. 

Across  Fibre. 

• 

In  ordinary  condition  

20.45 

39-73 

22.75 

19.20 
36.00 
21.72 

per  cent. 

4-4 

O.I 

8.0 

per  cent. 
2.6 
0.0 

6.0 

Cold-rolled      

Cold-rolled  and  annealed  

Cold-rolling  nearly  doubled  the  strength,  but  annihilated  the  elongation. 
By  annealing,  all  but  2^  tons  per  inch  of  the  extra  strength  was  lost;  but 
the  original  elongation  was  doubled. 

Strength  of  Iron  Plates  as  Affected  by  Galvanizing. 

Fourteen  specimens  of  Glasgow  best  boiler  plate,  from  3/l6  to  ^  inch 
thick,  were  prepared  for  trial,  half  the  number  having  been  galvanized. 
There  was  no  perceptible  difference  in  any  respect  between  the  galvanized 
and  the  ungalvanized  plates. 

Specific  Gravity  of  the  Irons  Tested. 


Armour-plate 7-6134 

Angle-iron 7.6006 

Iron  plates 7.6287 


Yorkshire  rolled  bars 7.7600 

Staffordshire  rolled  bars...    7.6178 
Lanarkshire  rolled  bars ...    7.6280 

Crank-shaft 7.6307 

The  specific  gravity  was  diminished  by  cold-rolling,  though  the  tensile 
strength  was  increased ;  as  follows : — 

Ordinary.  Cold-rolled. 

Bar  iron,  specific  gravity 7-636  7-582 

Boiler-plate,         „  7.566  7.539 

The  specific  gravity  of  iron  was  also  diminished  by  stretching  under 

tensile  Stress  : —  SPECIFIC  GRAVITY. 

Before  Stretching.      After  Stretching. 

Three  i-inch  Yorkshire  bars,  stretched  to  .90  inch...  7.752  7-674 

Two  .83-inch  Blochairn  bars,  „          .76     „   ...7.636  7.569 

Average  for  five  bars 7.760  7-632 

showing  an  average  reduction  of .  1 28,  or  1.65  per  cent.,  in  the  specific  gravity. 


STRENGTH   OF   WROUGHT   IRON. 


579 


EXPERIMENTS  OF  THE  STEEL  COMMITTEE  OF  CIVIL  ENGINEERS.     1870. 

The  Steel  Committee,  who  will  be  again  noticed  in  treating  of  the 
strength  of  steel,  tested  the  strength  of  a  number  of  wrought-iron  bars 
i  y2  inches  diameter,  consisting  of  twelve  bars  of  Lowmoor  iron,  six  bars 
of  best  Yorkshire  iron,  and  six  bars  of  usual  S.  C.  Crown,  or  Stafford- 
shire iron.  Table  No.  189  gives  condensed  results  of  the  experiments 
for  the  tensile  strength  of  wrought-iron  bars,  in  10  feet  of  length;  and 
table  No.  188,  the  same  for  their  compressive  strengths. 

Note  to  Tables  Nos.  188,  189. — The  lowest  elastic  strength  in  any  group 
of  bars  did  not  exceed  i  ton  per  square  inch  less  than  the  average  elastic 
strength;  say,  not  more  than  10  per  cent,  less  than  the  average  for  iron  bars. 

A  chemical  analysis  of  these  irons  is  given  with  that  of  the  steels  tested 
by  the  committee,  in  table  No.  203,  page  603. 

Table  No.  188. — COMPRESSIVE  STRENGTH  OF  WROUGHT-!RON  BARS.    1870. 

i%  inches  in  diameter.     Observations  made  on  lo-feet  lengths. 
(Reduced  from  results  of  experiments  made  by  the  Steel  Committee.) 


Mark  and  Description. 

Elastic  Strength 
(Compressive) 
in  Tons  per 
Square  Inch. 

Elastic  Compression. 

Elastic  Compression 
per  Ton  per  Square 
Inch,  in  parts  of 
the  Length. 

L  S  3  Lowmoor 

tons. 

13-5 
13-5 
13.0 

per  cent. 
.101 
.106 
.101 

length  =  i. 

L  S  s 

L  S  6 

Averages  

13-3 

.102,  or  i  in  977 

.000077,  or  1/12,987 

L  i  Lowmoor  

12.5 
10.5 
II.5 

.090 

L  2 

L  3 

Averages  

II.5 

.089,  or  i  in  1130 

.000077,  or  1/12,987 

K  C  I  Yorkshire. 

13.0 
13.0 

.100 

.103 

K  C  2 

K  C  ^ 

Averages 

I3.0 

.101,  or  i  in  987 

.000078,  or  Vi2,82i 

F  R  i,  usual  S.  C.  Crown 
F  R  2, 
F  R  3,         „            „ 

Averages 

II.5 
II.5 
12.0 

•093 
.095 
.103 

II.7 

.097,  or  i  in  1030 

.OOOOSO,  Or  T/i2,5oo 

Yorkshire  

Summary 

12.6 

11.7 

Averages. 

.097,  or  i  in  1030 
.097,  or  i  in  1030 

.000077,  or  1/12,987 
.000083,  or  1/12,048 

S   C  Crown 

Total  averages  

12.  1 

.097,  or  i  in  1030 

.000080,  or  Via.sco 

THE   STRENGTH   OF   MATERIALS. 


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STRENGTH  OF  WROUGHT  IRON. 


58l 


HAMMERED  IRON  BARS  (SWEDISH). 

Table  No.  190  contains  a  selection  of  results  of  trials  made  by  Mr. 
Kirkaldy  of  the  tensile  and  compressive  strength  of  hammered  bar-iron 
manufactured  by  Messrs.  Gammelbo  &  Co.,  Nericia,  Sweden. 

Table  No.  190. — STRENGTH  OF  SWEDISH  HAMMERED  IRON  BARS.     1866. 

TENSILE  STRENGTH. 


Size  of  Specimens  (2,  3, 
or  4  of  each  scantling). 

Length  for 
Elongation. 

Elastic  Strength  per 
Square  Inch. 

Absolute 
Strength 
per  Square 
Inch 

Elongation 
in  parts  of 
Length. 

Lowest. 

Highest. 

Average. 

tons. 

tons. 

tons. 

tons. 

per  cent. 

l^-inch         round,  ) 

turned  down  from  / 

15  inches 

10.75 

12.00 

11.05 

18.80 

24.6 

bars  2  inches     .       j 

I  -inch  square  

20.35 

22.9 

3-inch  round  

10 

18.85 

33-1 

2-inch  round  

10 

18.87 

32.5 

I  -inch  round 

IO 

18.92 

^-inch  round  

10 

23.90 

5-9 

Flat,  3  x  yz  inch  

15 

23.00 

12.  1 

Flat,  2  x  Yz  inch  

15 

20.62 

21.6 

Flat,  lYit-Yt  inch  ••• 

15 

20.55 

16.4 

I  -inch    square    iron  } 

converted  into  blis-  > 

12.77 

1.2 

tered  steel  ) 

2  -inch    round,    case  ) 
hardened  j 

22.50 

— 

I  -inch    round,    case  | 

hardened  } 

19-35 

^-inch  round,   case  ) 

hardened  \ 

23-2 

COMPRESSIVE  STRENGTH. 


Size  of  Specimens  (2,  3, 
or  4  of  each  scantling). 

Length  for 
Compression. 

Elastic  Strength  per 
Square  Inch. 

Absolute 
Strength 
per  Square 
Inch. 

Compression 
in  parts  of 
Length. 

Lowest. 

Highest. 

Average. 

i^-inch  round  
I  y^,  -inch  round  
i^-inch  round  
I  -inch  square    

1%  inches 
15 

3.     » 
i  inch 

tons. 
10.10 

8-94 
8.94 

tons. 
12.64 

10-75 
9.84 

tons. 
10.74 

9-45 
10.42 

tons. 
66.45 
12-53 
37-90 
82.20 

per  cent. 

45-4 
3-7 
33-i 
53-3 

I  -inch    square    con-") 
verted  into  blistered  > 
steel                 .     ...) 

i  inch 

83.40 

48-3 

I  ^-inch  turned  

SHEARING  S' 

II 

FRENGTH 

||    .S.» 

582 


THE   STRENGTH   OF   MATERIALS. 


For  transverse  strength,  four  2-inch  square  bars  were  tested,  on  a  span 
of  25  inches,  with  the  following  results:— 


Breadth. 

Depth. 

Elastic  Stress. 

Deflection. 

Ultimate  Stress. 

Deflection. 

inches. 

inches. 

pounds. 

inches. 

pounds. 

inches. 

I 

2.04    X     2.02 

7,5°° 

.089 

15,888 

5-18 

2 

2.  02     X    2.O4 

7,000 

.088 

14^75 

4.98 

3 

1.95     X    2.02 

6,000 

.072 

I35965 

5-85 

4 

2.00    X     2.00 

6,000 

.078 

I'3»338 

5.38 

Averages  . 

6,625 

.082 

^S16 

5-35 

or  2.96  tons 

or  6.48  tons 

The  bars  remained  uncracked  under  the  ultimate  stress. 

For  torsional  strength,  the  averages  for  four  bars  turned  to  i  ^  inches  in 
diameter,  with  a  length  of  7  diameters,  the  stress  being  applied  at  the  end 
of  a  i2-inch  lever,  were  as  follows: — 

Elastic  stress 1062  pounds,  or  .474  ton. 

Deflection  in  parts  of  a  revolution  =  i...  .on  turn. 

Ultimate  stress 2677  pounds,  or  1.195  tons- 

Ultimate  deflection 4.70  turns. 

One-inch  square  bars  of  the  same  manufacture  were  tested.  For  tensile 
strength,  they  broke  with  an  average  of  20.34  tons  per  square  inch.  Under 
bending  stress,  the  average  results  of  four  bars,  1.04  inches  wide  by  1.05 
inches  deep,  showed  that  they  bore  an  elastic  stress  of  1250  pounds,  with 
.216  inch  deflection.  The  ultimate  stress  was  1978  pounds,  with  6.60 
inches  of  deflection. 


MR.  J.  TANGYE'S  EXPERIMENTS  ON  THE  COMPRESSIVE  RESISTANCE 
OF  WROUGHT  IRON. 

A  i-inch  round  bar  of  soft  Lowmoor  iron,  8  or  9  inches  long,  was  planed 
on  two  opposite  sides  to  a  thickness  of  ^  inch,  and  was  subjected  to 
pressure  on  one  side  under  a  steel  die  ^  inch  square,  having  an  area  of 
Y^  square  inch. 

The  following  are  the  results  of  the  tests;  and  they  prove  clearly  that  a 
unit  of  iron  has  a  much  greater  power  of  resistance  when  it  forms  a  portion 
of  a  larger  mass,  than  when  it  is  isolated  in  the  manner  customary  in  making 
experiments  on  resistance  to  compression: — 

Load  per  square  inch. 

12  tons  no  impression. 

16 


20 
24 
28 
32 
36 
40 


slightest  indentation,  sensible  to  the  finger-nail, 
distinctly  visible,  edge  followed  by  finger-nail. 


indented  about 


inch. 


STRENGTH   OF   WROUGHT   IRON. 


583 


KRUPP  AND  YORKSHIRE  IRON  PLATES.     1875. 

Mr.  Kirkaldy  made  an  experimental  inquiry  into  the  relative  properties  of 
wrought-iron  plates  manufactured  by  Herr  Krupp,  Essen,  and  plates  manu- 
factured in  Yorkshire.  The  results  are  detailed  in  a  valuable  report  by 
Mr.  Kirkaldy,  from  which  the  following  particulars  have  been  extracted. 
Twenty-seven  plates  in  all,  not  less  than  4 
feet  by  3  feet,  of  three  thicknesses,  2/3  inch, 
^  inch,  and  ^  inch,  were  obtained  from 
Mr.  Krupp  and  from  six  Yorkshire  manufac- 
turers; from  which  the  specimens  were  cut 
out.  The  Yorkshire  brands  were: — Low- 
moor,  Bowling,  Farnley,  Taylor's,  Cooper  &  Co.,  and  Monkbridge. 

Tensile  Strength. — Table' No.  191  gives  condensed  results  of  the  tests  for 
tensile  strength.  Each  entry  for  Krupp  iron  is  an  average  for  nine  speci- 
mens; and  for  Yorkshire  iron,  for  eighteen  specimens.  The  results  for  the 
three  thicknesses  are  nearly  alike.  The  form  of  the  specimens  is  shown  in 
Fig.  202. 


Fig.  202.— Test  Specimens  for  Tensile 
Strength. 


Table  No.  191. — KRUPP  AND  YORKSHIRE  IRON  PLATES — TENSILE 
STRENGTH.     1875. 

Thicknesses  ^,  ^,  and  %  inch.     Breadth,  2  inches;  length  for  extension,  10  inches. 
(Reduced  from  Mr.  Kirkaldy's  Reports.) 


Description. 

Elastic 
Strength 
per  square 
inch. 

Ultimate 
Strength 
per  square 
inch. 

Ratio  of 
Elastic  to 
Ultimate 
Strength. 

Extension. 

Sectional 
Area  of 
Fracture. 

At 
30,000  Ibs. 
per  square 
inch. 

Ultimate. 

LENGTHWAY. 
Unannealed  — 
KruDD 

tons. 

1  1.6 
12.4 

tons. 

22.7 
21.3 

per  cent. 

51-3 
58.4 

52.5 
59.6 

per  cent. 

1.30 
.65 

per  cent. 

25.4 
16.7 

per  cent. 

60.4 
794 

Yorkshire  

Annealed  — 
Krupn  ... 

II.O 
12.0 

21.0 
20.1 

2.72 
1.42 

28.2 
18.4 

56.3 

77.8 

Yorkshire 

CROSSWAY. 
Unannealed  — 
Krupp        .    ... 

II.4 
12.4 

21.7 
20.3 

52.6 
61.4 

i-35 
•5i 

174 
II.  2 

75.2 

85.3 

Yorkshire  

Annealed  — 
Krupp 

10.8 

12.  1 

20.4 
I9.2 

52.7 
62.8 

2-37 
.81 

19.7 
12.8 

73-0 
83.1 

Yorkshire 

AVERAGES. 
Krupp  

Yorkshire 

II.  2 

12.2 

21.5 
20.2 

52.1 
60.4 

1.94 
.85 

22.6 
14.8 

66.2 
81.4 

THE   STRENGTH   OF   MATERIALS. 


Effect  of  Drilled  Holes  and  Punched  Holes  on  the  Tensile  Strength.— 
Specimens  8  inches  wide  were  prepared  according  to  Fig.  203,  with  two 
rows  of  rivet  holes,  .85  inch  in  diameter,  in  the  central  portion,  2^  inches 
apart,  and  the  holes  were  at  2  inches  pitch.  The  punched  holes  were 
conical,  as  usual.  The  reduction  of  width  of  solid  metal  by  the  holes 


Before  Bulging. 


After  Bulging. 


Fig.  203. — Specimen  Plate  to  test 
Effect  of  Drilling  and  Punching. 


Figs. 


05.  —  Test  Specimen  Plates  for  Bulging  Stress. 


amounted  to  (.85  x  4  = )  3.40  inches,  and  the  net  section  was  (8  —  3.4  =  )  4.6 
inches  wide,  or  57.5  per  cent,  of  the  total  width.  Four  Krupp  specimens 
and  nine  Yorkshire  specimens  of  varying  thickness,  were  tested  for  each 
result.  Table  No.  192  gives  some  deductions  from  the  elaborate  results 
reported  by  Mr.  Kirkaldy. 

Table  No.  192. — KRUPP  AND  YORKSHIRE  IRON  PLATES — TENSILE 
STRENGTH  OF  DRILLED  AND  PUNCHED  PLATES.     1875. 

Thickness  Y%,  ^,  and  ^  inch.     Holes,  .85  inch  in  diameter. 
(Deduced  from  Mr.  Kirkaldy's  Reports.) 


Description. 

Reduced 
Section,  in 
parts  of  Total 
Section. 

Reduced 
Strength,  in 
parts  of  Total 
Strength. 

Tensile 
Strength  per 
square  inch  of 
Net  Section,  in 
parts  of  that 
of  Entire 
Section. 

Total 
Elongation 
of  Holes. 

DRILLED  HOLES. 
Length  way  — 
KruoD 

per  cent. 

C7  C 

per  cent. 
6?  8 

per  cent. 

per  cent. 

Yorkshire  

5/O 

C7  C 

c6  o 

3°-7 
TQ  o 

->/•-> 

3D-y 

99-O 

Crossway  — 
Krupp... 

C7  c 

6r  T 

1  06  2 

22  8 

Yorkshire 

J/O 

C7  C 

3/o 

57.2 

L3-4 

PUNCHED  HOLES. 
Lengthway  — 
Krupp 

C7  r 

r  r    r 

806 

T  7  7 

Yorkshire 

.)/•;> 

C7  r 

51O 

CQ  O 

87  o 

1  >7 
8  i 

jt  O 

°*j 

Crossway  — 
Kruop 

C7  r 

CO  O 

87  o 

ill 

Yorkshire  

5/O 

C7  c 

AH  6 

82  8 

7  O 

AVERAGES. 
Krupp 

C7  c 

c6  ^ 

08  o 

19  6 

Yorkshire.  ...         ... 

C7.; 

C2  Q 

Q2  O 

117 

STRENGTH  OF  WROUGHT  IRON. 


585 


Resistance  to  Bulging  Stress. — Discs  12  inches  in  diameter,  cut  in  the 
lathe  out  of  plates,  were  pressed  into  an  aperture  10  inches  in  diameter,  by 
a  bulger-ram  about  5  inches  in  diameter,  of  which  the  end  was  turned  to  a 
radius  of  5  inches.  The  preparation  for  the  trial,  and  the  object  after 


Figs.  206,  207,  208. — Specimens  for  Resistance  to  Bending  Stress. 

having  been  bulged,  are  shown  in  Figs.  204  and  205.  A  selection  of  the 
results  is  given  in  table  No.  193.  Each  result  for  Krupp  iron  is  an  aver- 
age from  four  specimens;  and  for  Yorkshire  iron  an  average  from  six 
specimens.  The  stress  was  gradually  increased  until  the  specimen  was 
pushed  through  the  aperture,  or  until  the  specimen  gave  way  either  by 
cracking  or  bursting. 

Table  No.  193. — KRUPP  AND  YORKSHIRE  IRON  PLATES — RESISTANCE 
TO  BULGING  STRESS.     1875. 

Discs  12  inches  in  diameter,  pressed  into  lo-inch  apertures. 
(Selected  from  Mr.  Kirkaldy's  Table.) 


Stress  —  Bulging  in  Inches. 

Ultimate. 

J.  nicRncss 
of 
Plate. 

Lbs. 
25,000. 

Lbs., 

100,000. 

Lbs., 
200,000. 

Bulge. 

Stress. 

Effects. 

inches. 

inches. 

inches. 

inches. 

inches. 

tons. 

Unannealed. 

Krupp,  440 

.82 

2.14 

— 

3-27 

62.10 

uncracked 

»      533 

.64 

1.86 

— 

340 

73.20 

uncracked 

„      653 

.50 

1.59 

2.82 

3.36 

97.06 

i  burst 

Yorkshire,  .390 

•S3 

(  4  plates  ) 
t     2.50     J 

— 

2.65 

41.00 

3  burst,  i  cracked 

„          .510 

.61 

f  5  plates  ( 
I     1.85     ( 

— 

2.72 

6l.03 

5  burst 

.625 

•35 

)  4  plates) 
1     1-57     J 

J  3  P^tes  ) 
(     2.83     J 

2.52 

73.83 

burst 

Annealed. 

Krupp,  440 

•83 

2.25 

— 

3-27 

5540 

uncracked 

„      533 

.72 

1.96 

— 

3-39 

71.28 

uncracked 

„      653 

.56 

i-75 

{  i  plate  ) 

!    3-18    j 

345 

88.62 

i  burst 

Yorkshire,  .390 

•93 

2-73 

3-19 

47-35 

i  burst,  i  cracked 

.510 

.72 

(  4  plates  1 

(    2.05    5 

— 

2.88 

55-85 

2  burst,  i  cracked 

.625 

43 

1.65 

(  3  Plates  ) 
I     3-14     J 

2.82 

77.30 

3  burst,  i  cracked 

586 


THE   STRENGTH   OF   MATERIALS. 


Resistance  to  Bending  Stress. — Specimens,  2*1/2,  inches  wide,  of  plates  of 
the  three  thicknesses,  were  bent  double  both  hot  and  cold.  First,  by  bend- 
ing them  between  supports  10  inches  apart  to  a  right  angle,  as  in  Fig.  206; 
then  the  cold  specimens  were  doubled,  as  in  Fig.  207,  to  a  distance  apart 
of  four  times  the  thickness;  whilst  the  hot  specimens  were  doubled  flat,  as 
in  Fig.  208.  Of  the  specimens  of  Krupp  iron,  thirty-six  in  number,  all 
bore  the  test,  except  six  which  were  more  or  less  cracked.  Of  the  speci- 
mens of  Yorkshire  iron,  seventy- two  in  number,  twenty-five  only  passed  the 
tests  uncracked. 

PRUSSIAN  IRON  PLATES.     1874. 

Two  large  iron  plates,  .64  inch  thick,  manufactured  by  Mr.  Borsig,  of 
Berlin,  were  tested  for  tensile  strength  by  Mr.  Kirkaldy  in  1874.  The 
following  abstract  contains  the  averages  of  four  experiments  to  each 
result: — 


Unannealed. 

Annealed. 

With  Fibre. 

Across  Fibre. 

With  Fibre. 

Across  Fibre. 

Elastic  strength  

13.00  tons 

23.40    „ 

54-5  % 
23-8  % 

1  2.  60  tons 
22.70    „ 

55-5  % 
U.6  % 

12.73  tons 
22.53       „ 
56.5   % 
24-7   % 

12.04  tons 
21.70    „ 
55-4  % 
15-1  % 

Ultimate  strength  
Ratio 

Elongation 

The  greatest  deviation  from  the  average  elastic  tensile  strength  was  half  a 
ton  below  it. 

The  plates  were  tested  for  bulging  strength.  Discs,  12  inches  in 
diameter — two  annealed  and  two  unannealed — were  cut  out  of  the  plates 
and  pressed  through  an  aperture  10  inches  in  diameter,  the  same  as  shown 
in  Figs.  204  and  205,  page  584. 


Bulging. 


Pressure. 

22.32  tons 

Unannealed. 

;  90  inch   

44.64  „ 

66.96  „ 
89.28  „ 

(ultimate)  104.18    ., 
104.20    „ 

1.49   »    

i-95    »      
2.39    „      
2.69    „      

Annealed. 

.95  inch. 


2.07 
2.60 


burst, 
one  burst. 


TENSILE  STRENGTH  OF  IRON  WIRE. 

Mr.  Barlow  deduced  from  experiments  by  Mr.  Telford  on  the  strength 
of  iron  wire  from  T/IO  inch  to  I/20  inch  in  diameter,  that  the  ultimate  tensile 
strength  was  equivalent  to  36  tons  per  square  inch  of  section. 

The  tensile  strength  of  Warrington  iron  wire  is  given  at  page  247 : — 
Unannealed,  about  36  tons  per  square  inch;  annealed,  about  24  tons  per 
square  inch. 


STRENGTH   OF  WROUGHT  IRON. 


587 


American  Wire. — Mr.  Roebling  states  that  bar  iron  of  from  i  inch  to 
i  y%  inches  square,  fit  to  make  the  best  quality  of  wire,  should  have  a  tensile 
strength  of  60,000  Ibs.,  or  27  tons  per  square  inch.  The  same  iron,  reduced 
to  No.  9  wire,  bears  100,000  Ibs.,  or  44^  tons  per  square  inch;  and,  if 
drawn  to  No.  20  wire,  it  will  bear  from  20,000  Ibs.  to  30,000  Ibs.,  or  9  to 
13  tons,  more.  From  these  data  it  would  appear  that  wire  made  of  the  best 
qualities  of  iron  has  about  the  same  strength  as  some  qualities  of  steel  wire. 

The  tensile  strength  of  American  iron  wire,  together  with  that  of  wire 
from  the  cables  of  a  suspension  bridge  after  having  been  32  years  in  use, 
according  to  Professor  Thurston,  are  as  in  table  No.  194. 


Table  No.  194. — TENSILE  STRENGTH  OF  AMERICAN  IRON  WIRE.     1875. 


Breaking  Weight. 

. 

_                         J      A 

Actual. 

Per  Square  Inch. 

inch. 

pounds. 

tons. 

per  cent. 

.029 

75 

56.90 

IOO 

•0535 

238 

47.26 

98.9 

.071 

368 

40.35 

91.7 

.08 

474 

42.10 

98.7 

.1205 

963 

37-70 

96.7 

•134 

1310 

41.47 

98.5 

Wire    from    suspension  \ 

bridge,  .1236  diameter.  > 

1081 

40.17 

56 

Average  of  1  2  tests  1 

SHEARING  AND  PUNCHING  STRENGTH  OF  WROUGHT  IRON. 

Swedish  bar  iron  bore  an  average  shearing  stress  of  15.20  tons  per  square 
inch;  the  ultimate  tensile  strength  was  18.80  tons  (page  581). 

The  shearing  resistance  of  bars  3  inches  by  j£  inch  and  i  inch  thick, 
flatwise,  with  parallel  cutters,  and  to  punching  i-inch  and  2-inch  holes 
through  bars  %  inch,  i  inch,  and  i  ^  inches  thick  —  the  power  being  applied 
through  a  hydraulic  shearing  press,  —  was  found  by  Mr.  C.  Little1  to  be  :  — 


Per  square  inch 
of  area  cut. 


BARS. 


inch  thick,  shearing,  and  punching  i-inch  holes  .........  22.35 

i  „  „  „          i        „         .........  21.83 

^         „  punching  2-inch  holes  ...........................  19.00 

i  „  „         2         „         ...........................  19.90 

1/4  „  „  2  „  ...........................  19.50 


The  shearing  resistance  of  "  ordinary  round  bar  iron  of  commerce,"  by 
direct  pull,  was  ascertained  by  Chief  Engineer  W.  H.  Shock,  of  the  United 

1  Proceedings  of  the  Institution  of  Mechanical  Engineers,  1858;  page  73. 


588  THE   STRENGTH   OF   MATERIALS. 

States  Navy.  The  following  are  averages  of  the  results  from  12  specimens 
to  each  average.  The  diameters  were  exactly  measured;  the  attachments 
were  slightly  rounded  at  the  edges,  and  hardened : l — 

DIAMETER  OF  BOLTS.  Resistance  in  tons  per  square  inch  cut. 

inch.  single  shear.  double  shear. 

y2  19.68    18.32 

y% 17-41   17-23 

2<  17-61    17-76 

y%   18.50     16.88 

i       !7.9o     16.78 


Averages  .................     18.22  17.40 

Mean  of  the  averages  ................  17.81  tons. 

The  averaged  results  of  experiments  on  the  strength  of  rivetted  joints,2 
showed  that  whilst  the  plates  broke  with  a  load  of  19.44  tons  per  square 
inch,  the  rivets  were  sheared  by  a  stress  of  17.45  tons  per  square  inch  of 
section. 

The  shearing  strength  of  wrought  iron,  in  view  of  the  foregoing  data,  is 
taken  at  80  per  cent.,  or  four-fifths  of  the  ultimate  tensile  strength. 

TRANSVERSE  STRENGTH  OF  WROUGHT  IRON. 

Rectangular  Bars  of  Wrought  Iron.  —  Wrought-iron  bars  are  not  readily 
ruptured  by  transverse  stress.  Their  transverse  elastic  strength,  therefore, 
naturally  constitutes  the  chief  matter  of  investigation.  Actual  data  are 
extremely  scarce.  Mr.  Barlow  gives  the  approximate  elastic  tensile  and 
transverse  strengths  of  four  bars  of  iron;  of  which  the  elastic  tensile 
strength  of  the  first  bar  was  9.5  tons  per  square  inch;  and  of  the  others, 
10  tons  per  square  inch.  The  elastic  transverse  strengths  of  these  bars 
are  here  given,  as  approximately  observed,  and  as  calculated  from  the 
observed  tensile  strength  by  formula  (  i  ),  page  507. 

Elastic  Transverse  Strength. 
BARS-  Calculated.  Observed. 

2    inches  x  2  in.  deep.  33  in.  span.  2.66  tons.  2.50  +  10115. 

i-5     »  x  3        »  33       »  4-72     „  4-25+    „ 

i-5     »  x  3        »  33       »  4-72     „  4-25+    ,, 

!-5     »  x  2.5     „  33       „  3.28     „  3-°°+    » 

The  limit  of  elastic  strength  was  not  closely  ascertained,  but  it  was 
known  to  be  greater  than  the  observed  strengths  here  noted.  The  calcul- 
ated strengths  appear,  therefore,  to  be  substantially  correct.  The  following 
is  the  form  of  the  calculation,  as  exemplified  for  the  first  of  these  bars  :  — 


,. 
=  2.661  tons. 


33 

Mr.  Edwin  Clark  tested  three  bars  of  wrought-iron,  one  i  inch  square, 
and  two  i^  inches  square,  for  transverse  strength,  as  follows: — 

1  Journal  of  the  Franklin  Institute,  1874. 

2  Transactions  of  the  North  of  England  Mining  Institute. 


STRENGTH   OF   WROUGHT   IRON.  589 

Elastic  Transverse  Strength. 
BARS>  Calculated.  Actual. 

i    inch    x    i  in.  deep.         12  in.  span.         1.155  tons.         1.117  tons. 
i-5    »       x    i-5     »  36       »  I-299     »  1-275     » 

The  Swedish  bars,  noticed  at  page  581,  2  inches  square,  had  an  ultimate 
tensile  strength  of  18.8  tons  per  square  inch,  and  the  i-inch  square  bars, 
20.34  tons.  By  the  formula  (  i  ),  page  507, 

Ultimate  Transverse  Strength. 
SWEDISH  BARS.  Calculated.  Observed. 

2.04  in.  x  2. 02  in.  deep.     25  in.  span.     7.230  tons.     7.093  tons,  uncracked. 
2.02  „   x  2.04       „  25       „  7.302    „        6.646     „ 

1.95    „     X  2.02  „  25  „  6.9II       „  6.234       „ 

2.00    „     X  2.00          „  25  „  6.948      „  5.955        „ 


Average  for  2  -inch  bars,  .........   7.098    „        6.482     „  „ 

Average  for  i-inch  bars  — 

1.04  in.  x  1.05  in.  deep.     25  in.  span.     1.077  tons-     0.883  tons>  uncracked. 

The  calculated  strength  of  the  2-inch  bars  averaged  9.5  per  cent,  in  excess 
of  the  observed  strength;  and  that  of  the  i-inch  bars,  22  per  cent.  But 
the  bars  were  not  broken,  nor  even  cracked,  and  they  would  of  course  have 
borne  a  greater  load  before  breaking. 

There  is  a  regular  and  close  correspondence  between  the  calculated  and 
the  observed  transverse  strengths  of  wrought-iron  bars  above;  contrasting 
with  the  diversity  observed  with  cast-iron  bars.  The  regularity  results  from 
the  more  nearly  uniform  texture  and  strength  of  wrought  iron. 

Formula  (  i  ),  page  507,  in  its  general  form,  may  be  adapted  for  wrought 
iron  by  assuming  an  average  tensile  strength  of  22.5  tons  per  square  inch 
for  the  value  of  s.  Then  1.155  s=  1.155  x  22.5  —  26. 

Transverse  Strength  of  Rectangular  Bars  of  Wrought  Iron,  average  quality, 
Loaded  at  the  middle,...  W-^M2  ............................  (i) 

Loaded  at  one  end,  ......   W  =  ^l—   ...........................  (2) 

Round  Iron.  —  The  strength  of  round  wrought-iron  bars,  taking  the  same 
tensile  strength,  22.5  tons,  is  found  from  the  general  formula  (  15  ),  page 
510;  in  which  .7854x.y  =  .7854x  22.5  -  17.7. 


Transverse  Strength  of  Round  Wrought-iron  Bars,  average  quality. 
Loaded  at  the  middle,...  W=17'7^  ...........................  (3) 

Loaded  at  one  end,  ......  W  =  4'4  d*.   ...........................  (4) 

W  =  the  breaking  weight  in  tons;  b  the  breadth,  d  the  depth,  and  /  the  span, 
all  in  inches. 


590 


THE   STRENGTH   OF   MATERIALS. 


TRANSVERSE  DEFLECTION  AND   ELASTIC  STRENGTH  OF  WROUGHT  IRON. 

Wrought-Iron  Rectangular  Bars. — The  deflections  of  a  few  bars  under 
given  weights,  applied  at  the  middle,  were  observed  by  Mr.  Barlow  and  Mr. 
Edwin  Clark.  To  collate  with  these  the  observations  of  Mr.  Kirkaldy  on 
bars  of  Swedish  iron,  page  582,  the  results  are  here  grouped  together, 
and  the  value  of  E,  the  coefficient  of  elasticity,  calculated  by  formula  (  6  ), 
page  529,  are  added: — 


BARS. 

Span. 

Load. 

Deflection. 

1 

E. 

Barlow,     2  inches  square,  

inches. 

33 
33 
33 
33 
33 

12 

% 

3^ 
36 

36 
25 
25 
25 
25 

•  the  fir 

tons. 
2.50 
2.00 
4.00 
4.00 
4.OO 
.711 
1.275 

1.  1  45 
1.695 
1.695 
3.348 

3-125 
2.679 
2.679 

st  10  bars  (Ic 

inches. 
.100 
.077 
.088 
.102 
.104 
.026 
.320 
.290 
.400 

•350 
.089 
.088 
.072 
.072 

mg  span), 

I2,T54 

16,616 

3,730 
7,532 
12,764 
10,228 
7,948 

8,220 

8,454 
9,638 
7,566 
7,004 
7,830 
7,260 

10,228 

2 

»              *         »    .          "      ,    

5x3  inches  deep  

C  X  2  1J 

E.  Clark,     inch  square  (3  bars),.... 
c 

c 

Swedish    2.04  x  2.02  deep,  

2.O2  X  2.  02 

I   QC   X  2  O^ 

„              2.00X2.00      „      

Average  coefficient  of  elasticity,  E,  foi 

Adapting  the  general  formula  ( 8  ),  page  530,  the  numerical  constant 
becomes  4.62  x  10,228  =  47,253,  say  47,000. 

Elastic  Deflection  of  Uniform  Rectangular  Bars  of  Wrotight  Iron,  loaded 

at  the  middle. 


W/3 


(5) 


47,  ooo  bd* 

D  =  the  deflection,  b  the  breadth,  d  the  depth,  /  the  span,  all  in  inches; 
W  the  weight  in  tons. 

Round  Iron. — For  round  bars,  substitute  the  above-found  value  of  E,  in 
the  general  formula  (  24),  page  533.  Then,  3.1416  x  £  =  3.1416  x  10,228 
=  32,116,  say  32,000. 

Elastic  Deflection  of  Round  Wrought-Iron  Bars,  loaded  at  the  middle. 

W/3 


D=  

32,000  a4 

TORSIONAL  STRENGTH  OF  WROUGHT  IRON. 


(6) 


Data  are  extremely  scarce.     The  Swedish  iron,  page  581,  gave  a  shear- 
ing  resistance   of   15.2   tons   per   square  inch.     By  the  general  formula 


STRENGTH   OF   WROUGHT   IRON.  5QI 

(  i  ),  page  534,  the  breaking  force  at  the  end  of  a  1  2-inch  lever,  applied  to 
a  i  i/o,  -inch  round  bar,  is 


12 

The  actual  force  was  found  to  be  (page  582)  =  1.195  tons- 

Taking  the  shearing  strength  of  wrought  iron  at  80  per  cent,  of  the 
tensile  strength,  as  decided  at  page  588,  put  s  =  the  ultimate  tensile  strength, 
then  h  =  .So  s,  and,  by  substitution  in  the  general  formulas  for  torsional 
strength  (i),  page  534,  and  (6),  page  536, 

For  wrought  iron  round  shafts,...  W  =  - 


R  4.5  R 


For  wrought  iron  square  shafts,...  W  =  l  =  --  ...........   (8) 

W  =  the  force  in  tons. 
R  =  the  radius  of  the  force  in  inches. 
W  R  =  the  moment  of  the  force  in  statical  inch-tons. 
tf^the  diameter  of  the  round  shaft  in  inches. 
b  =  the  side  of  the  square  shaft  in  inches. 
s  =  the  ultimate  tensile  strength  in  tons  per  square  inch. 

Take  the  tensile  strength  s  equal  to  22.5  tons  per  square  inch  as  an  average 
value  j  then,  by  substitution  and  reduction:  — 

Torsional  Strength  and  Sizes  of  Wroiight-Iron  Shafts  of  average  quality. 

Round  shafts  :—  W  =  l^  ..........................................  (  9  ) 

R 


Square  shafts  :—W=7i?A3 (  n  ) 

R 


ELASTIC  TORSIONAL  STRENGTH  AND  DEFLECTION  OF  WROUGHT-!RON  BARS. 

The  results  of  experiments  with  Swedish  bars,  page  582,  show  that  the 
elastic  torsional  strength  was  40  per  cent,  of  the  ultimate  torsional  strength. 
The  elastic  shearing  stress  is  found  by  formula  (  3  ),  page  535, 

,       WR 


For  Swedish  hammered   bars,  page  582,  WR  =  .474  ton  x  12  inches  - 

/TOO 

—  ^—   '• 

«  2  7  o  X   I  • 


5.688,  the  moment  of  the  force;  and  h-  —  —  '•  —  =  6.06  tons  per  square 


inch,  the  elastic  limit  of  shearing  stress. 


592  THE   STRENGTH   OF   MATERIALS. 

The  value  of  E',  the  coefficient  of  torsional  elasticity,  as  defined  at  page 
536,  is  found  for  the  Swedish  bar,  by  the  general  formula  (12),  page  537  :  — 

TJ,,  WR/  .474  X  T2  X  10.5 

JH,   —  _  =  __LLZ  _  —  -=I22Q 

.873  x  i.5*x.oii 


By  inversion  and  reduction,   the  equation  for  torsional  deflection  is  ob- 
tained :  — 

Elastic  Torsional  Deflection  of  Round  Wrought-Iron  Bars. 

n      WR/ 
=  7^-^  ...................................  <I4> 

D  =  the  total  angular  deflection  in  parts  of  a  revolution. 
W  —  the  twisting  force  in  tons. 
R  =  the  radius  of  the  force  in  inches. 
W  R  =  the  moment  of  the  force. 

/  =  the  length  of  the  shaft  in  inches. 
d  -  the  diameter  of  the  shaft  in  inches. 


STRENGTH   OF  STEEL. 


593 


STRENGTH   OF  STEEL. 
MR.  KIRKALDY'S  EARLY  EXPERIMENTS. 

In  the  course  of  the  experiments  already  noticed,1  page  571,  Mr.  Kirk-. 
aldy  tested  a  great  number  of  bars  and  plates  of  steel,  the  general  results 
of  which  are  given  in  a  condensed  form  in  tables  Nos.  195  and  196.  The 
bars  were  from  ^  inch  to  i  inch  in  diameter,  and  possessed  an  average 
tensile  strength  of  from  60  tons  per  square  inch  for  tool-steel,  to  28  tons  per 
square  inch  for  puddled  steel.  The  greatest  observed  strength  was  66.2 
tons. 

The  steel  plates  were  from  s/l6  to  s/l6  inch  thick.  Their  tensile  strength 
ranged  from  45^  to  32  tons  per  square  inch,  with  the  fibre.  The  average 
tensile  strength  was  40  tons  with  the  fibre;  and  across  the  fibre  the  tensile 
strength  was  36^  tons,  or  91  per  cent,  of  the  tensile  strength  in  the  direc- 
tion of  the  fibre. 

Table  No.  195. — TENSILE  STRENGTH  OF  ROUND  STEEL  BARS.     1861. 

(Mr.  David  Kirkaldy.) 


NAME. 

Condition. 

Size. 

Breaking  Weight 
per  square  inch. 

Elonga- 
tion in 
parts  of 
length. 

Lowest. 

Highest. 

Average. 

Turton's  cast  steel,  for  tools  \ 
Jowitt's        do.            do.       I 
Do.            do.          chisels  \ 
Do.    double  shear  steel...  / 
Do.    cast  steel  for  drifts...  1 
Bessemer  tool  steel.  .           / 

Forged, 
reheated, 
and 
cooled 
gradu- 
ally. 

Rolled. 

J5 

Forged. 
Rolled. 

inch. 

•53to.59 
.5610.58 
.56  to.  60 
.56&.57 

.6510.75 
•75 

•75 

.5  7  to  .60 

•57&-S9 
.91  to  .93 
.56 

•75 
•55to.57 
•75 
75  to.i.o 

•75 
•77 

tons. 
50.10 

52.55 
50.15 

47.65 

43-15 
46.10 

45.27 
45.27 

39-97 
37.51 
38.42 
36.70 
38.30 
29.07 
29.94 
24.55 
19.00 
20.50 

tons. 
64.90 

66.20 
61.60 

55-95 
58.25 
54-97 
52.12 

50.12 

51.87 
49.42 
42.92 

44-45 
42.30 
36.92 
33-62 
33.54 
31-93 
31.40 

tons. 

59.32 
59.10 

5575 
52.87 
51.76 

49-75 
47.90 

47-60 

46.56 

45-15 
41.08 
40.47 
40.05 
32.37 
3i.9i 
31.32 
28.24 
28.05 

per  cent. 

54 
5.2 

7-i 
13-5 
13-3 

5-5    ' 
12.4 

8.7 

9-7 
10.8 

15-3 
137 
11.9 
1  8.0 
'19.1 
ii-3 

12.0 
9.1 

Moss  &  Gamble,  cast  steel 
for  rivets 

Naylor,  Vickers,  &  Co.,  cast 
steel  for  rivets 

Wilkinson  blister  steel  . 

Towitt's  cast  steel,  taps  

Krupp's  cast  steel,  bolts  
Homogeneous  metal  

Do.             do  

Forged. 

?> 

Rolled. 
Forged. 

?j 

Towitt's  spring  steel  

Mersey  Co.'s  puddled  steel... 
Blochairn              do. 
Do.                    do. 
Do.                    do. 

Averages 

38.50 

46.80 

42.66 

II.  2 

1  Experiments  on  Wrought  Iron  and  Steel. 


38 


594 


THE   STRENGTH   OF  MATERIALS. 


Table  No.  196. — TENSILE  STRENGTH  OF  STEEL  PLATES.     1861. 

(Mr.  David  Kirkaldy.) 


DESCRIPTION  OF  STEEL. 

Thickness 
of 
Plate. 

Breaking  Weight 
per  square  inch. 

Elongation  in 
parts  of  length. 

With 
Fibre. 

Across 
Fibre. 

With 
Fibre. 

Across 
Fibre. 

Turton  &  Sons,  cast  steel  
Shortridge  &  Co      do           

inch. 

3/i6 

!    3/l6  &  i/4 
3/8 
'/8  &  3/l6 

3/i6 
5/i6 

I 

tons. 
42.10 
42.97 
36.48 

33-75 

45-28 
45-80 
45.64 
43.00 
32-32 
3440 
31-93 

tons. 

43-00 
43-37 
38-90 
30.84 
43-30 

37-93 
38.11 

37.67 
32.90 
32.85 
30.22 

per  cent. 

5-7 
8.6 

17-5 
19.8 

2.8 

4i 
3-6 

8.2 

5-9 

6.2 

3.6 

per  cent. 

9.6 
8.9 

17-3 
19.6 
14.4 
i-3 
3-3 
2.7 
4.1 
3-2 
5-7 

Naylor,  Vickers,  &  Co.,  cast  steel... 
Moss  &  Gamble,                   do. 
Shortridge  &  Co.,                  do.      ... 
Mersey  Co    puddled  steel 

Mersey  Co.,  "  hard  "  puddled  steel.. 
Blochairn  puddled  steel  

Blochairn,          do.          

Shortridge  &  Co    do 

Mersey  Co.,  "mild"  puddled  steel- 
Mersey  Co.,        do.            do. 

Total  averages 

39-42 

37-17 

7.8 

8.2 

Averages  for  comparison  of  strengths  ) 
lengthwise  and  across     I 

40.17 

36.56 

8.2 

7.6 

Mr.  Kirkaldy  discovered  that  the  strength  of  steel  was  materially  in- 
creased by  hardening  the  metal  in  oil;  and  that  it  was  materially  reduced 
by  hardening  in  water.  Three  pieces  from  a  bar  of  chisel-steel  were  so 
treated,  with  the  following  results  :  — 

Tensile  Strength. 

Soft  steel  ............................................  541^  tons. 

*„        cooled  in  water 
cooled  in  oil 


Coal-tar  and  tallow  were  used  for  cooling  steel,  and  with  good  effect; 
but  they  were  not  so  efficacious  as  oil. 

Steel  plates,  similarly  treated  in  oil,  acquired  a  gain  of  strength  varying 
from  56.4  per  cent,  for  the  highest  temperature  at  which  they  were  cooled 
to  12.8  per  cent,  for  the  lowest. 

The  shearing  strength  of  steel  rivet-iron  was  found,  from  seventeen 
tests,  to  average  74  per  cent,  of  the  ultimate  tensile  strength  of  the  same 
bar. 

STRENGTH  OF  HEMATITE  STEEL.     1866. 

Mr.  Kirkaldy  tested  the  strength  of  bar-steel  manufactured  by  the 
Barrow  Hematite  Steel  Company.  Four  samples  were  tested  for  each 
kind  of  stress:  —  For  tensile  stress,  cast  steel,  forged  and  turned  to  i^ 
inches  diameter;  length,  14  inches.  For  compressive  stress,  hammered  cast 
steel,  forged  and  turned  to  i^  inches  diameter;  length,  14  inches.  For 
shearing  stress,  hammered  cast  steel,  forged  and  turned  to  ij^  inches 


STRENGTH   OF   STEEL. 


595 


diameter.  For  transverse  stress,  hammered  cast  steel,  i^  inches  square; 
span,  25  inches.  For  torsional  stress,  Bessemer  cast  steel,  forged  and 
turned  to  i^  inches  diameter;  length,  8  diameters. 


ELASTIC, 
per  square  inch. 


Tensile  strength 1 8. 63  tons. 

Compressive  strength 23.21     „ 

Shearing  strength — 


ULTIMATE, 
per  square  inch. 

32.27  tons. 

71.24   „ 
25.21    „ 


Elongation. 

19.2  per  cent. 


{Elastic  load 3.80  tons. 
Ultimate  load  ) 
(uncracked)    j     7'35     " 
Torsional    strength  )  F1ocfiV  ^ ^ 

•   i-°3     » 


Deflection  .122  inch. 
„          6.64  ins. 

„         .008  turn. 
1-54    „ 


STRENGTH  OF  KRUPP  STEEL.     1867768. 

Blocks  of  Krupp's  cast  steel  from  the  heads  of  broken  crank-shafts  of 
the  "  Jeddo  "  and  the  "  Sultan,"  were  cut  up  into  numerous  specimens  by 


Fig.  209.—  Krupp  Steel  Crank-shaft,  "Sultan." 

Mr.  Kirkaldy,  and  tested  for  strength.     The  annexed  Fig.  209  shows  how 
the  broken  crank  of  the  "  Sultan  "  was  divided  and  cut  up.1 


Specimens. 


For  tensile  strength,  ....... 

For  compressive  strength, 


For  transverse  strength,...  <  Depth  .... 
(  Span 

For  torsional  strength,  .....  <  T  1 


JEDDO. 

SULTAN. 

leter  ...   1.25  inches. 

1.128  inches. 

th  8.5 
icter  ...    — 

10.0 

1.128 

th  — 

1.128 

1th  1.37       „ 
h  1.76       „ 
...  .           10 

1.50 
1.91 

10 

1.128 
2  diamete 

rs. 

leter...   1.25       „ 
th  2  diameters. 

JThe  author  is  indebted  to  Mr.  Longsdon  for  copies  of  the  "  Results  of  Experiments," 
from  which  the  above  particulars  have  been  reduced. 


596 


THE   STRENGTH   OF   MATERIALS. 


Average  Results. 

For  tensile  strength: —  JEDDO.  SULTAN. 

Elastic  strength 18.53  tons.  19.10  tons. 

Do.     extension 541  per  cent  .586  per  cent. 

Do.         do i  in  185  i  in  171. 

Elastic  extension  per  ton  per  )       ,  ,  , 

i     i       ,1  f     7^28?  or  .000202.      /o266»  or  .000300. 

square  inch ;  length  =  !..../     ' 34 

Breaking  weight 41.18  tons.  42.07  tons. 

^weight  .elaStiC.t0.bre.aking  }    «  Per  cent.  45-4  per  cent. 

Permanent  extension 12.6      „  7.9          ,, 

Sectional  area  of  fracture 77.4     „  76.9        „ 

For  compressive  strength: — 

Elastic  strength —  21.13  tons. 

Do.     compression .798  per  cent. 

Do.  do i  in  125. 

Elastic  strength,  per  ton  per  )  , 

square  inch;  length  =  i  ...  }  '/**•  or  •°°°377- 

Breaking  weight —  89.30  tons. 

For  transverse  strength : — 

Elastic  stress 7.94  tons.  10.74  tons. 

Ultimate  stress 21.31    „  27.14     „ 

Ratio 37.2  per  cent.  39.6  per  cent. 

Elastic  deflection 055  inch.  .082  inch. 

Ultimate  deflection 1.49     „  1.19     „ 

For  torsional  strength : — 

>    .491  ton.  .497  ton. 

Ultimate  stress 1.280  „  1.068  „ 

Ratio 38.4  per  cent.  47.3  per  cent. 

Elastic  torsion 005  turn.  .on  turn. 

Ultimate  torsion 441     „  .339     „ 

The  lowest  ultimate  tensile  strength  of  the  steel  of  the  "  Jeddo  "  was 
nearly  10  tons  per  square  inch  below  the  average;  of  that  of  the  "Sultan" 
it  was  2^  tons  below  the  average.  The  strength  of  the  specimens  cut 
from  the  interior  of  the  blocks  averaged  very  little  less  than  that  of  those 
from  the  exterior. 

The  crank-shaft  of  the  "  Jeddo  "  was  supplied  to  replace  a  broken  shaft 
of  wrought  iron,  noticed  at  page  576,  of  which  the  tensile  strength  averaged 
about  20  tons,  as  against  41.2  tons  for  the  steel  shaft. 

EXPERIMENTS  OF  THE  STEEL  COMMITTEE. 

A  Committee  of  Civil  Engineers1  instituted  and  completed  a  series  of 
experiments  on  the  strength  of  steel  bars,  in  1868-70.  They  were  con- 

1  The  Committee  consisted  of  Messrs.  W.  H.  Barlow,  George  Berkley,  John  Fowler, 
Douglas  Galton,  C.B.,  and  J.  Scott  Russel.  Mr.  Berkley,  Secretary;  Mr.  W.  Parsey, 
Assistant-  Secretary. 


STRENGTH   OF   STEEL.  597 

ducted  with  every  provision  for  insuring  accuracy;  and  the  results  were 
printed  in  two  reports,  from  which  the  following  particulars  are  derived. 

First  Series  of  Experiments. 

The  first  series  of  experiments,  203  in  number,  were  conducted  by  Mr. 
Kirkaldy,  under  the  instructions  of  the  Committee,  with  his  testing  machine, 
in  which  the  amounts  of  extension,  compression,  and  deflection  were  read 
off  a  dial.  The  experiments  were  directed  to  test  the  resistance  of  steel 
bars  to  tension,  compression,  transverse  strain,  and  torsion.  Twenty-nine 
samples  of  steel  bars,  2  inches  square  and  15  feet  long,  of  the  best  marketable 
quality  ordinarily  made,  were  obtained  from  ten  manufacturers;  of  these, 
1 8  were  of  Bessemer  steel,  and  1 1  of  crucible  steel.  Each  bar  was  parted 
into  lengths  by  a  shaping  machine,  for  bending,  twisting,  pulling,  and  thrusting, 
as  shown  by  Fig.  210.  For  pulling,  the  specimen  was  prepared  as  in  Fig.  211, 
and  divided  into  inches  of  length;  for  bending,  as  in  Fig.  212 ;  for  twisting,  as 
in  Fig.  213;  and  for  thrusting,  or  compression,  as  in  Fig.  214.  For  tensile, 

Bending.       Twisting.  Pulling.  Thrusting.  Do. 

Spare    i>  i  i  >  j        11  Spare. 

Fig.  210. — Specimen  Bars — how  divided. 


SO  INCHES- 


Fig.  2ii.—  Graduation  of  Bars  for  Pulling  Stress. 

~     =l         3=        =3B  ffl 


Fig.  212.  —  For  Bending  Stress.  Fig.  213.  —  For  Twisting  Stress. 


Fig.  214.  —  For  Thrusting  or  Compressive  Stress  —  how  divided. 

SPECIMEN  BARS  OF  THE  STEEL  COMMITTEE,     ist  Series. 

compressive,  and  torsional  tests,  the  bars  were  turned  down  to  1.382  inches  in 
diameter,  having  a  sectional  area  of  1.5  square  inches,  and  highly  polished. 
For  bending,  or  transverse  tests,  they  were  planed  to  a  section  of  1.9  inches 
square.  The  final  results  have  been  condensed  from  the  report,  and  are 
worked  out  in  the  following  tables  :  — 

Table  No.  197  shows  the  tensile  strength  of  the  steel  bars. 

Table  No.  198  shows  the  elastic  compressive  strength  of  the  steel  bars. 
The  ultimate  compressive  strength  of  short  specimens,  which  is  always 
an  indefinite  quantity,  is  not  given  here;  but  it  may  be  stated  that  the 
short  specimens  required  a  great  deal  to  crush  them;  and  that  the  long 
specimens,  36  diameters  in  length,  failed  by  buckling,  when  the  elastic 
limit  of  stress  was  arrived  at. 

Tables  Nos.  199  and  200  show  the  transverse  strength  and  the  torsional 
strength  of  steel  bars,  —  distinguishing  the  elastic  from  the  ultimate  stress. 


598 


THE   STRENGTH   OF   MATERIALS. 


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STRENGTH   OF   STEEL. 


599 


Table  No.  198. — COMPRESSIVE  STRENGTH  OF  STEEL  BARS.     1868. 

Two-inch  square  bars  turned  to  1.382  inches  in  diameter  (1.5  square  inches  of  area). 

Lengths,  various. 
(Reduced  from  the  Experiments  of  the  Steel  Committee,  ist  Series.) 

BESSEMER  STEEL. 


DESCRIPTION, 
with  respective  number  of 
samples. 

Elastic  Strength  (compressive)  in  tons  per 
square  inch. 

Elastic  Compression  per 
ton  per  square  inch,  in 
parts  of  the  length  of 
36  diameters. 

Length, 
i  diam.  ; 
1.38  in. 

Length, 
2  diams.  ; 
2.76  in. 

Length, 
4  diams.  ; 
5-53  in. 

Length, 
36  diams.  ; 
50  in. 

5,  Hammered,  tyres,. 
5,  Hammered,  axles,. 
4,  Hammered,  rails,.. 
4,  Rolled,  

tons. 
23.03 
23-84 
23.88 
18.98 

tons. 
22.32 
22.76 
23.32 
18.30 

tons. 
22.23 
21.34 

21.77 
18.07 

tons. 
19.15 
18.51 
18.95 
17.20 

Length=i. 

.000065,  or  i/15>38S 
.000062,  or  i/x6>fa 
.000065,  or  i/IS,38S 
.000065,  or  1/15,385 

5,  Hammered,  tyres,. 
4,  Hammered,  axles,. 
I,  Hammered,  rails,.. 
i,  Rolled  axles 

CR 

24.02 
26.90 
26.78 
21.87 

UCIBLE 

23.93 
26.11 
26.34 
19.64 

STEEL. 
21.74 

23.99 
20.55 

18.75 

21.  II 

22.30 
19.54 
18.77 

.000065,  or  1/15,385 
.000065,  or  1/15,385 
.000065,  or  1/15,385 
.000069,  or  1/14,493 

1  8,  Bessemer  steels,...! 
n,  Crucible  steels,  

29,  Steels,  

SUMI 
22.43 
24.89 

dARY    A\ 
21.67 
24.01 

fERAGES. 
20.85 
22.26 

18.45 
20.43 

.000064,  or  1/15,625 

.000066,  Or  x/i5,i52 

23.66 

22.84 

21.55 

19.44 

.000065,  or  Vi5,385 

Table  No.  199. — TRANSVERSE  STRENGTH  OF  STEEL  BARS.     1868. 

Two-inch  square  bars  planed  to  1.9  inches  square.     Distance  of  supports,  20  inches. 

BESSEMER  STEEL. 


DESCRIPTION, 
with  respective  numbers  of  samples. 

Elastic 
Stress. 

Ultimate 
Stress. 

Ratio  of 
Elastic  to 
Ultimate 
Stress. 

Ultimate 
Deflection. 

REMARKS. 

AVERAGES. 
5,  Hammered   tyres, 

tons. 

t? 

7-99 
6.61 

tons. 

I3-I7 
13-20 
12.85 

n-75 

per  cent. 

57-3 
61.5 
6l.2 
56.3 

inches. 
3-82 
4.08 

3-94 
4-03 

Bent  to 
6  inches; 
uncracked. 

5,  Hammered,  axles,  

4    Hammered    rails       . 

4,  Rolled  ;  tyres,  axles,  rails,  .  .  . 

CRUCIBLI 
5,  Hammered,  tyres,  II     8.36 
4,  Hammered,  axles,  8.38 
lf  Hammered,  rails,         ...       1     i  XT 

:  STEEL 
14.65 

17.91 
12.06 

57-4 

53-9 
43-6 

53-8 

3-32 
3-35 
3-65 
3-84 

\     In  most 
f    cases  bent 
f  to  6  inches; 
)  uncracked. 

I,  Rolled,  axles,  

Su 

1  8    Bessemer  steels 

MMARY  j 

IVERAGI 
12.74 
15.04 

s. 

59-5 
52.2 

3-97 
3-54 

1  1,  Crucible  steels, 

29,  Steels,    

7-74 

13.89 

55-7  ' 

3-76 

6oo 


THE   STRENGTH   OF    MATERIALS. 


Table  No.  200. — TORSIONAL  STRENGTH  OF  STEEL  BARS.     1868. 

Two-inch  square  bars,  turned  to  1.382  inches  in  diameter  (1.5  square  inch  of  section). 

Length  for  torsion,  8  diameters  =  1 1  inches. 

(Reduced  from  the  Experiments  of  the  Steel  Committee,  1st  Series.) 
BESSEMER  STEEL. 


DESCRIPTION, 
with  respective  number 
of  samples. 

Elastic 
Stress  at 
the  end 
of  a 
i2-inch 
lever. 

Elastic 
Torsion. 

Ultimate 
Stress  at 
the  end 
of  a 
i2-inch 
lever. 

Ratio  of 
Elastic  to 
Breaking 

Stress. 

Ultimate  Torsion, 
*  uncracked,  for  3.75  turns. 

Least. 

Greatest. 

Average. 

AVERAGES. 

tons. 

i  turn  =  i. 

tons. 

per  cent. 

i  turn=i. 

i  turn=i. 

i  turn=i. 

5,  Hammered,  tyres, 

.701 

.014 

i-54 

45-4 

I.SO 

2-73 

2.21 

5,  Hammered,  axles, 

.667 

.Oil 

1.47 

44.9 

2-33 

3-75* 

3-07 

4,  Hammered,  rails, 

.688 

.012 

1-45 

46.8 

2.10 

3-75* 

2-73 

4.  Rolled;    tyres,   ) 
axles,  rails,...  \ 

.569 

.008 

1.44 

39.5 

2.6l 

3.75* 

3-" 

CRUCIBLE  STEEL. 

5,  Hammered,  tyres, 

•736 

.014 

1.59 

46.6  II    1.77 

3-39 

2.33 

4,  Hammered,  axles, 

•731 

.013 

1.69 

43-4          1-07 

2.32 

1.79 

i,  Hammered,  rails, 

.714 

.016 

1.81 

40.0           .86 

.86 

.86 

i,  Rolled,  rails  

•554 

.012 

i-34 

42.7         1-73 

2.14 

1.94 

SUMMARY  AVERAGES. 

1  8,  Bessemer  steels,  .  .  . 

.656 

.Oil 

1.47 

44.6 

— 

— 

2.78 

1  1  ,  Crucible  steels,  .... 

.684 

.014 

1.61 

42.5 

— 

— 

i-73 

29,  Steels,  

.670 

.013 

1-54 

43-6 

— 

— 

2.26 

The  lowest  tensile  and  compressive  elastic  strengths,  ranged,  for  each 
group  in  the  first  series,  about  5  tons  per  square  inch  below  the  averages 
given  in  the  tables; — say  20  per  cent,  below  the  averages.  The  same 
proportionate  range  is  found  in  the  elastic  resistances  to  torsional  and  trans- 
verse stress. 

Second  Series  of  Experiments  (made  at  Woolwich  Dockyard]. 

The  object  of  the  second  series  of  experiments  by  the  Steel  Committee, 
was  to  make  experiments  on  the  tension  and  compression  of  long  steel 
and  iron  bars,  measuring  the  changes  of  length  directly  from  the  bars.  For 
this  purpose,  91  bars  of  steel  and  iron,  each  14  feet  long  and  i%  inches  in 
diameter,  were  obtained,  consisting  of  33  bars  of  Crucible  steel,  34  bars  of 
Bessemer  steel,  1 2  bars  of  Lowmoor  iron,  6  bars  of  best  Yorkshire  iron,  and 
6  bars  of  usual  S.  C.  Crown,  or  Staffordshire  iron. 

The  extensions  were  measured  on  10  feet  length  of  each  bar;  and  for 
compressive  tests,  the  bars  were  cut  to  a  length  of  1 2  feet,  and  the  measure- 
ments made  on  a  length  of  10  feet. 

The  bars  were  tested  in  their  natural  skins.  Before  they  were  tested, 
they  were  thoroughly  examined  and  straightened,  and  the  diameters  checked 
by  means  of  vernier  callipers,  capable  of  showing  a  variation  of  a  icooth 
part  of  an  inch.  The  results  for  iron  bars  have  been  given  at  page  579. 

Table  No.  201  gives  the  condensed  results  of  the  experiments  for  the 
tensile  strength  of  steel  bars,  and  table  N<o.  202  gives  the  same  for  their 
compressive  strength.  The  bars  are  here  distinguished  by  letters. 


STRENGTH   OF   STEEL. 


601 


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Crucible  steels 
Bessemer  steel 

1 

^3 

602 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  202. — COMPRESSIVE  STRENGTH  OF  STEEL  BARS.     1870. 

\l/2  inches  in  diameter  ;  lo-feet  lengths. 

(Reduced  from  the  Experiments  of  the  Steel  Committee,  2d  Series. ) 
CRUCIBLE  STEEL. 


Description,  and  Reference  Letter. 

Elastic 
Strength 
(compressive) 
in  tons  per 
square  inch. 

Elastic  Compression. 

Elastic  Compression  per 
ton  per  square  inch,  in 
parts  of  the  length. 

a   Chisel;  3  samples  

tons. 
26.33 
26.2 
25.5 
26.0 

18.0 
16.2 
24.0 
19.5 

27.0 
24.0 

per  cent. 

.198,  or  i  in  506 
.202,  or  I  in  496 
.186,  or     in  537 
.198,  or     in  506 
.137,  or     in  731 
.126,  or     in  791 
.180,  or     in  555 
.138,  or     in  722 
.204,  or     in  490 
.185,  or     in  541 

length=i. 

.000075,  or  1/13,333 
.000077,  or  Vxa^ty 
.000073,  or  '/i3,699 
.000076,  or  '/xs.xsl 
.000076,  or  Vis.iss 
.000080,  or  '/".soo 
•000075,  or  */I3,333 
.000071,  or  Vi^oSs 

.000075,  Or  '/X3.333 
.000077,  Or  Viz.gSj 

b   Tyre;  3  samples  
c    2  samples  

d  Rods;  2  samples  

e 

f  Gun-barrels;  3  samples 
f  Hammered  ;  2  samples 
2  samples  

/    Rods  

j   Rolled 

k   Fagotted,  hammered  ) 
and  rolled  ;  3  samp.  ) 
1    3  samples  

BESSEMEI 
1  8.0 

21.2 

16.0 
1  6.0 

.  STEEL. 
.133,  or  i  in  751 

.163,  or  i  in  612 
.125,  or  i  in  801 
.125,  or  i  in  801 

.000074,  or  x/i3,si4 

.000077,  or  i/I2)987 
.000078,  or  x/12,821 
.000078,  or  i/I2>8ai 

m  2  samples  

"  n  Tyres,  axles  ;  3  samp. 

Crucible  steels 

SUMMARY 

23-3 
17.8 

AVERAGES. 
.175,  or  i  in  570 
.137,  or  i  in  732 

.000076,  Or  z/i3,25o 
.000077,  Or  Vi3,040 

Bessemer  steels  
All  steels  

20.5 

.156,  or  i  in  641 

.000076,  or  VIS.HO 

BARS  TESTED- 
o    Crucible  steel  ;  axles,  ) 
rails,  tyres  (3  samp.)  f 
p   Bessemer  steel  ;  axles,  ) 
rails,  tyres  (3  samp.)  j 

—for  Compres 
23-0 
24.0 

sion,  but  not  for  Ext 
.172,  or  i  in  581 
.182,  or  i  in  550 

ension. 
.000073,  Or  '/I3.700 

.000074,  or  1/13,514 

The  lowest  elastic  strength  in  any  group  of  bars,  in  the  second  series, 
did  not  exceed  one  ton  per  square  inch  less  than  the  average  elastic 
strength  of  the  group — say  not  more  than  5  per  cent,  less  than  the  average 
for  steel  bars. 

Table  No.  203  shows  the  chemical  composition  and  specific  gravity  of 
fourteen  of  the  bars  subjected  to  tests. 

Tensile  Strength  of  Tempered  Steel. 

The  Steel  Committee  publish  the  results  of  experiments  made  at  H.  M. 
Gun  Factory,  Woolwich,  on  the  comparative  strength  of  untempered  and 
tempered  steel.  A  summary  of  the  results  is  given  in  table  No.  204.  The 
specimens  were  .50  and  .53  inch  in  diameter,  and  from  i  inch  to  two  inches 
in  length. 


STRENGTH  OF  STEEL. 


603 


Table  No.  203. — CHEMICAL  ANALYSIS  AND  SPECIFIC  GRAVITY  OF  STEEL 

AND  IRON  BARS. 

(Tested  by  the  Steel  Committee,  1870.) 
CRUCIBLE  STEEL. 


CHEMICAL  CONSTITUENTS. 

Ultimate 

Tensile 

Reference. 

Iron. 

Carbon. 

Silicon. 

Man- 
ganese. 

Sulphur. 

Phos- 
phorus. 

Gravity. 

Strength 
per  square 
inch. 

p.  cent. 

p.  cent. 

p.  cent. 

p.  cent. 

p.  cent. 

p.  cent. 

tons. 

a 

98.86 

•79 

.115 

.19 

trace 

.01 

7.839 

52.76 

b 

98.67 

•67 

.20 

•44 

trace 

.02 

7.831 

51.01 

c 

98.87 

•57 

.14 

•37 

.01 

.04 

7.851 

43-48 

d 

98.63 

.90 

•39 

.02 

.06 

7.844 

41.85 

e 

98.87 

.58 

.22 

•30 

.02 

.01 

7.825 

40.50 

f 

98.88 

•47 

— 

.61 

.02 

.02 

7.845 

38.51 

0 

99.22 

•59 

•03 

.14 

trace 

.02 

7.859 

h 

99.16 

•44 

.14 

•23 

.01 

.02 

7.850 

35-47 

i 

99.16 

•52 

.10 

.19 

trace 

•03 

7.850 

Averages, 

98.89 

.62 

.114 

•34 

.01 

.026 

7.842 

42.15 

BESSEMER  STEEL. 

P 

99.24 

•34 

trace 

•35 

.04 

•03 

7-857 

— 

I 

99-21 

•32 

.01 

.40 

.04 

.02 

7.857 

34.19 

m 

99.22 

.05 

.38 

.02 

.02 

7.853 

33-68 

n 

99-13 

•35 

•03 

•44 

.04 

.01 

7.852 

33-66 

Averages, 

99.20 

•33 

.022 

•39 

•035 

.02 

7-855 

33-84 

YORKSHIRE  IRON. 

II   99-49 

•23 

.10 

.08 

.02       j       .08       ||     7.758      ||     23.69 

Table  No.  204. — TENSILE  STRENGTH  OF  TEMPERED  STEEL. 


Average  Breaking  Weight  per  Square  Inch. 

Manufacturer, 
or 

MATERIAL. 

Number 
of 

After  being  Tempered  in  Oil  at 

Contractor. 

Specimens. 

Tempered 

As 

Received. 

High 

Medium 

Low 

Water. 

.  ^ 

Kelt. 

Heat. 

Heat. 

? 

tons. 

tons. 

tons. 

tons. 

tons. 

tons. 

Krupp,  
Firth,  

Cast  Steel,.... 
Steel,  

9 
3 

32.1 

34-4 

"T 

65.4 

54-4 

56.4 

Firth,  

Homog.  Steel, 

217 

31.6 

47-7 

48.0 

47-0 

44.4 

Cammell,  .. 

Steel,  

2 

26.6 

Cammell,  .. 

Homog.  Steel, 

61 

29-5 

54-6 

45-7 

51-5 

49.0 

51-1 

— 

Steel,  

4 

3!-7 

36.6 

— 

Homog.  Steel, 

36 

29.0 

37-2 

39-3 

53-9 

— 

Styrian  Steel,. 

8 

56.2 

82.2 

Moser  

Steel,  

7 

33-5 

54-6 

55-8 

Whitworth, 

Steel,  

2 

38.2 

48.2 

6o4 


THE   STRENGTH   OF   MATERIALS. 


STRENGTH  OF  FAGERSTA  STEEL.     1873. 

Mr.  Kirkaldy  made  a  comprehensive  set  of  experiments  on  the  strength 
of  steel  manufactured  at  the  Fagersta  Works,  Sweden. 

First  Series  of  Experiments. 

Twelve  hammered  bars,  2  inches  square,  in  four  groups  of  different 
degrees  of  hardness,  here  distinguished  as  a,  b,  c,  d,  were  tested  for  tensile, 
compressive,  shearing,  torsional,  and  transverse  strength — three  samples  for 
each  test.  For  the  transverse  tests,  the  specimens  were  planed  to  1.9  inches 

i 


20' 


Fig.  215. — Fagersta  Steel — Test  Specimen  for  Bending  or  Transverse  Stress. 

square;  for  the  other  tests,  they  were  turned  to  a  diameter  of  1.128  inches, 
having  i  square  inch  of  sectional  area.  The  forms  of  the  specimens  are 
shown  in  Figs.  215,  216,  217.  The  condensed  results  are  given  in  tables 
Nos.  205  to  207. 

Table  No.  205. — FAGERSTA  STEEL  BARS — TRANSVERSE  STRENGTH.    1873. 

1.9  inches  square;  span,  20  inches.     Load  applied  at  the  middle. 


BARS. 

Elastic 
Stress. 

Ultimate 
Stress. 

Ratio 
of  Elastic 
to  Ultimate 
Stress. 

Ultimate 
Deflection. 

Remarks. 

a 

tons. 

Q.  1-2 

tons. 
14.  ^ 

per  cent. 

66.0 

inches. 
78 

fractured 

b    

y--}--} 

Q.OQ 

IQ.C7 

4Q.6 

1.4-0 

fractured 

c 

8.l8 

17.  0'? 

4.8  0 

3.  31 

uncracked 

d    

7.O4. 

11.28 

62.3 

c.I  I 

uncracked 

Averages  

8.58 

15.61 

56.5 

2.67 

Table  No.  206. — FAGERSTA  STEEL  BARS — TORSIONAL  STRENGTH.     1873. 

Diameter  1.128  inches  (i  square  inch  section).     Length  for  torsion,  8  diameters. 
Stress  applied  at  the  end  of  a  1 2-inch  lever. 


BARS. 

Elastic 
Stress. 

Breaking 
Stress. 

Ratio  of 
Elastic  to 
Breaking 
Stress. 

Ultimate  Angular  Torsion. 

Least. 

Greatest. 

Average. 

a     

tons. 
.507 
.502 
.484 
•341 

tons. 
.946 
1.043 
1.009 
.679 

per  cent. 

53-9 

48.2 

48.3 
50.2 

i  turn=i. 
.207 
.625 
.897 
3-053 

i  turn=i. 
410 
.938 
1.255 

3725 

i  turn=i. 
.29I 

•793 

1.  02  1 
3.219 

b            .    . 

c     

d 

Averages  

.458 

.919 

50.2 

I.I95 

1.528 

I-33I 

STRENGTH  OF  STEEL. 


605 


Table  No.  207. — FAGERSTA  STEEL  HAMMERED  BARS — TENSILE, 

COMPRESSIVE,    AND    SHEARING   STRENGTH.       1873. 

2-inch  square  bars  turned  to  1.128  inches  in  diameter  (i  square  inch  of  section). 

TENSILE  STRENGTH. 


BARS. 
10.  15  inches  long. 
(9  diameters.) 

Elastic 
Strength  in 
tons  per 
square  inch. 

Breaking 
Weight  in  tons 
per  square 
inch. 

Ratio 
of  Elastic 
to  Breaking 
Strength. 

Permanent 
Extension. 

Sectional 
Area  of 
Fractujer--- 

/,        >''' 

a     

tons. 
27.7O 

tons. 
^S.OA 

per  cent. 
71.1 

per  cent. 
1.8 

petipent^y 

b      . 

28.Ii; 

S 

47.6o 

CQ.4. 

c.i 

QW&7 

c     

2C..Q4 

45.82 

5X  7 

;6.6 

6.6 

8c  h 

d 

IQ.24 

27.77 

70  i 

i6c. 

385 

J'-'O 

Averages  

25.25 

39.16 

64.8 

7-5 

78.9 

COMPRESSIVE  STRENGTH  (per  square  inch). 


BARS. 

Length, 
i  diameter, 
1.128  ins. 

Length,  2  diameters, 
2.25  inches. 

Length,  4  diameters, 
4.51  inches. 

Length,  8  diameters, 
9.02  inches. 

Elastic 
Strength. 

Elastic 
Strength. 

Destroying 
Weight. 

Elastic 
Strength. 

Destroying 
Weight. 

Elastic 
Strength. 

Destroying 
Weight. 

a 

tons. 
28.57 
27.98 
26.78 
1741 

tons. 
28.27 
26.19 

25-I3 
18.75 

tons. 
75.85 

77-37 
69.64 

54.15 

tons. 
28.27 
26.19 
23.81 
18.30 

tons. 
59.92 
52.50 
47.01 
36.50 

tons. 

27-53 
25.89 

23.51 

18.16 

tons. 
45.62 
42.50 

37.87 
21.05 

b          

c     

d    

Averages 

25.18 

24.70 

69.24 

24.03 

48.88 

2377 

36.76 

SHEARING  STRENGTH  (per  square  inch). 


BARS. 

Ultimate  Shearing  Strength. 

Detrusion  before  Rupture, 
as  a  measure  of  hardness  inversely. 

Per  square 
inch. 

Per  cent, 
of  Ultimate 
Tensile  Strength. 

Actual. 

In  parts  of 
the  diameter. 

a     

tons. 
27.42 
35.60 

31-99 
20.28 

per  cent. 

73.3 
75-2 
69.5 
74.0 

inch. 

•193 

.249 
.281 
.323 

per  cent. 

17 
21 

25 
29 

b 

c     

d    

Averages.  . 

28.82 

73-5 

.261 

23 

For  Shearing  Stress. 


For  Pulling  or  Tensile  Stress. 


or 


Figs.  216,  217.— Fagersta  Steel— Test  Specimens. 


6o6 


THE   STRENGTH   OF   MATERIALS. 


Second  Series  of  Experiments  on  Fagersta  Steel. 

To  test  the  influence  of  hammering,  and  of  annealing  steel  bars.  Four 
ingots  6  inches  square,  differing  in  hardness,  <?,  f,  g,  h,  were  cut  into 
specimens  alike,  in  duplicate,  and  hammered  down  to  four  sizes,  namely, 
5,  4,  3,  and  2  inches  square. 

In  table  No.  208,  are  given  results  showing  the  comparative  tensile 
strength  of  each  ingot,  and  of  the  2-inch  hammered  bars  formed  from 
them.  They  prove  that  the  steel  was  made  considerably  stronger  by 
hammering.  In  the  original  Report,  it  is  made  apparent  that  the  strength 
was  proportionally  increased  as  the  bars  were  reduced  in  size. 

Table  No.  208. — FAGERSTA  STEEL  INGOTS  AND  HAMMERED  BARS — 
COMPARATIVE  TENSILE  STRENGTH.     1873. 

UN  ANNEALED. 


Ingots  and  Hammered  Bars  of 
Various  Degrees  of  Hardness. 

Elastic 
Strength 
(Tensile),  in 
tons  per 
square  inch. 

Breaking 
Weight  in 
tons  per 
square  inch. 

Ratio  of 
Elastic  to 
Breaking 
Weight. 

Permanent 
Extension. 

Sectional 
Area  of 
Fracture. 

Ingots,  6  inches  square. 
e 

/ 
1 

Averages 

tons. 
21.27 
17.21 
12.64 
9.68 

tons. 
29.98 
30.48 
24.65 
12.46 

per  cent. 
71.0 

58.3 
51.2 
42.1 

per  cent. 
I.I 
2.0 

3-5 
n.6 

per  cent. 
98.5 

97-5 

15.20 

24-39 

55.6 

4-5 

95.0 

Hammered  Bars,  2  inches 
square. 
e 

/ 
f 

Averages 

29.69 
21.30 
17.50 
15.71 

44-03 
43-70 
33-48 
26.76 

67.4 
48.7 

52.3 
58.7 

2.2 
10.2 

17.9 
22.5 

96.8 
71.6 

47-5 
38.7 

21.05 

37.00 

56.8 

13-2 

63.6 

Ingots,  6  inches  square. 
e 

/ 
I 

Averages 

ANN 

17-34 
16.55 
11.68 
9.00 

EALED. 

28.37 

33-05 
23.66 

23.57 

61.2 
5i.3 
49.5 
38.2 

i-7 

7-2 

4.2 
18.2 

97-8 
85.0 
94.8 
72.8 

13.64 

27.16 

50.0 

15.6 

87.6 

Hammered  Bars,  2  inches 
square. 
e 
f 

i 
Aver  acres 

21.20 
20.67 
16.30 
1478 

38.42 
40.99 
31.60 
25.15 

55.2 
50.4 
51.6 
58.7 

5-5 
12.7 
19.1 

22.2 

91.9 
54.0 
42.4 
35-9 

18.24 

34-04 

54.0 

14.9 

56.0 

STRENGTH   OF   STEEL. 


607 


Table  No.  209. — FAGERSTA  STEEL  HAMMERED  AND  ROLLED  BARS — 
COMPARATIVE  TENSILE  STRENGTH.    1873. 

HAMMERED — UNANNEALED. 


Bars  of  Various  Degrees 
of  Hardness. 

Elastic  Strength 
in  tons  per 
square  inch. 

Breaking  Weight 
in  tons  per 
square  inch. 

Ratio  of  Elastic 
to 
Breaking  Weight. 

Permanent 
Extension. 

3-inch  square  bars. 
i 
k 
I 

Averages,  

tons. 
...      2745      ". 
...      17.32      ... 

...    12.68    ... 

tons. 
...      39.49     ... 
...      31.29     ... 
...      25.28      ... 

per  cent. 
...      87.5       -. 
...      55.4      ... 
...       50.2       ... 

per  cent. 
.4       ... 
...          2.3       ... 
...       25.2       ... 

...    19.15    ... 

...      32.02      ... 

...      64.4       ... 

...         9.3       ... 

>£-inch  square  bars. 
i 
k 
I 

Averages,  

...    42.05    ... 
...    34.96    ... 
...    25.67    ... 

...      60.59     ••• 
...      42.84     ... 
...      32.21      ... 

...      69.4       ... 
...      51.6       ... 
...      79.7       ... 

...          5.7       ... 

16.0     ... 

IO.I       ... 

...    34.23    ... 

...      45.21      ... 

...      66.9       ... 

10.6     ... 

3-inch  square  bars. 
i 
k 
I 

Averages,  

HAMMERE 

...    20.85    ... 
...    13.30    ... 
...    12.68    ... 

D  —  ANNEALED 

...    31.66    ... 
...    31.09    ... 
...    25.28    ... 

...      65.8       ... 
...      42.8       ... 
...       50.2       ... 

1.7     ... 
...       7.7     ... 
...     25.2     ... 

...    15.61     ... 

...    29.34    ... 

...       52.9       ... 

...     11.5;    ... 

>£-inch  square  bars. 
i 
k 
I 

Averages,  

...    31.12    ... 
...    21.34    ... 
...    14.24    ... 

...    54.92    ... 
...    36.66    ... 

...       56.8       ... 

...       8.3     ... 

...    25.39    ... 

...      56.1       ... 

...       12.6      ... 

...    22.23    ..- 

...    38.99    ... 

...     57.0     ... 

...     9.5   ... 

3-inch  square  bars. 
i 
k 
I 

Averages  . 

ROLLED— 
...    2687 

UNANNEALED. 
•59.71; 

...     79-6     ... 
...     48.7     -. 
...     43.2     ... 

.6     ... 
...       2.5     ... 
...     31.1     ... 

...    13.57    ... 

...      10.22      ... 

...    27.85    ... 
...    23.64    ... 

16  89 

...    28.41     ... 

...     57.2     ... 

...     11.4     ... 

>£-inch  square  bars. 
i 
k 
I 

Averages,  

...      35.09      ... 
...     20.89      ". 
...      15.09      ... 

...     59.82    ... 
...    40.50    ... 
...    27.13    ... 

...     56.2     ... 
...     51.6     ... 
...     55-6     ... 

16.0     ... 

...       22.2       ... 

...     23.69      ... 

...    42.48    ... 

...     54.4     ... 

...       15.2       ... 

6o8 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  209  (continued}. 
ROLLED — ANNEALED. 


Bars  of  Various  Degrees 
of  Hardness. 

Elastic  Strength 
in  tons  per 
square  inch. 

Breaking  Weight 
in  tons  per 
square  inch. 

Ratio  of  Elastic 
to 
Breaking  Weight. 

Permanent 

Extension. 

3-inch  square  bars. 
i 
k 
I 

tons. 
...      19.78      ... 
...      12.32      ... 
...      10.22      ... 

tons. 

...      32-55      ." 
...      26.87      -.. 
...      23.64     ... 

per  cent. 
...       60.7        ... 
...       45-8        -. 
...       47.7        ... 

per  cent. 
2.0       ... 
...          3.8       ... 
...       26.0       ... 

...      14.11       ... 

27.6Q      . 

"U.4 

10.6 

j^-inch  square  bars. 
i 
k 
I 

...      28.93      .« 
...      I8.39      - 
12.  AC 

...      57.14     ... 
...      35.81      ... 
23.CO      . 

...       50.6       ... 
...       51.4       ... 
C^  o 

...       8.5     ... 
...       9.8     ... 

27  I 

Averages.  ., 

...      2O.OO      ... 

38.82      , 

ci.7 

13.8 

Third  Series  of  Experiments  on  Fagersta  Steel. 

To  compare  the  tensile  strength  of  steel  bars,  reduced  by  hammering  and 
by  rolling.  Bars  of  three  degrees  of  hardness  were  tested,  say  /,  k,  I.  Of 
each  degree,  six  3-inch  square  hammered  bars  were  tested,  five  of  which 
were  reduced  by  hammering  to  2%,  2,  ij^,  i,  and  ^  inch  square,  and 
then  turned  to  given  diameters. 

In  table  No.  209  are  given  comparative  results  for  the  3-inch  and  ^-inch 
square  bars,  hammered  and  rolled.  The  original  table  shows  that  the 
strength  was  proportionally  increased  as  the  bars  were  reduced  in  size. 


Fourth  Series  of  Experiments  on  Fagersta  Steel. 

To  test  the  tensile  and  compressive  strength  of  Fagersta  steel  plates,  of 
/^>  ~/i,  Y%)  YZI  and  y%  inch  thickness,  cut  into  strips  2^  inches  wide. 
Table  No.  210  gives  the  comparative  results  of  the  trials. 


Fifth  Series  of  Experiments  on  Fagersta  Steel. 

To  show  the  variations  in  results  for  tensile  strength,  arising  from  differ- 
ences in  the  form  and  proportions  of  specimens. 
Two  sets  of  specimens  were  prepared  according  to 
Figs.  218;  one  set  was  10  inches  wide  and  10 
inches  long  at  the  parallel  middle  portion;  and 
the  smaller  set  i*^  inches  wide  and  4}^  inches 
}ol[iS  a*  the  middle.  Condensed  results  are  given 
in  table  No.  211;  and  the  results  of  the  i  oo-inch 
bars,  from  table  No.  210,  are  added,  for  comparison. 


STRENGTH   OF   STEEL. 


609 


Table  No.  210. — FAGERSTA  STEEL  PLATES — TENSILE  AND  COMPRESSIVE 

STRENGTH.     1873. 

Specimens  2j^  inches  wide,  100  inches  long. 
TENSILE  STRENGTH — UNANNEALED. 


PLATES. 
Thickness. 

Elastic 
Strength 
(Tensile) 
in  tons  per 
square  inch. 

Elastic  Extension. 

Breaking 
Weight  in 
tons  per 
square  inch. 

Permanent 
Extension. 

Ratio 
of  Elastic 
to  Breaking 
Strength. 

Sectional 
Area  of 
Fracture. 

inch. 

tons. 

per  cent. 

tons. 

per  cent. 

per  cent. 

per  cent. 

H 

17.37 

.136 

24.61 

5.21 

70.6 

62.1 

X 

15.89 

.124 

24.17 

10.17 

65.7 

46.3 

3/B 

H.34 

.091 

21.84 

20.64 

51.9 

29.0 

YZ 

12.28 

.082 

22.39 

16.30 

54-8 

38.8 

H 

11.65 

.078 

22.00 

17-95 

52.9 

39-3 

Averages 

13.71 

.102  or  i  in  980 

23.00 

14.03 

59.2 

43.i 

ANNEALED. 

1A 

11.92 

.096 

20.30 

10.98 

58.7 

364 

X 

13-30 

.098 

22.14 

16.88 

60.  i 

H 

11.56 

.096 

20.86 

18.19 

55-4 

30.4 

/4. 

12.19 

.088 

22.09 

19.15 

55.2 

357 

* 

11.25 

.088 

21.  l8 

1745 

36.9 

Averages 

12.04 

.093  or  i  in  1020 

21.31 

16.53 

56.5 

344 

Elastic  extension  per  ton  )  Unannealed 0000744,  or  1/i3,43s. 

per  square  inch, J  Annealed 0000772,  or  */I8»a8, 

COMPRESSIVE  STRENGTH — UNANNEALED. 


PLATES. 
Thickness. 

Elastic  Strength 
(Compressive) 
in  tons  per  square 
inch. 

Elastic 
Compression. 

Elastic  Compression 
per  ton  per  square  inch, 
in  parts  of  the  length. 

inch. 

tons. 

per  cent. 

Length  =  i. 

X 

I7.8l 

.106 

X 

1  6.20 

.115 

X 

11.83 

.089 

% 

I3.30 

.099 

H 

11.38 

.088 

Averages 

14.10 

.100,  or  i  in  looo 

.000071,  Or  1/14,100 

ANNEALED. 

# 

10.40 

•063 

X 

12.28 

.088 

H 

11.25 

.083 

% 

10.13 

.074 

H 

8.98 

.066 

1  0.6  1 

.075,  or  i  in  1333 

.0000707,  or  1/14,143 

6io 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  211. — FAGERSTA  STEEL  PLATES — TENSILE  STRENGTH  AS 

AFFECTED  BY  THE  FORM  AND  PROPORTIONS  OF  THE  SPECIMENS.   1873. 

Averaged  results  of  specimens  from  ^  to  %  inch  thick. 

UNANNEALED. 


SPECIMENS. 

Elastic 
Strength 

^  (Tensile) 
in  tons  per 
square  inch. 

Breaking 
_  Weight 
in  tons  per 
square  inch. 

Ratio 
of  Elastic 
to  Breaking 
Weight. 

Permanent 
Extension. 

Sectional 
Area  of 
Fracture. 

Length  =  breadth....  Figs.  218 
Length  =  3  breadths        „ 
Length  =  44  breadths      

tons. 
16.05 
15.56 
I3-7I 

tons. 
26.39 
25.56 
23.00 

per  cent. 
60.0 
60.3 
59.2 

per  cent. 
29.7 

35-o 
14.0 

per  cent. 
48.8 

43-2 
43-1 

Length  =  breadth....  Figs.  218 
Length  =  3  breadths        „ 
Length—  /|/|  hffadths 

ANNEAI 

13-53 
12.98 
12.04 

,ED. 

23.65 
23.16 
21.31 

57.0 
56.0 
56.5 

33-i 
39-3 
16.5 

39-1 
36.5 

34-4 

Sixth  Series  of  Experiments  on  Fagersta  Steel. 

To  test  the  influence  of  holes  drilled  and  holes  punched  in  steel 
plates.  Specimens  were  formed  12^  inches  wide, 
and  otherwise  like  the  broad  specimen,  Fig.  219, 
for  comparison.  There  were  three  rows  of  rivet 
holes,  3  inches  apart;  and  five  holes  in  each  row  at 
2^-inch  centres.  The  holes  were  .77  inch  in  dia- 
meter, and  made  .77  x  5  =  3.85  inches  of  blank;  the 
net  section  was  (12.5-3.85  =  )  8.65  inches  wide, 

or  69.2  per  cent,  of  the  total  width.     Table  No.  212  gives  some  deductions 

from  the  reported  results. 

Table  No.  212. — FAGERSTA  STEEL  PLATES — TENSILE  STRENGTH,  WITH 
RIVET  HOLES  WHEN  DRILLED  AND  WHEN  PUNCHED. 

Specimens  12^  inches  wide.     Three  rows  of  holes  .77  inch  in  diameter. 
UNANNEALED. 


DRILLED  HOLES. 

PUNCHED  HOLES. 

PLATES. 
Thickness. 

Reduced 

Section  in  parts 
of  the  Total 
Section. 

Reduced 
Strength  in 
parts  of  Total 
Strength. 

Tensile 
Strength  per 
square  inch 
of  Net  Section, 
in  parts  of  that 
of  Unreduced 

Reduced 
Strength  in 
parts  of  Total 
Strength. 

Tensile 
Strength  per 
square  inch 
of  Net  Section, 
in  parts  of  that 
of  Unreduced 

Section. 

Section. 

inch. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

% 

69.2 

74.9 

108.5 

67.0 

97.1 

X 

69.2 

76.6 

III.O 

68.5 

99.2 

% 

69.2 

77.0 

1  1  1.6 

69.5 

100.4 

'A 

69.2 

78-3 

H3.5 

54.8 

79-4 

H 

69.2 

77.2 

1  1  2.0 

51.0 

74.0 

Averages  .  .  . 

69.2 

76.8 

III.3 

62.2 

90.0 

STRENGTH   OF   STEEL. 


611 


Table  No.  212  (continued}. 
ANNEALED. 


DRILLED  HOL^S. 

PUNCHED  HOLES. 

PLATES. 
Thickness. 

.  Reduced 
Section  in  parts 
of  the  Total 
Section. 

Reduced 
Strength  in 
parts  of  Total 
Strength. 

Tensile 
Strength  per 
square  inch 
of  Net  Section, 
in  parts  of  that 
of  Unreduced 

Reduced 
Strength  in 
parts  of  Total 
Strength. 

Tensile 
Strength  per 
square  inch 
of  Net  Section, 
in  parts  of  that 
of  Unreduced 

Section. 

Section. 

inch. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

1A 

69.2 

72.9 

105.7 

67.I 

97.1 

X 

69.2 

74-5 

108.0 

67.8 

98.0 

H 

69.2 

75-i 

108.8 

66.4 

96.1 

% 

69.2 

77.0 

1  1  1.6 

69.9 

105.6 

X 

69.2 

75-9 

IIO.O 

68.7 

99.2 

Averages... 

69.2 

75-1 

108.8 

68.0 

98.6 

Unannealed.  Annealed, 

per  cent.  per  cent. 

Note. — The  average  elongation  with  drilled  holes,  14.9  18.0 

Do.                do.             punched  holes,  6.3  16.6 

Do.                 do.             solid  plate, 29.7  33.1 

Seventh  Series  of  Experiments  on  Fagersta  Steel. 

To  test  rolled  steel  plates  under  bulging  stress.  The  specimens  were 
discs,  12  inches  in  diameter,  cut  out  in  the  lathe,  and  pressed  through 
an  aperture  10  inches  in  diameter.  The  bulger  or  ram  was  cylindrical, 
about  5  inches  in  diameter;  and  the  preparation  for  the  trial  is  shown  in 
Fig.  204,  page  584,  and  the  finished  article,  after  bulging,  in  Fig.  205. 

Table  No.  213. — FAGERSTA   STEEL   PLATES — RESISTANCE   TO   BULGING 

STRESS. 

Discs  12  inches  in  diameter;  aperture  10  inches  in  diameter. 
UNANNEALED. 


Stress  —  Bulging  in  inches. 

Ultimate. 

Disc. 



Effect. 

Thickness. 

Lbs., 
25,000. 

Lbs., 

100,000. 

Lbs., 

200,000. 

Bulge. 

Stress. 

inch. 

inches. 

inches. 

inches. 

inches. 

tons. 

M 

X 

1.86 
1.09 

z 

— 

3.00 

3-i  i 

14.50 
31-94 

buckled 

uncracked 

H 

.89 

2.68 

— 

3.22 

46.92 

)) 

X 

.68 

1-93 

— 

3-33 

71.83 

» 

* 

•44 

1.61 

2.77 

3-44 

97.90 

» 

ANNEALED. 

X 

X 

2.25 
1.32 

— 

— 

3-04 
3.12 

11-53 

26.77 

buckled 
uncracked 

X 

.94 

— 

— 

3-23 

43-67 

a 

y* 

•73 

2.06 

—    , 

3-34 

67.28 

» 

H 

.52 

1.72 

3-14 

3-45 

90.09 

» 

6l2 


THE   STRENGTH   OF   MATERIALS. 


SIEMENS-STEEL  PLATES  AND  TYRES.     1875. 

A  number  of  steel  plates  manufactured  to  the  specification  of  the 
Admiralty,  by  the  Landore  Siemens-Steel  Company,  were  tested  for  the 
Company  by  Mr.  Kirkaldy. 

By  the  terms  of  the  specification,  it  was  required  that  the  ultimate 
tensile  strength  should  be  not  less  than  26  tons,  nor  more  than  30  tons, 
per  square  inch,  with  an  extension  of  20  per  cent,  in  a  length  of  8 
inches.  Strips  cut  lengthwise  of  the  plate,  ij^  inches  wide,  heated  uni- 
formly to  a  low  cherry-red  heat,  and  cooled  in  water  at  82°  F.,  were  to 
sustain  bending  double  in  a  press,  to  a  curve  of  which  the  inner  radius 
was  to  be  one-and-a-half  times  the  thickness  of  the  plate.  Abstracts  of 
the  results  of  the  tests  for  tensile  strength  are  given  in  table  No.  215, 
together  with  tests  for  the  tensile  strength  of  steel  tyres.  Twelve  specimens 
of  the  plates  of  the  2d  series  in  the  table,  were  tested  for  bending,  length- 
wise and  crosswise,  between  supports  at  10  inches  apart.  All  the  specimens 
bore  the  test  uncracked. 

Plates  of  various  thicknesses  were  tested  for  resistance  to  bulging  stress, 
i2-inch  discs  having  been  forced  through  lo-inch  apertures,  in  the  manner 
before  described,  page  584.  All  the  plates  bore  the  test  without  cracking. 
Particulars  are  given  in  table  No.  214. 

Two  steel  tyres,  of  which  the  tensile  strengths  were  tested  (3d  series, 
table  No.  215),  were  respectively  43  and  37  inches  in  diameter,  and  2.32 
and  2.10  inches  in  thickness.  They  were  collapsed  under  transverse 
pressures  of  42.22  and  52.16  tons;  so  that  opposite  sides  of  the  hoop 
were  pressed  into  contact  with  each  other.  The  larger  tyre  burst  at  one 
of  the  bends ;  the  smaller  remained  unbroken. 


Table  No.  214. — SIEMENS-STEEL  PLATES — RESISTANCE  TO 
BULGING  STRESS.     1875. 

Discs  12  inches  in  diameter,  pressed  into  lo-inch  apertures. 
(Reduced  from  Mr.  Kirkaldy's  Reports.) 


Stress.            Bulging  in  inches. 

Ultimate. 

Thickness  of  Plates. 

Ib. 
25,000. 

ib. 
100,000. 

Ib. 

200,000. 

Bulge. 

Stress. 

EFFECTS. 

inch. 

inch. 

inches. 

inches. 

inches. 

tons. 

Unannealed. 

•37 

.42 

I.7I 

— 

3-15 

63.750 

uncracked. 

•7i 

.05 

1.09 

I.96 

3-48 

145.500 

do. 

Annealed. 

•37 

.67 

2.02 

— 

3-17 

60.357 

uncracked. 

.41 

.56 

1.84 

— 

3.22 

68.191 

do. 

.41 

•59 

1.89 

— 

3-23 

68.080 

do. 

.50 

.29 

1.45 

2.79 

3.31 

101.920 

do. 

.62 

•15 

1.40 

2.51 

3-38 

II5.III 

do. 

.70 

.10 

1.26 

2.18 

3-42 

123.260 

do. 

STRENGTH  OF  STEEL. 


6l3 


Table  No.  215. — SIEMENS-STEEL  PLATES — TENSILE  STRENGTH. 

From  .37  to  .71  inch  in  thickness. 

(Reduced  from  Mr.   Kirkaldy's  Reports.) 

SERIES  i.     PLATES  OF  DIFFERENT  THICKNESSES. 


1875- 


Treatment,  and  Thickness 
of  Plates. 

Elastic 
Strength 
per  square 
inch. 

Ultimate 
Strength 
per  square 
inch. 

Ratio  of 
Elastic  to 
Ultimate 
Strength. 

EXTENSION. 

Sectional 
Area  of 
Fracture. 

At  60,000 
Ibs.  per 
sq.  inch. 

Ultimate. 

LENGTHWAY.    inch. 
Unannealed....  .37 
Do  71 

Means 

tons. 
15.446 
13.572 

tons. 

32.535 
29.870 

per  cent. 

47-4 

45.4 

per  cent. 
4.50 

6.75 

per  cent. 
22.3 
24.5 

per  cent. 
62.5 

55-3 

14.509 

31.202 

46.4 

5.62 

23-4 

58.9 

Annealed  37 

14.062 
13.929 

13.303 
13.125 
11.741 
10.937 

30.143 
29.647 
29.491 
29.388 

27-595 
26.821 

46.6 
46.9 

45i 

44.6 

42.5 
40.7 

8.00 
8.08 
8.50 
8.66 
13.80 
17.72 

24.8 
21.  1 
24.8 
20.4 

25.5 
25.0 

56.9 

55.3 
61.5 

55-5 
56.7 
54-5 

Do                  4.0 

Do.              .  .40 

Do                  50 

Do        .    .      62 

Do.             .    70 

Averages 

12.848 

28.848 

444 

10.79 

24.6 

56.8 

CROSSWAY. 
Unannealed....  .37 
Do  71 

Means  

I5-3I4 
13.571 

32.442 
30.062 

47-2 
45.1 

4.52 
7.07 

22.4 
247 

62.5 
56.4 

14.442 

31.250 

46.1 

5-79 

23.5 

59-5 

Annealed  37 
Do                  40 

13.928 
13.840 
13-393 
13.303 
11.741 
10.937 

29.856 

29'82! 
29.366 

29.705 
27.040 
26.885 

46.6 

46.3 
45.6 
44-8 
43-4 
40.6 

9.39 
9.07 
7.81 
8.50 
16.61 
17.30 

26.4 
26.3 
20-4 
20.2 
22.7 
26.0 

53-4 
50.4 
61.0 
53-3 
647 
49-3 

Do.                        A2 

Do.              .    C2 

Do  62 

Do                  70 

Averages  

12.856 

28.788 

44-5 

11.44 

23.6 

59.1 

SERIES 
A  nnealed  64 

2.     PLAI 

'ES  ANNE. 

25-483 
26.996 

\LED,    AN! 

>  HARDEN 

ED. 

24.1 
20.2 

47-5 
5i-3 

Do                  62 

Means  

26.240 

22.2 

494 

Hardened:  — 
Cherry-red,  and  \ 
cooled  in  water  >  .64 
at  82°  F  ) 

— 

28.867 
29.036 

— 

— 

22.4 

1  8.0 

50.7 
54-5 

j        Do                    62 

Means  

28.951 

20.2 

52.1 

614 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  215  (continued], 
SERIES  3.     TYRES. 


EXTENSION. 

Diameter 
of  Specimens. 

Elastic 
Strength 
per  square 
inch. 

Ultimate 
Strength 
per  square 
inch. 

Ratio  of 
Elastic  to 
Ultimate 
Strength. 

Sectional 
Area  of 
Fracture. 

At  60,000 
Ibs.  per 

Ultimate. 

sq.  inch. 

inches. 

tons. 

tons. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

istTyre,  specimen. 

1.511 

17.098 

29.853 

57-2 

6.58 

1  8.8 

55.8 

1.511 

17.321 

30.800 

56.2 

6.20 

23-6 

51.9 

2d  Tyre,  specimen. 

1.511 

18.482 

30.083 

61.4 

6.48 

17.7 

58.8 

1.511 

18.840 

3L075 

60.6 

5.59 

16.9 

70.4 

Averages         .  . 

17-935 

30-453 

58.8 

6.21 

19.2 

59-2 

WHITWORTH'S  FLUID-COMPRESSED  STEEL.* 

On  Sir  Joseph  Whitworth's  system  of  treatment,  a  pressure  of  6  tons  per 
square  inch  is  applied  as  quickly  as  possible  to  melted  steel,  after  it  is  taken 
from  the  furnace.  A  column  8  feet  high  is  reduced  i  foot  in  height  in  the 
course  of  five  minutes. 

Specimens  for  testing  tensile  resistance  are  cylindrical,  formed  as  in 
Fig.  220;  the  central  portion  has  a  sectional  area  of  %  square  inch,  being 


Figs.  220,  22T. — Whitworth's  Fluid-Compressed  Steel — Test  Specimens. 

.798  inch  in  diameter,  and  has  length  of  2  inches,  or  2^  diameters.  The 
upper  and  lower  portions  are  screwed,  and  are  seized  by  nuts.  The  usual 
appearance  of  broken  specimens  is  shown  at  Fig.  221. 

Table  No.  216  gives  results  of  tests  for  the  tensile  resistance  of  fluid- 
compressed  steel,  and  of  the  purest  and  best  irons  made  in  England. 

Sir  Joseph  Whitworth  states  that  he  can  produce,  with  certainty,  by  com- 
pression, steel  having  40  tons  ultimate  strength,  with  30  per  cent,  ductility. 
In  relation  to  this,  Mr.  F.  W.  Webb  says  that  he  has  no  difficulty  in  pro- 
ducing a  mild  cast  steel  having  30  to  32  tons  ultimate  strength,  and  33  or 
34  per  cent,  ductility. 

Sir  Joseph  Whitworth  considers  that  there  is  no  need  for  more  than  30 
per  cent,  of  ductility;  with  this  proportion,  steel  tears  when  ruptured,  and 
does  not  fly  to  pieces. 

He  expresses  the  value  of  steel  by  the  sum  of  the  tensile  strength  in  tons 
per  square  inch,  and  the  ductility  in  percentage  of  the  length,  found  by 
fracturing  specimens  of  the  form,  Fig.  220.  Thus,  for  steel  of  40  tons 
strength,  and  30  per  cent,  ductility,  the  resultant  value  is  (40  +  30  — )  70. 

1  The  materials  for  this  notice  are  derived  from  the  Proceedings  of  the  Institution  of 
Mechanical  Engineers,  1875,  page  268. 


STRENGTH  OF  STEEL  AND  IRON. 


6iS 


Table  No.  216. — WHITWORTH'S  FLUID-COMPRESSED  STEEL,  AND  BEST 
IRONS — TENSILE  STRENGTH. 

i.  FLUID-COMPRESSED  STEEL. 


Arbitrary  Distinguishing 
Colours  for  Groups. 

Ultimate 
Tensile 
Strength 
per  square 
inch. 

Ductility, 
or 
Elongation. 

Uses  to  which  the  Steel  is  applicable. 

tons. 

per  cent. 

(  Axles,    boilers,     connecting    rods, 

Red,  Nos.  i,  2,  3.... 

40 

32 

1      cross-heads,  crank-pins,  hydraulic 
j     cylinders,  cranks,  propeller  shafts, 

(      rivets,  tyres,  &c. 

C  Cylinder  linings,  slide-bars  for  loco- 

motives, shafting,  couplings,  drill- 

Blue,  Nos.  i,  2,3... 

48 

24 

-{      spindles,      eccentric  -  shafts     for 

punching  machines,  large  swages, 

[     hammers,  &c. 

(  Large  planing  and  lathe  tools,  large 

Brown,  Nos.  i,  2,  3, 

58 

I? 

<      shears,  drills,  smiths'  punches,  dies 

(      and  taps,  small  swages,  &c. 

Yellow,  Nos.  i,  2,  3, 

68 

IO 

(  Boring  tools,  finishing  tools  for  plan- 
\      ing  and  turning. 

Special  alloy  with  ) 
Tungsten  ) 

72 

H 

For  particular  purposes. 

Note. — In  each  group  No.  i  is  most  ductile,  No.  3  least  ductile. 
2.  IRON. 


DESCRIPTION. 

Ultimate  Tensile  Strength. 

ELONGATION. 

Several  Specimens. 

Averages. 

Several  Specimens. 

Averages. 

WROUGHT  IRON. 
Yorkshire     

tons  per  square  inch. 
(  31,  30,  29,  27,  ) 

{  27,  26.8,  26.8  1 

27,  24.8 
27,  26.8,  26,  25 
25,  24,  24,  24,  20 
26,  24,  24 

tons. 

28.3 

25.9 
26.2 

234 

24.7 

per  cent. 
(  23,  22,  31,  41,  ) 

(  22,  43,  42         f 

39,40 

39,  40,  41,  38 
35,  39,  34,  33,  '5 
30,  35,  28 

per  cent. 

32 
39-5 

39-5 
31.2 

3i 

Lowmoor  

Northamptonshire  .  .  . 
Staffordshire  

Do.    (Dudley  Ward). 
Averages  

25.7 

34-6 

CAST  IRON. 

(  13,  12,  II,  II,) 

\  10,  9-5,  7          } 

10.5 

(  .90,  1.  10,  1.00,  ) 
1  .6s,.75,.i2,.5o5 

.72 

6l6  THE    STRENGTH   OF   MATERIALS. 


CHERNOFFS   EXPERIMENTS   ON   STEEL.1 

Steel,  when  cast  and  allowed  to  cool  quietly,  assumes  a  crystalline  struc- 
ture. The  higher  the  temperature  to  which  it  is  heated,  the  softer  it  be- 
comes, and  the  greater  is  the  liberty  its  particles  possess  to  group  themselves 
into  crystals. 

Steel,  however  hard  it  may  be,  will  not  harden  if  heated  to  a  temperature 
lower  than  what  may  be  distinguished  as  dark  cherry-red  (temperature  a), 
however  quickly  it  is  cooled ;  on  the  contrary,  it  will  become  sensibly  softer, 
and  more  easily  worked  with  the  file. 

Steel  heated  to  a  temperature  lower  than,  say,  red  but  not  sparkling 
(temperature  &),  does  not  change  its  structure  whether  cooled  quickly  or 
slowly.  When  the  temperature,  in  rising,  has  reached  b,  the  substance  of 
steel  quickly  passes  from  the  granular  or  crystalline  condition,  to  the  amor- 
phous, or  wax-like  structure,  which  it  retains  up  to  its  melting  point  (tem- 
perature c}. 

The  points  #,  b,  and  c,  have  no  permanent  place  in  the  scale  of  temper- 
ature, but  their  positions  vary  with  the  quality  of  the  steel;  in  pure  steel, 
they  depend  directly  on  the  quantity  of  constituent  carbon.  The  harder 
the  steel,  the  lower  the  temperatures.  The  tints  above  specified  have 
reference  only  to  hard  and  medium  qualities  of  steel;  in  the  very  soft 
kinds  of  steel,  nearly  approaching  to  wrought  iron,  the  points  a  and  b 
range  very  high,  and  in  wrought  iron  the  point  b  rises  to  a  white  heat. 

The  assumption  of  the  crystalline  structure  takes  place  entirely  in  cooling 
between  the  temperatures  c  and  b-}  when  the  temperature  sinks  below  b 
there  is  no  change  of  structure.  For  successful  forging,  therefore,  the 
heated  ingot,  after  it  is  taken  out  of  the  furnace,  must  be  forged  as  quickly 
as  possible,  so  as  not  to  leave  any  spot  untouched  by  the  hammer,  where 
the  steel  might  crystallize  quietly,  but  that  the  formation  of  crystals  should 
be  hindered,  and  that  the  steel  should  be  kept  in  the  amorphous  condition 
until  the  temperature  sinks  below  the  point  b. 

Below  this  temperature,  if  the  piece  be  left  to  cool  in  quiet,  the  mass  will 
no  longer  have  a  disposition  to  crystallize,  but  will  possess  great  tenacity  and 
homogeneity  of  structure. 

When  steel  is  forged  at  temperatures  lower  than  b,  its  crystals  or  grains, 
being  driven  against  each  other,  change  their  shapes,  becoming  elongated 
in  one  direction  and  contracted  in  another;  whilst  the  density  and  the 
tensile  strength  are  considerably  increased.  But  the  available  hammer- 
power  is  only  sufficient  for  the  treatment  of  small  steel  forgings ;  and  the 
object  of  preventing  the  coarse  crystalline  structure  in  large  forgings  is 
more  easily  and  more  certainly  effected,  if,  after  having  given  the  forging 
the  desired  shape,  its  structure  be  altered  to  the  homogeneous  amorphous 
condition  by  heating  it  to  a  temperature  somewhat  higher  than  b,  and  the 
condition  be  fixed  by  rapid  cooling  to  a  temperature  lower  than  b.  The 
piece  should  then  be  allowed  to  finish  cooling  gradually,  so  as  to  prevent, 
as  far  as  possible,  internal  strains  due  to  sudden  and  unequal  contraction. 

1  Abstracted  from  Remarks  on  the  Manufacture  of  Steel,  and  the  Mode  of  Working  tt,  by 
D.  Chernoff,  1868;  translated  by  Mr.  William  Anderson,  C.E.,  1876.  Mr.  Anderson 
has  conferred  a  substantial  favour  upon  the  steel-manufacturing  and  steel-consuming  com- 
munity by  the  translation  and  circulation  of  this  valuable  document. 


TRANSVERSE   STRENGTH   OF   STEEL.  617 

STRENGTH  OF  STEEL  WIRE. 

Dr.  Pole  states  that  music-wire  has  a  resistance  equal  to  90  tons  per 
square  inch. 

Mr.  Roebling  states  that  steel  wire  has  been  manufactured  which  would 
resist  a  tensile  stress  of  300,000  Ibs.,  or  134  tons,  per  square  inch;  but 
not  in  large  quantity. 

Steel  wire,  No.  14  W.G.,  or  .085  inch,  about  X/I3  inch,  in  diameter,  made 
for  purposes  of  steam-ploughing,  has  a  tensile  resistance  of  from  2000  Ibs. 
to  2240  Ibs.,  equivalent  to  from  160  to  175  tons  per  square  inch. 

SHEARING  STRENGTH  OF  STEEL. 

The  ultimate  resistance  of  steel  to  shearing  stress  varies  from  69  to  78 
per  cent,  of  the  ultimate  tensile  strength  per  square  inch  of  section.  Mr. 
Kirkaldy  found,  for  16  specimens  from  a  bar  of  rivet  steel,  an  average  of 
73-5  Per  cent.;  and  the  same  for  12  specimens  of  Fagersta  steel.  Mr.  J.  T. 
Smith,  in  an  article  hereafter  noticed,  states  that  the  force  required 
to  punch  a  hole  ^  inch  in  diameter  through  the  |^-inch  webs  of  Bessemer 
steel  rails  varied  from  46^  tons  to  82^  tons,  according  to  the  hardness  of 
the  rail.  When  a  taper  of  x/i6  inch  was  allowed  in  the  hole,  the  shearing 
resistance  to  punching,  per  square  inch  of  surface  cut  through,  was  such  as 
to  average  70.14  per  cent,  of  the  tensile  strength  for  the  softer  steels,  and 
72.5  per  cent,  for  the  harder  steels. 

Upon  the  whole,  an  average  of  72  per  cent,  of  the  tensile  strength  may 
be  accepted  as  the  shearing  resistance  of  steel. 

TRANSVERSE  STRENGTH  OF  STEEL  BARS. 

The  instances  of  tests  for  the  transverse  strength  of  steel,  detailed  in 
previous  pages,  are  resumed  below,  showing  the  dimensions  of  specimens, 
with  their  average  ultimate  tensile  and  transverse  strengths.  The  transverse 
strengths,  also,  are  calculated  from  the  tensile  strength  by  formula  (  i  ), 
page  507,  and  entered  in  the  second  last  column  of  the  table.  The 
formula  is, 


W  =  the  breaking  weight  at  the  middle,  in  tons. 
#,  d,  /=the  breadth,  depth,  and  span,  in  inches. 

s  =  the  ultimate  tensile  strength,  in  tons  per  square  inch. 

Take  the  first  example  in  the  table:  —  1.75  inches  square,  25-inch  span, 
32.27  tons  tensile  strength. 

W  =  "55  *  '-75  3*  32.27  =  7<99  tons 

25 
Actual  weight  applied  =  7.35  tons  (uncracked). 

These  steels  show  a  still  closer  correspondence  of  the  calculated  to  the 
actual  strengths  than  was  shown  by  the  wrought  irons,  page  589.  Naturally, 
the  transverse  strength  for  uncracked  specimens,  as  calculated  above,  is 
somewhat  greater  than  the  observed  strengths,  since  the  strength  was  not 


6i8 


THE   STRENGTH   OF  MATERIALS. 


exhausted  by  actual  fracture.  The  averages  are  practically  identical,  and 
the  identity  of  the  calculated  with  the  experimental  strengths,  is  a  natural 
consequence  of  the  homogeneity  of  the  material. 


Number  and  Description 
of  Specimens. 

SECTION. 

Ultimate  Strength. 

Span. 

Tensile, 

Transverse. 

EFFECTS. 

Breadth. 

Depth 

per  sq. 

inch. 

Calculated. 

Actual. 

inch. 

inch. 

inch. 

tons. 

tons. 

tons. 

4,  Hematite,  page  591 
8,  Krupp,  "Jeddo,"' 
page  596,  

i-75 
1.50 

i-75 
1.91 

25 
10 

32.27 
42.07 

7-99 
26.59 

7.35 
27.14 

uncracked. 
fractured. 

4,  Krupp,  "Sultan," 
page  596,  

1-37 

1.76 

IO 

4I.I8 

20.19 

21.31 

fractured. 

18,  Bessemer,  p.  599,... 

1.90 

1.90 

20 

33-34 

13.24 

12.74 

uncracked. 

uncracked  in 

n,  Crucible,  p.  599,.. 

1.90 

1.90 

20 

36.30 

14.38 

15.04 

most 

instances. 

4,  Fagersta,  p.  604,  . 

1.90 

1.90 

20 

39.16 

I5-SI 

I5.6I 

half  fractured, 
half  uncracked. 

Averages, 

16.32 

16-53 

The  general  formula  (  i  )  may  be  adapted  for  steels  of  a  particular  tensile 
strength,  by  substituting  for  (1.155  s)  its  numerical  value.  Thus,  for  steel 
of  30  tons  tensile  strength,  1.155^=1. 155  x  30  =  34.6;  and 


w_34.6  bd* 

(  2  ) 

Ultimate 
Tensile  Strength, 
tons. 
•?O     . 

I 

Coefficient 
(1.155*)- 
.    34.6 

Ultimate 
Tensile  Strength, 
tons. 
42 

\  *  ) 

Coefficient 

(i-i55*). 

48  c; 

•?2 

•270 

44 

5°  8 

•24. 

•7Q    1 

A  e 

C  2  O 

OT-     ' 

?  r 

•    Oy'O 
4O  3 

T-J    

4.6 

5**" 

r  -3    T 

GO    • 

^6     . 

AI.6 

48 

DO'1 

r  r    A 

38  . 

.    4-3.0 

CQ 

JO-4 

e7  8 

40    . 

.    46.2 

Ov 

J  /<0 

RULE. — To  find  the  Ultimate  Transverse  Strength  of  Rectangular  Steel 
Bars. — Multiply  the  breadth  by  the  square  of  the  depth,  and  by  the  coeffi- 
cient (1.155  s)>  corresponding  to  the  ultimate  tensile  strength,  and  divide 
by  the  span.  The  quotient  is  the  breaking  weight  in  tons,  applied  at  the 
middle. 

Note. — To  find  the  coefficient  for  any  other  tensile  strength,  not  given 
above,  multiply  the  given  tensile  strength  by  1.155. 

TRANSVERSE  DEFLECTION  OF  STEEL  BARS. 

For  want  of  data,  it  is  assumed  that  the  deflection  of  steel  bars  is  to  that 
of  iron  bars  of  the  same  dimensions,  in  the  ratio  of  their  extensibilities, 
or  inversely  as  their  coefficients  of  elasticity.  From  the  results  of  experi- 
ments on  iron  and  on  steel  rails,  it  appears  that  the  coefficients  are  practi- 


TORSIONAL  STRENGTH   OF   STEEL. 


619 


cally  as  u  to  13.  Increasing,  therefore,  the  numerical  coefficients  for 
wrought-iron  bars,  in  formulas  (  5 )  and  (  6  ),  page  590,  in  this  ratio,  the 
following  formulas  are  deduced : — 

Elastic  Deflection  of  Uniform  Bars  of  Steel,  loaded  at  the  middle. 

Square  bars...  .  D  = (  i  } 

56,000  bd3 

Round  bars, D  =    ,W/3j7 (  4  ) 


38,000  d* 

D  =  the  deflection,  b  the  breadth,  d  the  depth,  /  the  span,  all  in  inches; 
W  =  the  weight,  in  tons. 

TORSIONAL  STRENGTH  OF  STEEL  BARS. 

The  torsional  resistances  of  steel,  already  recorded,  are,  with  the  ultimate 
tensile  and  shearing  strengths,  resumed  below,  and  the  calculated  resistances, 
by  formula  (  i  ),  page  534,  are  added  in  the  second  last  column.  The 
shearing  strength  is  taken  at  72  per  cent,  of  the  tensile  strength,  as  was 
settled,  page  617.  The  torsional  stress  was  applied,  in  the  following 
experiments,  at  the  end  of  a  1 2 -inch  lever.  The  formula  is, — 


W  =  the  breaking  stress  in  tons. 
h  =  the  shearing  strength  in  tons  per  square  inch. 
*/=the  diameter  in  inches. 
R  =  the  radius  of  the  force  in  inches. 
WR=  the  moment  of  the  force,  in  statical  inch-tons. 


Take  the  first  example  in  the  table  below:  —  i 
tons  shearing  strength,  and  a  1  2-inch  radius  :  — 


inches  in  diameter,  25.21 
Breaking  force,  by  formula^'278  x  I-253x25-2i  =  I>I4I  tons 


Description,  and  Number  of  Specimens. 

Diameter. 

Ultimate  Strength. 

Ultimate  Torsional 
Force. 

Tensile. 

Shearing. 

Calculated. 

Actual. 

4,  Hematite,  page  595, 
4,  KruPp,"Jeddo,"       „     596, 
4,  Krupp,  «  Sultan,"      „     596, 
1  8,  Bessemer,  „    600, 

inches. 
1.25 
1.25 
I.I28 
1.382 
1.382 
I.I28 
I.I28 
I.I28 
I.I28 
I.I28 

tons. 
32.27 
4I.l8 
42.07 

3343 
36.30 
38.04 
47.60 
45.82 

27.37 
39.16 

tons. 
25.21 

say  72% 
»      jj 
j>      )> 

V          » 

27.42 
35.60 

31.99 
20.28 
28.82 

tons. 
I.I4I 
1.342 
1.007 
1.472 

1.599 
.912 
I.I84 
1.064 
.674 
.958 

tons. 
1.030 
1.280 
1.068 
1.470 

1.610 

.946 
1.043 

1.009 

.679 
.919 

ii    Crucible,  „     600 

^  Fciffersta    a                     6oj. 

^,       Do.        b*  „     604, 

3       Do.        c*  ..             ,    604., 

3,       Do-        d,  „     604, 
12,       Do.        average,  

Averages  of  all 

I.I35 

1.105 

62O  THE   STRENGTH   OF   MATERIALS. 

The  results  of  experiment  and  of  calculation  show  a  close  correspondence. 

When  the  shearing   strength   is   not  known  experimentally,   substitute 

.72  s,  or  72  per  cent,  of  the  tensile  strength  s,  for  h  in  the  formula;  and 


(6) 

(7) 


R 

Ultimate  Torsional  Strength  of  Steel  Bars. 


When  the  tensile  strength,  s,  is  30  tons,  then  —  =  6,  and 


Generally,  for  tensile  strengths  of  from  30  to  50  tons,  the  values  of  the 
numerical  coefficients  in  formulas  (  7  )  and  (  8  ),  are  as  follows  :  — 

Tensile  Strength.  Coefficient.  Tensile  Strength.  Coefficient. 

tons.  (.20  s.)  tons.  (.20$). 


30  6 

32  6.4 

34  6.8 

35  7 

36  7-2 

38  7-6 


42  8.4 

44  8.8 

45  9 

46 9-2 

48  9.6 

50  10 


40     8 

Elastic  Torsional  Strength  of  Steel  Shafts. 

Hematite  steel 41.5  per  cent,  of  ultimate  strength. 

Krupp,  "Jeddo" 38.4 

Krupp,  "  Sultan  " 47.3 

Bessemer 44.6 

Crucible 42.5 

Fagersta  (average) 50. 2 


Average 44.1  percent. 

ELASTIC  TORSIONAL  SHEARING  STRESS  AND  DEFLECTION  OF  STEEL  BARS. 
The  elastic  shearing  stress  /z,  is  found  by  formula  ( 3  ),  page  535. 

h      WR  /     \ 

h  =  ^j*d* (9) 

For  Hematite  steel,  for  example,  page  595,  W  R  —  .428  ton  x  12  inches  = 
5.136,  the  moment  of  the  force,  and  h  —  — 5i_2 —  ^46  tons  per  square 

inch,  the  elastic  limit  of  shearing  stress.     The  coefficient  of  torsional  elas- 
ticity, E',  as  defined  at  page  536,  is  found  by  formula  (  12  ),  page  537 : — 

E'-  QW  */•   -     -4»8x  i,x  10  I2)forHematite  steel. 

.873^/4D        .873  X   I.254X  .008 


STRENGTH   VARIES   WITH   CONSTITUENT  CARBON. 


621 


For  the  several  steels,  the  elastic  shearing  stress  and  coefficient  of  elas- 
ticity, calculated  in  the  same  way,  are  as  follows : — 


Steels. 

Specimens. 

Diameter  and 
Length  for 
Observation. 

Elastic 
Shearing 
Stress  per 
square  inch. 

Coefficient 
of 
Elasticity. 

Hematite,  page  595 

inches.      inches. 

1.25    x  10 

tons. 
9.46 

E' 
3012 

Krupp,  "Jeddo,"  „     596 
Krupp,  "Sultan,"....     „     596 
Bessemer,  „     600 

1.25     X2.5 
I.I28  X2.25O 

1.382  x  ii 

10.85 

13-95 
10.73 

1382 
865 

2472 

Crucible,  „     600 

I.382X  ii 

11.19 

2025 

Fasrersta  a                           604 

.128  X  Q 

1C.  2^ 

Do        b                           604 

.128  x  o 

I  S.IO 



Do.       c,   „     604 

.128x9 

•>      .. 
14.56 



Do        d                           604. 

.128  X  Q 

IO.2C, 

Do.      average,  „     604 

.128x9 

13-77 

— 

Omitting  the  coefficients  of  elasticity  for  the  "Jeddo"  and  the  "Sultan," 
as  the  specimens  were  very  short,  the  average  of  the  remaining  three  co- 
efficients is  2503;  and  the  value  of  .873  E'  in  formula  (  10),  page  537,  is 
(.873  x  2503  =  )  2185;  say  2200.  Whence,  by  substitution: — 


Elastic  Torsional  Deflection  of  Steel  Bars. 
WR/ 


D 


2200  */4 


(10) 


D  =  the  total  angular  deflection  in  parts  of  a  revolution. 
W  =  the  twisting  force  in  tons. 
R  =  the  radius  of  the  force  in  inches. 
W  R  =  the  moment  of  the  force  in  statical  inch-tons. 

/=the  length  of  the  shaft  under  torsional  stress  in  inches. 
</=the  diameter  of  the  shaft  in  inches. 

STRENGTH  OF  STEEL  RELATIVELY  TO  THE  PROPORTION  OF  CONSTITUENT 

CARBON. 

Mr.  F.  W.  Webb  produces  steel  for  boiler  plates  having  a  tensile  strength 
of  28  tons  per  square  inch,  and  containing  J/sth  per  cent,  of  constituent 
carbon. 

Mr.  T.  E.  Vickers  tested  the  tensile  strength  of  steel  of  various  degrees  of 
carbonization,  ranging  from  No.  2,  having  0.33  per  cent.,  to  No.  20,  having 
1.25  per  cent,  of  carbon.1  The  specimens  were  turned  to  i  inch  in 
diameter,  and  to  a  length,  for  observation,  of  14  inches.  The  results  of  the 
tests  are  given  in  table  No.  217. 

The  table  shows  that  the  tensile  strength  of  steel  is  increased  by  the 
addition  of  carbon,  until,  with  i^  per  cent,  it  amounts  to  69  tons  per 
square  inch.  The  elongation  is,  at  the  same  time,  reduced.  But,  beyond 


1  See  Mr.  Vickers'  paj 
Mechanical  Engineers,  I  \ 


;r  on  the  "Strength  of  Steel 
>i,  page  158). 


(Proceedings  of  the  Institution  of 


622 


THE   STRENGTH   OF   MATERIALS. 


the  last  degree  of  carbonization,  i%  per  cent.,  the  steel  becomes  gradually 
weaker,  until  it  reaches  the  form  and  strength  of  cast  iron. 

Table  No.  217. — TENSILE  STRENGTH  OF  STEEL  CONTAINING  DIFFERENT 
PROPORTIONS  OF  CARBON. 

Mr.  T.  Edward  Vickers. 


Description  of 
Steel. 

Proportion  of 
Carbon! 
(approximate). 

Breaking 
Weight  per 
square  inch. 

ELONGATION. 

No.    2 

No     4 

per  cent. 

...         -33       ••• 

43 

tons. 
...       30.4       ... 
34  O 

inches. 

...     .37,  or  9.8  per  cent. 
•37  or  o  8 

No.    5 
No     6 

...         .48       ... 

C  -3 

...     37-5     ..- 

42  ? 

...     .25,  or  8.9       „ 
12  or  8  o 

No.    8 
No.  10 
No.  12 
No   15 

...              .63          ... 
...              .74          ... 
...              .84          ... 

I  OO 

...       45.0       ... 

...       45-5       -.. 
...       55.0       ... 
DO  O 

...     .00,  or  7.1       „ 
...     .69,  or  5.0       „ 
...     .12,  or  8.0       „ 
oo  or  5  o 

No  20 

I  25 

60  o 

62  or  4  4 

A  specimen  bar  was  turned  down  to  a  diameter  of  ^  mcn  at  the  middle, 
so  as  to  form  a  circular  notch.  On  being  tested,  it  broke  with  79^  tons 
per  square  inch,  whilst  the  ordinary  specimen  bar  of  the  same  steel  broke 
with  60  tons  per  square  inch. 

Mr.  Webb's  datum  above  given  is  in  harmony  with  Mr.  Vickers'  data. 

See  also  on  this  subject  RAILWAY  RAILS,  at  page  664. 


RESISTANCE  OF  STEEL  AND  IRON  TO  EXPLOSIVE  FORCE. 

Sir  Joseph  Whitworth  tested  iron  and  steel  by  the  explosive  force  of 
gunpowder.  The  specimens  were  cylinders  having  a  bore  of  ^  inch,  a 
diameter  outside  of  i  j£  inches,  and  a  length  of  4  inches.  They  were  made 
open  at  the  ends,  and  were  closed  for  the  purpose  of  the  experiments. 

Table  No.  218. — RESISTANCE  OF  IRON  AND  STEEL  TO  EXPLOSIVE  FORCE. 


METAL. 

Charge  of  Powder. 

Expansion 
in  diameter 
at  middle 
before 

Number 
of  pieces 
when 

bursting. 

burst. 

grains. 

ratio. 

inch. 

pieces. 

Cast  iron                                     

I  c 

I 

OOOO 

"^6 

Wrought  iron,  Staffordshire,  coiled  
Fluid  compressed  steel,  No.  3,  red  

95 

275 

6-3 
18.3 

.0997 
.1659 

5 

2 

Do.              do.           No.  3,  brown  .. 

325 

21.7 

.0950 

4 

1  The  intermediate  percentages  of  carbon  in  column  2,  from  No.  4  to  No.  15  inclusive, 
are  merely  approximate,  having  been  interpolated  in  proportion  to  the  Nos.  of  the  steel. 


RECAPITULATION   OF  DATA.  623 

RECAPITULATION   OF  DATA  ON  THE  DIRECT  STRENGTH 
OF   IRON   AND   STEEL. 

Cast  Iron,  pp.  553  to  561. — The  ultimate  tensile  strength  ranges  from 
5  to  7^  tons  per  square  inch:  first  meltings,  specimens  under  i  inch 
in  thickness.  For  thicker  castings  the  strength  diminishes.  The  com- 
pressive  strength  is  from  four  and  a  half  to  about  seven  times  the  tensile 
strength.  For  general  calculations,  say,  tensile  strength  7  tons,  compressive 
strength  49  tons. 

The  ultimate  tensile  strength  is  increased  by  repeated  remeltings  to  from 
15  to  20  tons  per  square  inch;  and  the  compressive  strength  to  from  70  to 
80  tons. 

The  elastic  strength  practically  is  equal  to  the  ultimate  tensile  strength. 

Wrought  Iron,  pp.  567  to  591. — The  ultimate  tensile  strength  of  rolled 
bar  iron  varies  from  22^  to  30  tons;  rivet-iron  from  24  to  27  tons. 
Plates  from  20  to  23  tons;  about  i  ton  less  crossway  than  length  way  of 
the  fibre.  The  strength  is  reduced  more  than  i  ton  by  annealing.  The 
resistance  to  compression  is  an  indefinite  quantity. 

The  elastic  tensile  strength  of  iron  bars  averages  not  less  than  50  per 
cent,  of  the  ultimate  strength;  and  that  of  iron  plates  is  generally  from 
55  to  60  per  cent,  of  the  ultimate  strength. 

The  elastic  strength  of  bars  and  plates,  both  tensile  and  compressive, 
may  be  taken  at  1 2  tons. 

The  elongation  of  wrought-iron  bars,  within  the  elastic  limit,  is  at  the 
rate  of  x/IO>ooo  to  VIS.QOO  part  of  the  length — say,  an  average  of  x/ia,o<x>  part — 
per  ton  per  square  inch;  or  a  total  of  X/IOOO  part  of  the  length.  The  same 
fraction  may  be  taken  for  compression  within  the  elastic  limit. 

Approximate  Strength  of  Wrought-iron  Bars  in  Terms  of  the  Circular 
Inch  (Mr.  E.  Clark). — "  A  strength  of  20  tons  per  square  inch  is  nearly 
equivalent  to  one  of  16  tons  per  circular  inch.  An  ordinary  i-inch  round 
rod  bears  tensilely  16  tons,  and  weighs  8  Ibs.  per  yard. 

"For  a  round  rod  of  any  diameter,  the  square  of  the  diameter,  in 
quarter-inches,  is  the  breaking  weight  in  tons. 

"  Half  this  quantity  is  the  weight  in  pounds  per  yard. 

"  A  rod  will  be  perceptibly  damaged  by  half  this  stress,  which  can  never 
be  safely  exceeded;  one-third  being  sufficient  in  practice." 

Steel,  pp.  593  to  615. — By  Mr.  Kirkaldy's  earliest  experiments,  it  was 
found  that  the  average  tensile  strength  of  bar  steel  varied  from  60  tons 
for  tool -steel,  to  28  tons  for  puddled  steel;  and  that  of  steel  plates 
from  3/l6  to  s/i6  inch  thick,  from  32  to  45%  tons. 

From  subsequent  experiments,  it  appears  that  the  ultimate  tensile  strength 
of  rolled  bar  steel  varies  from  30  to  50  tons  per  square  inch.  The  average 
tensile  strength  may  be  taken  at  35  tons,  and  the  elastic  strength,  tensile 
and  compressive,  at  20  tons.  The  tensile  and  the  elastic  compressive 
strength  of  hammered  steel  bars  is  from  4  to  5  tons  more  than  that  of 
rolled  bars. 

By  annealing,  the  elastic  strength  of  rolled  steel  bars  is  reduced  3  tons, 
and  that  of  hammered  bars  5  tons. 

Steel  plates  have  elastic  tensile  and  compressive  strengths  averaging  about 
14  tons;  the  ultimate  tensile  strength  is  from  22  to  32  tons,  according  to 


624 


THE   STRENGTH   OF   MATERIALS. 


the  proportion  of  constituent  carbon.  The  strength  is  the  same  lengthwise 
and  crosswise. 

Annealing  reduces  the  tensile  strength  of  steel  plates,  elastic  and  ultimate, 
by  i  ^  or  2  tons ;  and  the  elastic  compressive  strength  by  twice  as  much. 

The  elongation  and  the  compression  of  steel  bars  within  the  elastic  limit 
may  be  taken  at  Vi 3,000  part  of  the  length  per  ton  per  square  inch;  or  a 
total  of  Viooo  part  of  the  length. 


Inches 


Sfress  per  S<fiuzr& 


Fig.  222.  —  Relative  Elongation  of  ic-feet  Bars  of  Cast  Iron,  Wrought  Iron,  and  Steel. 

The  comparative  behaviours  of  bars  of  cast-iron,  wrought-iron,  and  steel, 
TO  feet  in  length,  under  tensile  stress,  as  previously  recorded,  is  shown 
diagrammatically  in  Fig.  222,  annexed.  It  is  seen  that  the  extension  of 
cast  iron,  which  breaks  under  a  tensile  stress  of  7  tons  per  square  inch,  is 
very  limited,  and  that  the  rate  of  extension  is  nearly  uniform  concurrently 
with  the  stress;  whilst  the  wrought-iron  and  the  steel  bars,  which  were 
broken  at  28  tons  and  52^  tons  per  square  inch  respectively,  suddenly 
acquire  a  greatly  increased  rate  of  elongation  at  the  yielding  points,  which 
are  arrived  at  when  about  half  the  breaking  stress  has  been  applied.  It  is 
not  necessary  to  enter  into  more  detail,  as  the  diagram  is  only  intended 
to  show  the  leading  characteristics  of  the  three  metals  under  tensile 
stress. 


FACTORS   OF   SAFETY.  625 

WORKING  STRENGTH   OF  MATERIALS— FACTORS   OF 

SAFETY. 

The  elastic  strength  of  materials,  cast-iron  excepted,  is,  in  general  terms, 
half  of  their  ultimate  or  breaking  strength.  For  cast-iron,  though  there  is 
no  clearly  defined  elastic  limit,  the  same  measure  may  be  adopted.  If  a 
working  load  of  half  the  elastic  strength,  or  one-fourth  of  the  ultimate 
strength,  be  accepted,  equal  range  for  fluctuation  within  the  elastic  limit  is 
provided.  But,  as  bodies  of  the  same  material  are  not  uniform  in  strength, 
it  is  necessary  to  observe  a  lower  limit  than  a  fourth  where  the  material  is 
exposed  to  great  or  to  sudden  variations  of  load. 

Cast-iron. — Mr.  Stoney  recommends  one-fourth  of  the  ultimate  tensile 
strength,  for  dead  weights;  one-sixth  for  cast-iron  bridge  girders;  and  one- 
eighth  for  crane-posts  and  machinery.  In  compression,  free  from  flexure, 
according  to  Mr.  Stoney,  cast-iron  will  bear  8  tons  per  square  inch;  for 
cast-iron  arches,  3  tons  per  square  inch;  for  cast-iron  pillars,  supporting 
dead  loads,  one-sixth  of  the  ultimate  tensile  strength;  for  pillars  subject  to 
vibration  from  machinery,  one-eighth;  and  for  pillars  subject  to  shocks 
from  heavy-loaded  waggons  and  the  like,  one-tenth,  or  even  less  where  the 
strength  is  exerted  in  resistance  to  flexure. 

Wrought-iron. — For  bars  and  plates,  5  tons  per  square  inch  of  net 
section  is  taken  as  the  safe  working  tensile  stress;  for  bar  iron  of  extra 
quality,  6  tons.  In  compression,  where  flexure  is  prevented,  4  tons  is  the 
safe  limit;  in  small  sizes,  3  tons.  For  wrought-iron  columns,  subject  to 
shocks,  Mr.  Stoney  allows  a  sixth  of  the  calculated  breaking  weight;  with 
quiescent  loads,  one-fourth.  For  machinery,  an  eighth  to  a  tenth  is  usually 
practised ;  and  for  steam-boilers,  a  fourth  to  an  eighth. 

Mr.  Roebling  says,  "Long  experience  has  proved,  beyond  the  shadow  of 
a  doubt,  that  good  iron,  exposed  to  a  tensile  strain  not  above  one-fifth  of 
the  ultimate  strength,  and  not  subject  to  strong  vibration  or  torsion,  may 
be  depended  upon  for  a  thousand  years." 1 

Steel. — A  committee  appointed  by  the  Board  of  Trade  recommended 
that  a  stress  of  6^  tons  per  square  inch  should  not  be  exceeded  in  bridge 
work  for  railways.  Mr.  Stoney  recommends,  for  mild  steel,  a  fourth  of  the 
ultimate  tensile  strength,  or  8  tons  per  square  inch.  The  limit  for  com- 
pression must  be  regulated  very  much  by  the  nature  of  the  steel,  and  whether 
it  be  unannealed  or  annealed.  Probably  a  limit  of  8  tons  per  square  inch, 
the  same  as  the  limit  for  tension,  would  be  the  safe  maximum  for  general 
purposes.  In  the  absence  of  experience,  Mr.  Stoney  recommends  that,  for 
steel  pillars,  an  addition  not  exceeding  50  per  cent,  should  be  made  to  the 
safe  load  for  wrought-iron  pillars  of  the  same  dimensions. 

Timber. — One-tenth  of  the  ultimate  stress  is  an  accepted  limit.  Timber 
piles  have,  in  some  situations,  borne  permanently  one-fifth  of  their  ultimate 
compressive  strength. 

Foundations. — According  to  Professor  Rankine,  the  maximum  pressure 
on  foundations  in  firm  earth  is  from  17  Ibs.  to  23  Ibs.  per  square  inch;  and 
he  says  that,  on  rock,  it  should  not  exceed  one-eighth  of  the  crushing  load. 

Masonwork. — Mr.  Stoney  says  that  the  working  load  on  rubble  masonry, 

1  Engineering,  August  16,  1867. 

40 


626 


THE   STRENGTH   OF   MATERIALS. 


brickwork,  or  concrete  rarely  exceeds  one-sixth  of  the  crushing  weight  of 
the  aggregate  mass;  and  that  this  seems  to  be  a  safe  limit.  In  an  arch,  the 
calculated  pressure  should  not  exceed  one-twentieth  of  the  crushing  pres- 
sure of  the  stone. 

Ropes. — For  round  ropes,  the  working  load  should  not  exceed  a  seventh 
of  the  ultimate  strength;  and  for  flat  ropes,  one-ninth. 

Dr.  Rankine l  gives  the  following  data  as  factors  of  strength : — 


Dead  Load. 

Factors  of  safety  for  perfect  materials  ) 

and  workmanship / 

For  good  ordinary  materials  and  work- 
manship : — 

Metals 3 

Timber 4  to  5 

Masonry 4 


Live  Load. 


6 
8  to  10 


A  dead  load  on  a  structure  is  one  that  is  put  on  by  imperceptible  degrees, 
and  that  remains  steady;  such  as  the  weight  of  the  structure  itself. 

A  live  load  is  one  that  is  put  on  suddenly,  or  is  accompanied  with  vibra- 
tion ;  such  as  a  swift  train  travelling  over  a  railway  bridge,  or  a  force  exerted 
in  a  moving  machine. 


STRENGTH    OF   COPPER   AND    OTHER   METALS. 

Table  No.  219. — ULTIMATE  TENSILE  STRENGTH  OF  COPPER  AND  ITS 
ALLOYS,  AND  OTHER  METALS. 


DESCRIPTION  OF  METAL. 

Speci- 
fic 
Gravity 

Ultimate 
Tensile  Strength 
per  square 

inch. 

Experimentalist. 

Copper,  wrought  

tons. 
I5.OO 

Dr.  Anderson 

Do      cast 

8.48  to  1  1.  67 

Do.      ordinary  bolts  

16  oo 

)) 

Do.     bolts,  with  i  per  cent,  phosphorus... 

8.202 

7.56 

}> 

Do.            do.        i                  „ 

8.592 

16.47 

Do.            do.        1.5               „ 

8.876 

17-13 

Do.            do.        i                   „ 

— 

19.20 

Do.            do.        2                  „ 

8.614 

20.25 

Do.            do.        i                  „ 

— 

20.34 

Do.            do.        2                  „ 

8.580 

20.41 

Do.            do.        2                  „ 

8.615 

20.27 

Do.             do.        3 

8.422 

21.38 

Do.             do.        4                  „ 

— 

22.32 

Gun  met3l   12  copper  i  tin 

12  Q/t 

i  ^.y^. 

j> 

Do         ii                i 

I  3  71 

Do         10                i 

1  j-  /  • 
14.  T\ 

» 

Do           Q                i 

T"  /  J 
I7.OO 

jj 

)> 

1  Useful  Rules  and  Tables,  page  205. 


STRENGTH   OF  COPPER   AND  OTHER  METALS. 
Table  No.  219  (continued). 


627 


DESCRIPTION  OF  METAL. 


Speci- 

fic 
Gravity 


Ultimate 

Tensile  Strength 

per  square 

inch. 


Experimentalist. 


Gun  metal,  average  strength  of  good  bronze 
Gun  metal,  average  results  of  tests  oH 

specimens   from  bronze  guns — elastic  > 

strength,  6.56  tons ) 

Gun  metal,  American  guns — 

Gun-heads 

Breach-squares 

Small  bars  cast  in  same  moulds  with  guns 

Small  bars  cast  separately  in  iron  moulds 
Do.  do.  in  clay  moulds 

Finished  guns 


Alloys  of  copper  and  tin,  unwrought  — 

Equivalents.  By  weight. 

iodi+     Sn,  84.  29  copper  +  15.71  tin,  gun  metal... 
Sn,  82.81  +17.19 


9Cu+ 
8Cu+ 
7Cu+ 

Cu+ 


Sn,  81.10 

Sn,  78.97 
Sn,  34.92 
Sn,  15.17 
Sn,  o 


+1.90 
+21.03 
+65.08 
+84.83 
+ioo. 


brasses  ...... 

small  bells.  . 

speculummet 

tin  ........... 


8.523 
8.765 
8.584 

8-953 
8.313 


8.561 
8.462 

8-459 
8.728 
8.056 

7-447 
7.291 


tons. 
1473 

12.19 


13.24 
20.76 
18.76 
16.82 
11.51 
10.3  to  23.3 


16.1 

15.2 
17.7 

13.6 

1.4 
3-1 
2-5 


Aluminium  bronze— 90  copper,  i  aluminium  32.67 

Do.                  maximum 43.00 

Tin,  cast .        2.11 

Do.  Banco 7.297             .95 

Lead,  cast —               .81 

Do.    sheet —              .86 

Lead  pipe i.oo 

Zinc,  cast —           1.336 

Soft  solder — 2  tin,  i  lead 

Brass,  fine  or  yellow —             8.02 

Brass,  fine  or  yellow,  2  copper,  i  zinc 12.90 

Brass  tube,  62  copper,  38  zinc 46.00 

Do-     ,,     70      „        30    „    36-00 

Do.  wire 

Muntz's  metal — 3  copper,  2  zinc —           22.00 

Alloys  of  copper,  zinc,  iron,  and  tin—"  Sterro- 
metal " — 

Copper  10,  iron  10,  zinc  80 7.000           3.17 

Do.    60,     „      3,     „    39,  tin  i. 5 24.00 

Do.    60,     „      4,     „    44,   „   2:— 

Cast  in  sand —           19.25 

Cast  in  iron,  annealed , 24.25 

Cast  in  iron,  forged  red  hot —           31.00 

Copper  60,    iron  2,      zinc  37,     tin  i —           34.0 

Do.    60,      „     2,        „     35,       „    2 —           38.0 

Do.     55.0,,,     1.77,,,     42.36,  ,,0.83:— 

Cast 27.0 

Forged  red  hot —           34.0 

Drawn  cold —           38.0 


Dr.  Anderson 


Wade 

)) 


Mallet 


Dr.  Anderson 

Rennie 

Wade 

Rennie 

Navier 

Jardine 

Stoney 

Rankine 

Rennie 

Dr.  Anderson 

Everitt 

» 

Dufour 
Dr.  Anderson 


Dr.  Anderson 


628 


THE   STRENGTH   OF   MATERIALS. 


Table  No.  220. — PHOSPHOR-BRONZE,  BRONZE,  AND  BRASS.     FROM  LIEGE. 

TENSILE  STRENGTH. 


Elastic 

Ultimate 

Ratio 

S< 

:t. 

Report.) 

Strength 
per 

Strength 
per 

of  Elastic 
to 

At  20,000 

Sectional 
Area  of 

DESCRIPTION. 

square 
inch. 

square 
inch. 

Ultimate 
Strength. 

Ibs.  per 
square 
inch. 

Ultimate. 

Fracture. 

PHOSPHOR-BRONZE. 
Lowest  values 

tons. 
A  777 

tons. 
Q  712 

per  cent. 
31  t\ 

per  cent. 
OQ 

per  cent. 
36 

per  cent. 
0,6  I 

Highest  values 

4-/// 
10.625 

22.73O 

685 

.wy 
c  I  3 

33  4. 

68  I 

Averages  of  1  2  specimens 

7482 

15.386 

48.6 

i-59 

II.4 

87.4 

ORDINARY  BRONZE. 
Lowest  values 

7-221 

Q  06  1 

667 

18 

I  2 

08  ; 

Highest  values 

8  7Q4. 

:7' 
I^.lSA 

S;.Q 

I   IO 

4.  O 

QI  6 

Averages  of  6  specimens 

"•/y*t 

8.095 

10.582 

°>V 

76.5 

.56 

2.23 

y*.v» 

95-6 

BRASS 

A   AIQ 

12  284 

367 

q  80 

16  i 

81  7 

TENSILE   STRENGTH   OF   WIRE   OF   VARIOUS    METALS. 

M.  Baudrimont,  in  1835,  tested  the  strength  of  annealed  metallic  wires 
at  various  temperatures,  from  32°  F.  to  392°  F.1  The  wires  of  gold, 
platinum,  copper,  silver,  and  palladium  were  about  Yeoth  inch  in  diameter; 


the  iron  wires  were   Vi45tn  mc^  m  diameter.     The  results  of  the  tests  are 

Table  No.  221.  —  TENACITY  OF  METALLIC  WIRES  AT 
VARIOUS  TEMPERATURES.     1835. 


Dia- 

Ultimate  Tensile  Strength. 

Tensile  Strength  per 
square  inch. 

METAL. 

of  wire 

Area. 

at  61°  F. 

At  32°  F. 

At 

212°  F. 

At 
392°  F. 

At32°F. 

At 

212°  F. 

392°  F. 

inch. 

sq.  inch. 

Ibs. 

Ibs. 

Ibs. 

tons. 

tons. 

tons. 

Gold  

.0162 

.000207 

|       S.6I 

i       5-42 

4.64 
4-49 

3-86 
3.80 

12.  1 
II.7 

10.0 

9-7 

8-3 
8.2 

Maximum 
Minimum 

Platinum 

.Ol6l 

.000205 

j      6.70 
f     6.59 

5-94 
5.61 

5-27 
5-03 

14.6 
14.4 

13.0 

12.2 

"•5 

II.  0 

Maximum 
Minimum 

Copper  ... 

.0177 

.OOO247 

IO.  II 
10.02 

8.80 
8.73 

7.92 
7.27 

18.3 

18.1 

16.0 

I5.8 

14.4 

13  * 

Maximum 
Minimum 

Silver  

•0157 

.OOOI93 

7.86 
7.78 

6.74 
6-39 

5-13 

5.10 

18.1 
18.0 

15-6 
I4.8 

II-9 

ii.  8 

Maximum 
Minimum 

Palladium 

.0156 

.000192 

IO.  12 
9.98 

9.00 
8.8q 

7-99 
7.41 

23-5 
23.1 

2O.9 
20.6 

18.5 
17.2 

Maximum 
Minimum 

Iron  

.0069 

.000373 

II.  12 

10.89 

10.66 
10.  16 

11.31 
11.15 

133-2 
I30-3 

127.6 
121.7 

J35-4 
133-5 

Maximum 
Minimum 

1  Annales  de  Chimie,  1850. 


STRENGTH,  OF   STONE,   BRICKS,   ETC. 


629 


arranged  in  table  No.  221,  and  it  is  shown  that,  ist,  the  tenacity  varies 
with  the  temperature;  2d,  it  decreases  as  the  temperature  rises,  except  for 
iron;  3d,  that  iron  presents  a  peculiar  case.  At  212°  F.  its  tenacity  is 
less  than  at  32°  F.,  but  at  392°  F.  it  is  greater. 

Table  No.  222. — TENSILE  STRENGTH  OF  WIRE — PHOSPHOR-BRONZE, 
COPPER,  BRASS,  STEEL,  AND  IRON. 


(Reduced  from  Mr. 
Kirkaldy's  Report.) 

Unannealed. 

Annealed. 

Diameter. 

Ultimate 
Tensile 

Strength. 

Diameter. 

Ultimate 
Tensile 
Strength. 

Ulti- 
mate 
Exten- 
sion. 

DESCRIPTION  OF  WIRE. 

Total. 

Per 

square 
inch. 

Total. 

Per 

square 
inch. 

Phosphor-Bronze  — 
Lowest  values  ... 

inch. 

.0585  or  1/17.1 
.0665  or  1/15 

.063    or  1/15.1 

Ibs. 

394 

tons. 

43-59 

71.21 
56.28 

inch. 

.1070  or  J/9.3 
.1125  or  1/9 

.1108  or  1/9 

Ibs. 
527 

tons. 

22.58 
28.82 

24.42 

p.  c'nt. 

33-o 
46.6 

39-° 

Highest  values  
Averages    of    20  ) 
specimens  .  .      \ 

CoDoer... 

.0640  or  1/15.6 
.  .0605  or  1/16.5 
.0600  or  1/16.7 
.0580  or  1/17.2 
j.  0580  or  i/I7.2 

203 

233 
342 
170 
174 

28.18 
36.23 
54.07 
28.71 
29.40 

.0640  or  i/is.e 
.0605  or  1/16.5 
.0600  or  Vi6.7 
.0580  or  1/17.2 
.0580  or  i/I7.a 

119 
148 
211 
162 
122 

l6.52 
23.01 
33-32 
27.36 

20.  6  1 

34-1 
36.5 
10.9 
17.1 
28.0 

Brass 

Steel.  . 

Iron,  galvanized,  BBC 
Do.         do.         BCE 

STRENGTH   OF   STONE,   BRICKS,   &c. 
Table  No.  223. — TENSILE  STRENGTH  OF  STONE,  BRICKS,  AND  CEMENT. 


DESCRIPTION  OF  MATERIAL. 

Weight  per 
cubic  foot. 

Ultimate  Ten- 
sile Strength 
per  sq.  inch. 

Experimentalist. 

Sandstone  

Ibs. 

tons. 
KO 

Buchanan 

Whinstone  ,  

.i}^ 

6iu 

Arbroath  pavement  

.\j^ 
c6^ 

Caithness      do  

O'-'J 

4.71 

White  Marble 

•322 

Do  

.$6* 

24.6 

Hodgkinson 

Flint-glass  rod,  annealed  

I  O7 

Fairbairn 

Green  glass  rod  

I.2Q 

White  crown-glass  rod  

I  14. 

Thin  glass  globes,  cohesion  

2  2"? 

Plaster  of  Paris  

Ibs. 
71 

Rondelet 

Mortar  of  quartzose  sand  and  hydraulic  ) 
lime  \ 

— 

136  to  85 

Vicat 

Mortar  of  quartzose  sand  and  ordin-  ) 
ary  lime                                                  C 



21  tO  51 

» 

630 


THE   STRENGTH   OF   MATERIALS. 
Table  No.  223. — (continued). 


DESCRIPTION  OF  MATERIAL. 

Weight  per 
cubic  foot. 

Ultimate  Ten- 
sile Strength 
per  sq.  inch. 

Experimentalist. 

Adhesion  of  Plaster  of  Paris  to  brick  ) 
or  stone  —  average  .  .                           ( 

Ibs. 

Ibs. 
50 

Rondelet 

Adhesion  of  bricks  cemented  with  Port-  } 
land  cement,  12  months  old,  and  i  > 
cement  to  i  sand                  .               \ 

neat 

I  tO  I 

Gault-clay  bricks,  pressed;  in  air... 
Do.                 do.               in  water 
Gault-clay  bricks,  wire  cut  ;  in  air... 
Do.                 do.               in  water 
Gault-clay  bricks,  perforated;  in  air.. 
Do                         do.        in  water 
Stock  bricks  in  air 

41 
46 

68 

47 
108 
84 
78 

44 
46 
43 
39 
83 

g 

Grant1 
j> 
)) 
)) 
)) 
» 

Do            in  water  . 

/° 
06 

WJ 

7O 

» 

Staffordshire  blue  brick,  pressed  ) 
with  frog  ;  in  air  J 

y° 

74 

/y 

56 

» 
)) 

Do    in  water 

7fi 

07 

Do.  rough,  without  frog;  in  air.... 
Do.     do.               do.            in  water 
Fareham  red  bricks;  in  air  

/u 

48 

40 

126 

Ol 

47 
29 
8^ 

)) 
)) 

jj 

Do.            do.          in  water  

123 

?* 

62 

•>•> 

Portland  cement  :  — 
Seven-day  tests  

l^J 

862  to  408 

)> 

Average  of  do.  per  bush.  115.2  Ibs... 
Portland   cement,    123   Ibs.  per  bushel, 
mixed  with  equal  weight  of  Thames 
sand  :  — 
Age  in  water,  7  days  

90 

neat  cement. 

-,5-? 

358.5 

cement  &  sand 

TC7 

)) 
5? 

Do.          i  month  

^  3 

AID 

*^/ 

2OI 

Do.          6     do  

C2"? 

284 

Do.        12     do  

CA7 

3IQ 

Do.          2  years  

t' 
OOO 

•3  C  I 

Do.          4    do  

eg? 

*9 

l6  1 

Do.         7    do  

j'-'j 

COO 

l8A 

Portland   cement,   112  Ibs.  per  bushel, 
mixed  with  various   proportions   of 
sand;  12  months  old:  — 
3  sand,  i  cement  

J7^ 

241 

5   do.     i     do  

2T/I 

7   do.     i     do  

161 

55 

Roman  cement  —  averages  :  — 
Age  in  water,  7  days  

•yj 

QO 

)) 

Do.          i  month  

T  T  C 

Do.          6    do  

1  Ly 
2io 

JJ 

Do.        12    do  

286 

Do.          2  years  

2/1  "2 

» 

Do.          4    do  

281 

J5 

Do.          7    do  

•5  T  C 

Jl  J 

)> 

1  Proceedings  of  the  Institution  of  Civil  Engineers,  vols.  xxv.  and  xxxii. 


STRENGTH   OF   STONE,   BRICKS,   ETC.  63! 

Table  No.  224. — CRUSHING  STRENGTH  OF  STONES  AND  BRICKS. 


DESCRIPTION  OF  MATERIAL. 

Specific 
Gravity. 

Tons  per 
square  inch. 

Experimentalist. 

Granite  :  — 
Aberdeen  blue.            

2  625 

d.87 

Rennie 

Peterhead 

3  7O 

Cornish                                       

2662 

>/u 
2  8^ 

>j 

Dublin    .                         

466 

\Vilkinson 

Wicklow 

I  C2 

N  e  wry                                     

*  86 

j> 

Mount  Sorrel  .                

2  67  1; 

3-uw 
57A 

Fairbairn 

Whinstone  Scotch      .       

•3  7O 

Buchanan 

Greenstone  Irish    

Q  3O 

"Wilkinson 

Sandstones  and  Grits  :  — 
Arbroath  pavement 

yow 

•3    C2 

Buchanan 

Craigleith  freestone    

2  AC2 

J'j* 
2  6l 

Rennie 

Derby  grit,  friable  sandstone  

*^!H 

2  3l6 

I  4.O 

Yorkshire  pavin0" 

2  ?O7 

2  c  c 

Red  sandstone  Runcorn  

•*OD 
Q7 

L  Clark 

Quartz  rock,  Holyhead,  across  lami-  ) 
nation                                            C 



•7/ 

11.40 

Mallet 

Do    parallel  to  lamination  . 

62C 

Marble  :— 
Statuary  

vf.-O 

T  AA 

Rennie 

Italian  white  veined 

A.  "32 

Irish     

4-J^ 

675  to  Q  oo 

Wilkinson 

Limestone  :  — 

2.S84. 

•2  AA 

Rennie 

Purbeck  .    . 

2   CQQ 

A.  OQ 

Magnesian  

**yryy 

i  ^6 

Fairbairn 

Anglesea  

2  72O 

^  ^8 

L.  Clark 

Irish       

>j° 
5  06  to  7  56 

"Wilkinson 

Chalk  

22/1 

Rennie 

Slates  :  — 
Irish,  on  bed  of  strata  \ 

10  60 

Wilkinson 

Do     on  edge  of  strata... 

623 

Bricks  :  — 
Red  

.7C8 

Rennie 

Yellow-faced,  baked.  . 

•J3V 

AAO 

Do.           burnt  

.qq.\.j 
64.^ 

Gault-clay  pressed 

I  1  1  1 

Grant 

Do         wire-cut  

884 

Do.         perforated  

l.lSo 

Stock         

I  OA.A 

Fareham  red  

2  SOO 

Staffordshire  blue,  pressed  with  frogs 
Do.          rough,  without  frogs 
Stourbridge  fire-clay  

— 

3.100 

^8 

J5 

L.  Clark 

Do.            do  

/ 
.670 

J.  R.  Walker 

Tividale  blue  

620 

Brickwork  in  cement  not  hard      

2^2 

E  Clark 

THE   STRENGTH   OF   MATERIALS. 
Table  No.  224  (continued}. 


DESCRIPTION  OF  MATERIAL. 

Specific 
Gravity. 

Tons  per 
square  inch. 

Experimentalist. 

Portland  cement  3  months  old         

I  7O 

Grant 

i  do  to  i  sand                             

I.  II 

i  do  to  %  sand 

A? 

Portland  cement  Q  months  old 

•T-J 
267 

}> 

j  (Jo  to  i  sand                             

2  O4. 

)> 

i  do  to  5  sand                 „           

.71 

» 

Portland  cement,  concrete  blocks  —  12- 
inch  cubes  compressed,  12  months  old:  — 
i  cement  to  i  sand  and  srravel 

Total  Crushing 
Weight, 
tons. 
I7O  £ 

» 

i                to  3 

1  1  C  C 

>j 

i               to  6 

QI  O 

?> 

Mortar  :  — 
Lime  and  river  sand 

per  square  inch. 
IQA 

Rondelet 

Do             do        beaten  

266 

Lime  and  pit  sand  

2;8 

Do            do       beaten 

•3C7 

Glass                    

•jj/ 

12.31  to  14.23 

•>•) 
Fairbairn 

STRENGTH    OF    ELEMENTARY 
CONSTRUCTIONS. 


RIVET-JOINTS 
IN  IRON  PLATES. 

There  are  two  elements  by  which  the  strength  of  rivetted  joints  is  deter- 
mined:— the  tensile  strength  of  the  perforated  plate,  and  the  grip  and 
shearing  strength  of  the  rivets. 

Strength  of  Perforated  Iron  Plates. — The  usual  effect  of  perforation  by 
punching,  is  a  weakening  of  the  metal  about  the  holes ;  so  that  the  tensile 
strength  per  unit  of  section  between  the  holes  is  less  than  that  of  the 
unpierced  plate.  Yorkshire  iron  (table  No.  192,  page  584)  loses  from  13 
to  17  per  cent,  of  its  tensile  strength,  and  Krupp  iron  from  10  to  13  per 
cent.,  by  punching.  It  is  generally  assumed,  according  to  Mr.  Wilson, 
that  hard  plates  of  fair  quality  lose  from  20  to  24  per  cent,  of  their  tensile 
strength  by  punching  for  steam-joints,  but  that  many  soft  plates  do  not 
lose  more  than  8  per  cent.  When  plates  are  drilled,  on  the  contrary,  it  is 
considered  that  the  tensile  strength  remains  unimpaired. 

But,  Mr.  J.  Cochrane  found  from  experiments  with  bar  iron  that  there 
was  no  loss  of  tensile  strength  by  punching  holes  in  the  bars.1  Low- 
moor  and  Staffordshire  bars  were  planed  down  to  a  nominal  thickness  of 
YZ  inch,  and  shaped  to  a  width  of  2  inches.  One-inch  holes  were  made 
through  the  bars  in  three  ways: — ist,  by  drilling;  2d,  by  punching  ^  inch 
too  small  and  rimering  out  to  the  size;  and  30!,  by  punching  at  once  to 
the  full  size.  The  tensile  resistances  per  square  inch  were  as  follows : — 

Formation  of  Holes.  Lowmoor  Iron.  Staffordshire  Iron. 

[/  Drilling 24.72  tons.         23.15  tons. 

Punching  and  rimering 25.51     „  23.15     „ 

Punching 24.53     „  23.69     „ 

No  doubt  the  holes  were  punched  with  a  wide  clearance  in  the  die — 
a  provision  which  very  much  facilitates  the  separation  of  the  metal,  eases 
the  punch,  and  eases  the  stress  on  the  metal. 

Strength  of  Rivetted  Joints. — Sir  William  Fairbairn,  in  1838,  deduced 
from  experiment  with  small  specimens  of  i^-inch  iron  plate,  that  double- 
ri vetted  lap-joints  were  stronger  than  single-rivetted  joints,  and  that  their 
relative  values  were  as  follows : — 

Tensile  strength  of  the  solid  plate,  as 100 

Do.  double-rivetted  lap-joint,  as 70 

Do.  single-rivetted  lap-j oint,  as 56 

^Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xxx.  1869-70,  p.  265. 


634 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


Mr.  Bertram's  Experiments. — At  Woolwich  Dockyard,  Mr.  W.  Bertram 
tested  various  plate-joints.  The  results  were  reported,  and  they  were  inves- 
tigated by  the  author,  in  i860,1  in  connection  with  Mr.  Bertram's  method 
of  welding  joints — the  scarf-weld  and  lap-weld,  Figs.  224  and  225,  in  which 
the  lap  is  i  ^  inches.  Staffordshire  plates  of  good  quality  were  selected  for 


Fig.  223. — Entire  plate. 


Fig.  224. — Scarf- welded  joint. 


Fig.  225. — Lap-welded  joint. 


Fig.  226. — Single-rivetted  joint,  by  hand. 


O 


Fig.  227. — Single-rivetted  joint,  by  hand,  snap-headed.          Fig.  228. — Single-rivetted  joint,  by  machine. 


©  0 
0  G 

m 


Fig.  229. — Single-rivetted  joint,  with  countersunk  head.         Fig.  230. — Double-rivetted  joint,  snap-headed. 


!©  0 


Fig.  231. — Double-rivetted  joint,  countersunk  and 
snap-headed. 


Fig.  232. — Double-rivetted  joint,  with  single  welt, 
countersunk  and  snap-headed. 


Boiler- Plate  Joints,  tested  by  Mr.  Bertram. 

the  trials:  of  three  thicknesses,  ^-inch,  7/l6-inch,  and  ^-inch;  and  made  up 
into  ten  varieties  of  specimens,  4  inches  broad  and  24  inches  long,  in  which 
the  rivet-joints  were  made  with  24 -inch  rivets  at  a  pitch  of  2  inches.  Three 
specimens  of  each  variety  of  joint,  for  each  thickness  of  plate,  were  tested, 
and  the  results  averaged  for  each  set  of  three  specimens.  These  joints  are 
illustrated  and  described  at  Figs.  223  to  232.  The  net  sectional  area  of 

1  Recent  Practice  in  the  Locomotive  Engine,  1858-59;  also  Railway  Locomotives,  1860, 
by  D.  K.  Clark.     Blackie  &  Son.     See  these  works  for  an  extended  notice  of  plate-joints. 


RIVET-JOINTS. 


635 


plate,  in  the  line  of  the  rivets,  was  62.5  per  cent,  of  the  solid  section.  The 
sectional  area  of  a  ^-inch  rivet  is  .4417  square  inch,  giving  for  two  rivets 
a  shearing  section  of  .8834  square  inch. 


Shearing  Section 
of  Rivets. 


Net  Sectional 
Area. 

plate,       .94  sq.  ins.      .8834  sq.  in.,  or  94  per  cent,  of  net  section. 
7/l6       „  1.094    „  .8834      „       or  80.8     „ 

%         „  1-25      „  -8834      „        or  70.7     „ 


The  fractures  took  place,  in  nearly  all  cases,  in  one  of  the  plates,  in  the 
line  of  the  rivet-holes;  but,  in  a  few  cases,  the  rivets  were  shorn  across. 
The  normal  strength  for  the  solid  plates  was  nearly  uniform,  and  averaged, 
for  all  thicknesses,  20  tons  per  square  inch. 

Table  No.  225.  —  ULTIMATE  TENSILE  STRENGTH  OF  WELDED  AND 
RIVETTED  JOINTS  OF  BOILER-PLATE. 

Tensile  strength  of  the  entire  plate,  20  tons  per  square  inch. 
(Reduced,  in  1860,  from  Mr.  Bertram's  experiments.) 


Description  of  Joint. 

Form 
of 
Joint. 

Net  Ultimate  Tensile  Strength  of  Joint  ; 
that  of  the  entire  plate=ioo. 

M-inch 
Plate. 

7/i6-inch 
Plate. 

^-inch 
Plate. 

Average 
for  three 
thicknesses. 

i    Entire  plate  

Fig.  223 
Fig.  224 
Fig.  225 

per  cent. 
100 

faulty 
So 

per  cent. 
100 

106 

69 

per  cent. 
100 
102 

66 

per  cent. 
IOO 
104 
62 

2.  Scarf-  welded  joint  

3    Lap-welded  joint 

4.  Single  -  ri  vetted    joint,  ) 
by  hand                      ( 

Fig.  226 
Fig.  227 
Fig.  228 

Fig.  229 

40 

50 
40 

44 

50 
52 

54 
5o 

60 

56 
52 

52 

50 

53 
49 

49 

5.  Single-rivetted  joint,  by  J 
hand,  snap-headed..  ( 
6.  Single  -  ri  vetted    joint,  i 
by  machine..   .           \ 

7.  Single  -  rivetted    joint,  } 
with       countersunk  J- 
head  ) 

8.  Double-ri  vetted    joint,  ) 
snap-headed  \ 

Fig.  230 
Fig.  231 

Fig.  232 

59 
53 

52 

70 
72 

60 

72 

69 
65 

67 
65 

59 

9.  Double-ri  vetted    joint,  } 
countersunk        and  > 
snap-headed  ) 

10.  Double-rivetted    joint,  } 
with      single     welt,  ( 
countersunk        and  I 
snap-headed     .       .  j 

From  these  data,  it  appears  that  the  scarf-welded  joint  is  as  strong  as 
the  entire  plate,  and  that  the  strength  of  the  lap-welded  joint  averages  only 


636  STRENGTH   OF  ELEMENTARY  CONSTRUCTIONS. 

five-eighths,  or  62  per  cent,  of  that  of  the  entire  plate.  The  varieties  of 
single-rivetted  joints  average  nearly  equally  strong  for  each  variety;  and  they 
have  only  half  the  strength  of  the  entire  plate,  excepting  the  snap-headed, 
which  has  rather  more  than  half  the  strength.  Of  the  double-rivetted 
joints  the  ordinary  lap  is  the  strongest,  having  two-thirds  of  the  entire 
strength;  the  welt-joint  is  weakest. 

Comparing  the  different  thicknesses  of  plate,  the  averages  of  all  the 
lap-joints,  at  the  foot  of  the  table,  show  that  the  ^-inch  is  the  strongest, 
that  the  7/l6-inch  is  nearly  as  strong,  and  that  they  are  about  one-fourth 
stronger  than  ^-inch  lap-joints,  relatively  to  the  thickness  of  plate. 

Leaving  the  averages,  the  drift  of  the  evidence  is,  that  the  thinner  the 
plate  the  more  efficient  the  joint.  The  single-rivetted  joints,  No.  4,  have 
successively  40,  50,  and  60  per  cent  of  the  strength  of  the  entire  plates, 
and  the  double-rivetted  joints,  59,  70,  and  72  per  cent;  insomuch  that  the 
2/6-inch  single-rivetted  joint  is  absolutely  stronger  than  the  thicker  joints — 
the  actual  breaking  weights  being  successively  16,  17%,  and  18  tons  for  the 
j^-inch,  7/l6-inch,  and  ^-inch  joints.  For  the  double-rivetted  joints,  the 
actual  breaking  weights  are  about  23.5,  24.5,  and  21.5  tons;  showing  that 
the  7/l6-joint  is  absolutely  stronger  than  the  ^-inch,  and  that  the  3/^-inch 
joint  has  only  one-twelfth  less  absolute  strength  than  the  ^-inch  joint 
The  double-rivetted  welt -joint,  similarly,  is  more  efficient  for  the  thinner 
plates,  and  its  absolute  strength  is  practically  the  same  for  them  all. 

It  appears,  then,  that  ^4 -inch  rivetted  plates  are  practically  stronger 
than  7/l6-inch  and  ^-inch  rivetted  plates;  and  that,  of  the  ^-inch  joints, 
the  order  of  strength  is  as  follows : — 

Tensile  Strength. 

Entire  plate,  ^-inch  thick 100 

Double-rivetted  lap-joint,  average 71 

Double-rivetted  single  welt-joint 65 

Single-rivetted  lap-joint,  average 55 

These  proportions  do  not  differ  widely  from  those  that  were  given  by  Sir 
William  Fairbairn. 

It  appears  that  countersunk  rivetting  does  not  impair  the  strength  of  the 
joint,  as  compared  with  external  heads. 

To  bring  out  the  comparative  weakness  of  the  joints  of  the  thicker  plates, 
the  fourth  line  of  the  following  tablet,  which  is  obtained  by  dividing  the 
third  by  the  second  line,  shows  that  the  tensile  strength  per  square  inch  of 
net  section  of  the  ^-inch  single-rivetted  joint,  was  nearly  nine-tenths  of 
that  of  the  entire  plate;  whilst  that  of  the  7/l6-mch  joint  was  just  over  eight- 
tenths;  and  of  the  ^ -inch  joint,  seven-tenths. 

**"*  ^--h  ^nch. 

Net  section, 62.5  %  62.5%  62.5  %  of  that  of  entire  plate. 

Net  tensile  strength,  av.,...  43.5  „  51.5,,  55     „  „  „ 

Do.  per  square  inch  of ) 

net  section,  in  parts  of  >    70     „  82     „  88      „ 

that  of  entire  plate,...  j 

50     „       69     „      66     „  of  that  of  entire  p.ate. 


RIVET-JOINTS.  637 

The  lap-weld  joint  is  strikingly  weaker  than  the  body  of  the  plate, 
though  there  is  no  reduction  of  section.  The  weakness  arises  from  the 
indirectness  of  the  lap,  for  the  joint,  though  solid,  is  not  straight.  The 
experiment  proves  that  the  lap  is  essentially  an  element  of  weakness,  irre- 
spective of  the  loss  of  strength  by  rivet-holes:  the  thicker  the  plate,  the 
greater  is  the  distorting  leverage,  insomuch  that  the  absolute  strength  of 
the  ^-inch  lap-welded  joint  was  not  greater  than  that  of  the  2/6-inch  joint. 
The  annexed  Figs.  233  and  234,  show  the  ultimate  distortion  by  the 
oblique  stress  on  lap-joints. 

Scale,  One-half. 


Figs.  233,  234. — Ultimate  effects  of  Oblique  Stress  on  Lap-joints. 

On  the  principle  here  noticed,  one  may  account  for  the  practically  equal 
strength  of  the  joints  made  with  countersunk  rivets,  compared  with  those 
having  external  rivet-heads,  notwithstanding  the  greater  reduction  of  solid 
section  by  countersinking: — the  leverage  is  shortened,  and  it  may  be 
measured  from  the  centre  of  the  cylindrical  part  of  the  rivet  in  the  line  a  a, 
Fig.  235,  or  thereabouts,  towards  the  inner  side  of  the  plate.  On  the  same 
principle,  the  conical  form  of  punched  holes  reduces  the  leverage  and  the 
obliquity  of  the  pulling  stress.1 

Scale,  One-half. 


Fig.  235. — Diagram  to  show  Stress  on  Countersunk  Rivets. 

As  the  double-ri vetted  joints,  No.  8  of  the  series,  exhibited  respectively 
59,  70,  and  72  per  cent,  of  the  tensile  strength  of  the  entire  plate,  it 
appears  that  its  resistance  per  square  inch  of  net  section,  was  94,  112,  and 
115  per  cent,  of  that  of  the  entire  plate.  There  is  an  apparent  anomaly 
here : — it  may  be  supposed  that  the  normal  strength  of  the  particular  plates 
exceeded  20  tons  per  square  inch,  aided,  perhaps,  by  the  frictional  grip 
of  rivets,  first  pointed  out  by  Mr.  Edwin  Clark  (see  page  5  70). 

Mr.  J.  G.  Wright's  Experiments.'2' — Mr.  Wright  gives  the  strength  of  two 

1  The  author  believes  he  was  the  first  to  publish  the  rationale  of  the  strength  and  the 
weakness  of  ri vetted  joints,  as  the  cause  of  the  grooving  of  plates  at  such  joints. 

2  Discussion  upon  Mr.  W.  R.  Browne's  paper,  "  On  the  Strength  and  Proportions  of 
Rivetted  Joints,"  in  the  Proceedings  of  the  Institution  of  Mechanical  Engineers,  1872. 


638  STRENGTH   OF  ELEMENTARY   CONSTRUCTIONS. 

specimens  of  single-rivetted  square  lap-joints,  and  two  of  diagonal  joints, 
at  angle  of  45°,  which  were  tested  by  Mr.  Kirkaldy.  They  were  made  with 
^-inch  Staffordshire  plate,  exactly  .38  inch  thick,  12  inches  wide,  with 
2^-inch  lap,  punched  holes,  and  six  if-inch  rivets  in  the  square  joint,  at 
2  inches  pitch.  The  diagonal  joint  was  made  with  eight  rivets  of  the  same 
size  and  pitch.  The  ultimate  tensile  strength  of  the  solid  plate  was  19.69 
tons  per  square  inch  with  the  fibre,  and  16.80  tons  across.  The  section 
of  the  entire  plate  was  (i2x.38  =  )4.56  square  inches,  and  the  total  ulti- 
mate strength  with  fibre  was  (4.56  x  19.69  =  )  89.8  tons. 

Ultimate  Tensile  Strength. 

Entire  plate 89.8  tons. 

Square  joint 43.0     „     or  48  per  cent. 

Diagonal  joint 58.0     „      or  64       „ 

Square  Joint.  Diagonal  Joint. 

Net  sectional  area,     59.4  per  cent.     91.7  per  cent,  of  entire  plate. 
Net  tensile  strength,  48.0       „  64.0       „  „ 

Do.  per  square  inch  1 

of  net  section,  in  I  R 

parts  of  that  of  | 

entire  section...  j 

The  diagonal  joint  was  one-third  stronger  than  the  square  joint;  although, 
per  square  inch  of  net  section,  it  opposed  less  resistance,  because  its  resist- 
ance, which  was  necessarily  exerted  in  an  oblique  direction,  was  a  resultant 
compound  of  shearing  resistance  with  the  lengthwise  resistance  of  the  plates. 

The  net  sectional  area  of  the  square  joint  was  2.71  square  inches;  and 
the  shearing  section  of  the  rivets,  3.11  square  inches,  or  115  per  cent,  of 
the  net  section. 

Mr.  L.  E.  Fletcher's  Experiments. — In  these  experiments,  to  be  after- 
wards noticed,  a  double-rivetted  lap-joint,  made  with  punched  holes, 
zigzag,  of  7/l6-inch  Staffordshire  plate,  in  a  7-feet  Lancashire  boiler, 
was  burst  with  a  force  of  20.01  tons  per  square  inch  of  net  section 
between  the  rivets  in  line,  the  fracture  taking  place  in  the  plate.  With 
so  high  a  tensile  resistance,  it  is  probable  that  the  strength  of  the 
plate  was  very  little,  if  at  all,  impaired  by  punching.  The  rivets  were 
placed  at  2.44  inches  pitch  in  line,  and  had  an  average  diameter  of  T3/l6 
inch.  The  ultimate  strength  of  the  joint  may  be  taken  as  two-thirds  of 
that  of  the  solid  plate — being  in  the  ratio  of  the  net  sectional  area  to  the 
section  of  the  entire  plate. 

Messrs.  John  Elder  6°  Go's  Experiments. — A  double-rivetted  lap-joint  of 
2^ -inch  iron  plate  failed  with  a  force  of  15.06  tons  per  square  inch  of  the 
net  section  between  the  rivets,  the  strength  of  the  solid  plate  being  20.5 
tons;  also,  a  similiar  joint  of  9/l6-inch  plate  failed  with  a  force  of  14.28  tons 
per  square  inch  of  net  section,  whilst  the  strength  of  the  solid  plate  was  20.2 
tons.  Here,  it  was  found  that  the  net  tensile  resistance  of  the  plates  between 
the  holes  was  less  by  one-fourth  than  the  direct  strength: — confirmatory  of  the 
deductions  on  the  comparative  weakness  of  the  rivet-joints  of  the  thicker  plates. 

Mr.  Brunei's  Experiments. — Mr.  Brunei  made  experiments  on  double- 
rivetted  double-welted  plate-joints,  of  which  the  author  published  an 


RIVET-JOINTS. 


639 


analysis  in  1 85 8-5 9. l     The  specimens  were  of  ^-inch  best  Staffordshire 
plates,  20  inches  wide,  butt-jointed,  with  a  covering  or  fishing  plate  on  each 
side,  10  inches  deep,  put  to- 
gether   With    punched    holes  Scale,  One-twelfth. 

and  rivets,  as  in  Figs.  236-238, 
showing  chain  rivetting  and 
zigzag  rivetting.  See  tablet, 
p.  640. 

The  ist  specimen  failed 
with  153  tons,  shearing  10 
rivets;  and  the  2d  specimen 
failed  with  164  tons,  breaking 


a  plate  through   the   rivets; 
mean  strength,  158.5  tons,  = 
15.85  tons  per  square  inch  of 
the  entire  section.    Fig.  236. 

The  3d  and  4th  specimens 
failed  with  167  and  147  tons 
respectively,  through  a  line 
of  rivet-holes ;  mean  strength, 
157  tons,  =  15.7  tons  per 
square  inch  of  the  entire  sec- 
tion. Fig.  236. 

The  5th,  6th,  7th,  and  8th 
specimens  broke  with  158, 
1 60,  1 6 1,  and  168  tons  re- 
spectively; mean  strength,  162 
tons,  =  16.2  tons  per  square 
inch  of  the  entire  section. 
The  fractures  took  place  in 
the  plates,  following,  in  one 
case,  the  zigzag  course  of 
the  rivets.  In  two  cases,  the 
rivets  partly  failed.  Fig.  237. 

The  Qth  and  roth  speci- 
mens broke  through  the  plate 
with  171  and  176  tons  re- 
spectively; mean  strength, 
173^  tons,  =  17.35  tons  per 
square  inch.  Fig.  238. 

Five  solid  %-inch  plates, 
from  1 2  to  1 6  inches  in  width, 
of  the  same  quality  as  the  speci- 
mens, were  broken  by  from 
1 9. 4  to  2  2  tons  per  square  inch; 
mean  strength,  20.6  tons. 


O 


o 


O 


o 


c 


o 


0 


0 


o 


o 


o 


Figs.  236-238.— Rivetted  Plate-Joints.     Tested  by  Mr.  Brunei. 


The  third  line  following  the  tablet,  p.  640,  shows  that  the  strength  of  the 
plate  per  square  inch  was  impaired  by  from  i  to  7  per  cent,  by  punching. 
The  average  efficiency,  or  actual  strength,  of  the  double-welt  double- 

Recent  Practice  in  the  Locomotive  Engine,  1858-59;  also  Railway  Locomotives,  1860, 


page  2 


640 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


rivetted  joint  may  be  taken  as  80  per  cent,  of  that  of  the  entire  plate,  when 
the  net  sectional  area  is  83  ^  per  cent. 


BRUNEL'S  EXPERI- 
MENTS. 

Rivets, 

Pitch, 

Sectional  Area  of  Plate. 

Total  Shearing  Section 

SPECIMEN. 

dia- 
meter. 

versely. 

Entire. 

Net,  transversely. 

of  Rivets. 

Nos. 

inch. 

inches. 

sq.  in. 

sq.  inch. 

per  cent. 

square  inches. 

I  and  2,  Fig.  236, 

« 

4 

10 

8.28 

82.8 

7.42,  or   90  %  of  net  sect. 

3  and  4,  Fig.  236, 

3/ 

4 

10 

8.125 

81.25 

8.84,  or  1  10     „ 

5  to  8,  Fig.  237,.. 
gandio,  Fig.  238, 

X 
X 

4 
5 

10 
10 

8.125 
8.5 

81.25 

85 

8.84,  or  1  10     ,,         ,, 
io.6i,ori25     ,,         ,, 

Chain.  Chain. 

Nos.  i  and  2.  Nos.  3  and  4. 

Net  sectional  area 82.8  %  81.25  % 

Net  tensile  strength 77  76 

Strength  per  square  inch  of  j 

net  section,  in  terms  of  >   93  93.5 

that  of  entire  section...    .  I 


Zigzag. 
Nos.  5  to  8. 

85% 
78.6 


92.5 


Chain. 
Nos.  9  and  ic 

85% 
84 

99 


Mr.  R.  B.  Longridge  reported  the  results  of  tests  for  the  strength  of 
rivetted  joints  in  iron  boiler  plates,  made  for  him  by  Mr.  Kirkaldy. 


RIVET-JOINTS   IN   STEEL   PLATES. 

The  results  of  experiments  on  the  strength  of  rivet-joints  in  steel  plates, 
conducted  by  Messrs.  David  Greig  and  Max  Eyth,  Professor  A.  B.  W. 
Kennedy,  and  Mr.  C.  H.  Moberley,  have,  with  those  on  rivet-joints  in  iron, 
been  exhaustively  analysed  in  the  Strength  of  Materials,  in  preparation  by 
the  author;  and  noticed  in  summary  in  his  work  on  the  Steam  Engine. 

The  conclusions  arrived  at  on  the  proportions  and  strength  of  rivetted 
joints  in  boiler  plates  of  iron  and  of  steel,  ^-inch  thick,  are  collected  in 
the  table  No.  227.  The  relations  of  thickness  of  plate,  diameter  of  rivets, 
and  pitch  of  rivets,  here  shown  for  ^-inch  plate-joints,  are  applicable  to 
other  thicknesses  of  plates,  and  are  generalized  as  follows.  The  "spacing" 
denotes  the  distance  apart  of  the  two  rows  of  rivets  in  double-rivetting. 

Table  No.  226. — STANDARD  PROPORTIONS  OF  RIVETTED  JOINTS  IN 
IRON  AND  STEEL. 

Thickness  of  plates unity  or  i 

Diameter  of  rivets thickness  of  plate, 

Pitch  of  rivets  (single-r  i  vetting). ...  {  <;hickness  of  plate, 

(  diameter  of  rivets, 

Pitch  of  rivets  (double-rivetting)...  {  thickness  °f  Plate> 

(  diameter  of  rivets, 

Diagona,  pitch  (double-rivetting)..  {  SSf^St 
Spacing  (double-rivetting) longitudinal  pitch, 

Lap  (single-rivetting) {  thickness  of  plate, 

I  diameter  of  rivets, 

Lap  (double-rivetting) I  thi^ness  of  plate, 

I  diameter  of  rivets, 


5/3 


.56,  or 


6 

3 

10.48,  or 
5.24,01 


RIVET-JOINTS. 


641 


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642 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


On  these  proportions,  the  diameter  and  pitch  of  rivets  suitable  for  plates 
of  from  y%  inch  to  JI/i6  inch  in  thickness,  are  as  given  in  the  following 
table,  No.  227^.  The  calculation  is  not  extended  for  thicknesses  greater  than 
II/I6  inch,  for  which  the  corresponding  rivet  has  a  diameter  of  i  ^  inches, 
as  this  size  of  rivet  is  assumed  to  be  the  maximum  properly  available  in 
practical  operations. 

Table  No.  2270. — STANDARD  RIVETTED  JOINTS  OF  MAXIMUM  STRENGTH, 
IN  IRON  AND  STEEL  PLATES,  OF  VARIOUS  THICKNESSES. 

(Net  section  for  single-riveting,  62.5  per  cent,  of  whole  plate-section;  for  double-riveting, 

75  per  cent. ) 


PITCH  OF  RIVETS. 

LAP, 

Thickness 
of 
Plate. 

Diameter 
of 
Rivets. 

Single 

Double  Rivetting. 

Single 

Double 

Rivetting 

Lonei-         T^- 
tudinal.         D^gonal. 

Spacing. 

Rivetting. 

Rivetting. 

inch. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches 

# 

X 

2/3 

I 

X 

9/i6 

X 

I5/i6 

3/i6 

?/8 

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I/^ 

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Other  proportions  than  those  given  in  table  No.  2270  may,  of  course,  be 
adopted;  and,  in  fact,  must  be  adopted  for  plates  of  greater  thickness  than 
IZ/i6  inch.  By  means  of  the  following  general  formulas  the  pitches  of  rivets, 
and  their  diameters,  producing  joints  of  equal  resistance,  may  be  found 
for  plates  thicker  than  »/l6  inch. 

Pitch  of  Rivets  for  Equal  Resistance. 
_-7354 


P=^*  +  d 


ZLV.ZL 

I-57/      1-57 


•7854" 

/  =  thickness  of  plates,  in  inches. 
d=  diameter  of  rivets,  in  inches. 
p  =  pitch  of  rivets,  in  inches. 
r  =  ratio  of  shearing  section  of  rivets  to  net  section  of  plates. 


CO 


PILLARS   OR   COLUMNS.  643 

STRENGTH  OF  PILLARS  OR  COLUMNS. 

Mr.  Stoney  lucidly  develops  the  leading  principles  of  the  resistance  of 
columns  supporting  incumbent  loads,1  which  are,  no  doubt,  strictly  appli- 
cable to  columns  of  perfectly  homogeneous  material.  He  shows  that  the 
strength  of  very  long  square  or  round  pillars  varies  directly  as  the  4th 
power  of  the  diameter,  and  inversely  as  the  square  of  the  length;  that  it 
depends  not  on  the  direct  strength  of  the  material,  but  on  the  coefficient 
of  elasticity,  which  represents  the  stiffness  and  capability  of  resisting 
flexure;  that  the  strengths  of  similar  long  columns  are  as  the  squares  of  any 
linear  dimension,  or  as  the  sectional  areas,  whilst  their  weights  are  as  the 
cubes  of  the  dimension;  and  that,  if  the  strengths  of  long  pillars  of  similar 
section  are  the  same,  while  the  length  varies,  the  sectional  areas  vary  as  the 
lengths,  and  the  weights  vary  as  the  squares  of  the  lengths.  Finally,  that 
the  weight  which  produces  moderate  flexure  in  a  very  long  pillar  is  also 
very  near  the  breaking  weight,  as  a  trifling  additional  load  bends  the  pillar 
very  much  more,  and  strains  the  fibres  beyond  what  they  can  bear — a 
conclusion  of  great  practical  importance,  which  has  been  corroborated  by 
experience. 

MR.  HODGKINSON'S  INVESTIGATIONS. 

The  following  is  an  abstract  of  Mr.  Hodgkinson's  conclusions  on  the 
resistance  of  cast-iron  columns,  under  loads.  The  mode  of  fracture  of  cast- 
iron  struts  or  columns  under  compression,  is  the  same  when  the  height  is 
greater  than  the  diameter  of  the  specimen,  and  not  greater  than  four  or 
five  times  the  diameter.  When  the  height  is  greater,  the  specimen  bends. 
Fracture  usually  takes  place  by  the  two  ends  of  the  specimen  forming  cones 
or  pyramids,  splitting  the  sides,  and  throwing  them  out.  Sometimes  the  end 
slides  off  as  a  wedge,  the  height  of  which  is  somewhat  less  than  1.5 
diameters.  The  tensile  and  compressive  resistances  average  as  i  to  6.6;  or, 
as  the  specimens  were  of  unequal  quality,  the  ratio  should  be  i  to  7  or  8, 
giving  49  tons  per  square  inch  for  the  ultimate  compressive  resistance. 

Long  Columns. — Experiments  were  made  on  castings  of  Lowmoor  No.  3 
iron.  Of  three  cylindrical  columns  having  the  same  diameter  and  length, 
the  first  had  the  ends  rounded;  the  second,  one  flat  end  and  one  round 
end;  the  third,  both  ends  flat.  The  strengths  were  as  i,  2,  3  nearly. 

A  long  flat-end  column  has  the  same  strength  as  a  round-end  pillar  of 
half  the  length.  The  same  properties  apply  to  pillars  of  steel,  wrought-iron, 
and  wood.  Swelling  a  pillar  at  the  middle  adds  not  more  than  one-seventh 
to  the  strength.  The  power  of  resistance  is  as  the  3.6  power  of  the  dia- 
meter, and  as  the  1.7  power  of  the  length. 

These  remarks  apply  to  all  pillars  not  less  than  30  diameters  in  length, 
up  to  120  diameters;  and  the  following  are  the  formulas  for  the  breaking 
load  of  flat-ended  columns : — 

Solid  columns,     W  =  44 — — '. (  4  ) 

Hollow  columns,  W  =  44 — _ Z , (  5  ) 

L 

in  which  D  is  the  external  diameter  in  inches,  d  the  internal  diameter  in 
inches,  L  the  length  in  feet,  W  the  breaking  load  in  tons. 

1  The  Theory  of  Strains,  1873,  Page  244- 


644 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


Short  Flexible  Columns.  —  The  resistance  is  compounded  of  compressive 
resistance  and  transverse  resistance.  Let  W  =  the  breaking  load,  and  c  - 
the  direct  compressive  resistance  of  the  column  (say  49  tons  x  sectional 
area  in  square  inches),  then,  having  first  calculated  the  breaking  weight  W 
by  the  above  formula,  (  4  )  or  (  5  ),  the  strength  is  found  by  the  formula, 


This  formula  is  inferred  from  the  nature  of  the  compound  strain  •  the  results 
given  by  it  are  nearly  correct,  but  rather  excessive. 

The  strength  of  long  similar  columns  is  nearly  as  the  sectional  area,  or 
nearly  as  the  square  of  the  diameter;  it  is  as  the  1.865  power. 

The  strength  of  taper  columns  is  to  that  of  cylindrical  columns  as 
D2  D'2  to  D4,  the  extreme  diameters  of  the  taper  column  being  D  and  D'. 
If  the  two  columns  be  of  the  same  length  and  solid  content,  the  cylindrical 
one  is  the  stronger. 

The  strength  of  a  column  of  a  double-flanged  section  is  only  three-fourths 
of  that  of  a  uniform  hollow  cylinder  of  equal  weight;  and  that  of  a  column 
of  cruciform  section  is  less  than  half. 

The  strength  of  a  solid  square  cast-iron  column  is  50  per  cent,  more  than 
that  of  a  round  column  of  the  same  diameter. 

A  column  irregularly  fixed,  so  that  the  pressure  is  taken  diagonally,  has 
only  a  third  of  the  strength  when  squarely  fixed. 

Cast-iron  pillars,  with  discs  on  the  ends,  are  somewhat  stronger  than 
those  with  simply  flat  ends. 

Solid  square  cast-iron  pillars  bend  or  break  in  the  direction  of  a  diagonal. 

A  slight  inequality  in  the  thickness  of  hollow  cast-iron  pillars  does  not 
reduce  the  strength  materially. 

Square  is  the  strongest  section  for  timber  rectangular  in  form. 

COMPAKATIVE    STRENGTH    OF    LONG   COLUMNS. 

Cast  iron  .............................................  1000 

Wrought  iron  .....  ....................................  1  745 

Cast  steel  ............................................  2518 

Dantzic  oak  ..........................................  109 

Red  deal  ......................................  .  ......       78.5 

3.6  POWERS  OF  DIAMETERS. 


Diameter. 

Power. 

Diameter. 

Power. 

Diameter. 

Power. 

Diameter. 

Power. 

I 

I 

3 

52.196 

d 

632.91 

10 

3981.07 

i-5 

4-3 

3-5 

90.917 

7 

II02.4 

II 

5610.7 

2 

12.125 

4 

H7.03 

8 

1782.9 

12 

76745 

2.5 

27.076 

5 

328.32 

9 

2724.4 

1.7  POWERS  OF  LENGTHS. 

i 

i 

7 

27-33 

T3 

78.3 

21 

176.92 

2 

3-25 

8 

34.29 

H 

88.8 

22 

191.48 

3 

6.47 

9 

41.9 

15 

99.85 

24 

222.0 

4 

10.55 

10 

50.12 

16 

111.43 

26 

254-3 

5 

15.42 

ii 

58.93 

18 

136-13 

28 

288.5 

6 

21.03 

12 

68.33 

20 

162.84 

30 

324.4 

PILLARS   OR  COLUMNS. 


64S 


Mr.  F.  W.  Shields  gives  the  safe  load  on  hollow  cast-iron  columns  of 
good  construction,  with  flat  ends,  and  with  base  plates.1 


THICKNESS. 


inches. 


and  upwards. 


Load  per  square  inch  of  Sectional  Area. 


Length  20  to  24  Diameters. 


tons. 
2 


25  to  30  Diameters. 


The  reduction  of  the  load  per  square  inch  with  the  thickness,  is  devised  to 
allow  for  liability  to  weakness  from  inequalities  of  the  casting. 

Mr.  GORDON'S  RULES. 

The  first  and  second  formulas  were  deduced  by  Mr.  Lewis  D.  B.  Gordon 
from  the  results  of  Mr.  Hodgkinson's  experiments. 

As  here  given,  they  show  the  total  breaking  weight  of  a  cast-iron  column  with 
flat  ends.  The  succeeding  formulas  for  the  strength  of  columns  of  wrought- 
iron  and  steel  have  been  constructed  on  the  basis  of  Mr.  Gordon's  formulas. 

1.  For  solid  or  hollow  round  cast-iron  columns: — 

W-££- (    7    ) 

I+^ 

2.  For  solid  or  hollow  rectangular  cast-iron  columns: — 

W  =  -^ (   8   ) 

1+ 

500 

3.  For  solid  rectangular  wrought-iron  columns  (Mr.  Stoney): — 

W  =  -*£- (    9    ) 

l+— - 

3000 

4.  For  columns  of  angle,  tee,  channel,  or  cruciform  iron  (Mr.  Unwin): — - 

w=-^ do) 

I+-— 
900 

5.  Solid  round  columns,  of  mild  steel  (Mr.  Baker): — 

W=-££_ (ii) 

1+ 

1400 


1  Transactions  of  the  British  Association,  1861. 


646  STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 

6.  Solid  round  columns,  strong  steel  (Mr.  Baker):  — 


J  _J_  _ 

900 

7.  Solid  rectangular  columns,  mild  steel  (Mr.  Baker):  — 

W  =  -^-  .................................  (13) 

I  +  248o 

8.  Solid  rectangular  columns,  strong  steel  (Mr.  Baker):  — 

W  =  -^  ...................................  (14) 

I6OO 

W  =  the  breaking  weight  in  tons. 
a  =  the  sectional  area  of  the  material  in  inches. 

r=the  ratio  of  the  length  to  the  diameter.  The  diameter  for  calculation 
is  the  least  diameter  of  the  section,  or  that  in  the  direction  of  which 
the  piece  is  most  flexible. 

MR.  HODGKINSON'S  RULES  FOR  TIMBER  COLUMNS. 

When  both  ends  are  flat  and  well-bedded,  and  the  length  exceeds  30 
diameters  :  — 

Long  square  columns  of  Dantzic  oak  (dry):  — 

W=io.95^l  .................................  (  15  ) 

Long  square  columns  of  red  deal  (dry):  — 

W=7.8o^-  ..................................  (  16  ) 

Long  square  columns  of  French  oak  (dry):  — 


17 


W  =  the  breaking  weight  in  tons. 
d=  the  breadth  in  inches. 
/=  the  length  in  feet. 

When  timber  columns  are  less  than  30  diameters  in  length,  their  strength 
is  calculated  by  formula  (  6  ),  page  644,  for  which  the  value  of  W  is  to  be 
calculated  by  one  of  the  above  formulas. 

When  the  column  is  oblong  in  section,  multiply  the  result  as  found  for 
the  shorter  dimension  of  the  section  by  the  ratio  of  the  longer  to  the  shorter 
dimension. 

MR.  BRERETON'S  EXPERIMENTS  ON  TIMBER  PILES. 

Mr.  R.  P.  Brereton  gives  the  loads  that  could  be  borne,  as  found  from 
experiments,  by  large  fir  or  pine  timber  1  2  inches  square,  of  various  lengths  : 
10  feet  long  bore  120  tons;  20  feet  long,  115  tons;  30  feet  long,  90  tons; 
40  feet  long,  80  tons.  Mr.  Stoney  plotted  these  results,  and  constructed 
the  following  table  from  the  curve  :  — 


CAST-IRON    FLANGED   BEAMS. 


647 


i" 
Ratio  of  length  to  least  breadth  

10 

15 

20 

25 

30 

35 

40 

45 

50 

Weight  that  can  be  borne  in  tons  per  ) 
square  foot  of  section  ) 

120 

118 

"5 

100 

90 

84 

80 

77 

75 

Checking  this  table  by  Mr.  Kirkaldy's  experiments  (page  547)  on  balks 
of  Riga  and  Dantzic  timber,  having  a  length  of  20  feet,  which  was  18% 
times  the  width,  the  actual  breaking  weights  were — 

Total.  Per  square  foot  of  Section.        By  Mr.  Stoney's  Curve. 

Riga 148  tons,          or  126  tons,  116  tons. 

Dantzic 138     „  or  in     „       116     „ 


116 


Mean 119     „      

showing  a  close  correspondence  between  the  results  of  experiments  con- 
ducted independently.  Mr.  Hodgkinson's  rule  gives  results  which  are 
rather  less  than  those  that  are  given  in  Mr.  Stoney's  table. 

MR.  LASLETT'S  EXPERIMENTS  ON  COLUMNS  OF  WOOD. 

A  notice  of  Mr.  Laslett's  experiments  is  given  at  page  541.  He  deduces 
from  his  experiments,  repeated  for  many  kinds  of  wood :  that  the  maximum 
resistance  of  square  pieces  to  compression  is  exerted  when  the  sectional 
area  in  square  inches  is  to  the  length  in  inches  approximately  as  4  to  5, 
for  equal  seasoning  and  equal  specific  gravities.  According  to  this  deduc- 
tion, the  maximum  resistance  of  1 2-inch  square  balks  on  end,  would  be 
exerted  when  they  are  1 5  feet  in  length. 


CAST-IRON  FLANGED  BEAMS. 

Mr.  Hodgkinson  tested,  for  transverse  strength  and  deflection,  a  number 
of  cast-iron  model  beams  of  various  proportions,  and  he  discovered  that 
the  maximum  strength  of  double-flanged  beams,  for  a  given  sectional  area, 
was  realized  when  the  area  of  the  upper  flange  was  one-sixth  that  of  the 
lower  flange.  This  conclusion  harmonizes  with  the  fact  that  the  resistance 
of  cast  iron  to  compression  is  from  5  to  6  times  the  tensile  resistance;  and, 
as  a  scientific  fact,  it  has  its  value.  The  general  formula  (19),  page  511, 
for  flanged  beams,  is  as  follows : — 


in  which  a  is  the  sectional  area  of  the  lower  flange ;  a!'  that  of  the  web, 
d"  the  reputed  depth  of  the  beam  and  of  the  web,  taken  as  the  total  depth 
minus  the  thickness  of  the  lower  flange,  /  the  span,  all  in  inches,  and  W 
the  breaking  weight  at  the  middle,  in  tons.  Sections  of  cast-iron  beams 
which  have  been  tested,  are  given  in  Figs.  239-258,  comprising  17  model 
beams  tested  by  Mr.  Hodgkinson,  and  6  model  beams  by  Mr.  Berkley;  like- 
wise 1 1  large  beams.1  Mr.  Berkley's  model  beams  are  in  pairs,  and  have  the 

1  See  Hodgkinson  on  the  Strength  and  other  Properties  of  Cast  Iron,  1846;  and  the 
Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xxx.  page  252  (Mr.  Berkley's  paper 
on  the  "  Strength  of  Iron  and  Steel"). 


648 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


same  dimensions  as  Nos.  6,  7,  and  12  respectively.  Table  No.  228  con- 
tains, in  part  i,  the  needful  particulars  of  those  beams,  and  of  the  ultimate 
breaking  weight,  both  actual  and  as  calculated  by  formula  (i).  The 
deflection  and  the  elastic  strength  of  the  beams  are  given  in  part  2  of  the 
table.  A  tensile  strength  of  7  tons  per  square  inch  has  been  adopted  in  the 
calculation  of  the  ultimate  strength  of  Mr.  Hodgkinson's  model  beams, 
6^2  tons  for  the  large  beams,  and  10^/2  tons  for  Mr.  Berkley's  model 
beams;  for,  though  his  test-castings  bore  a  greater  tensile  load  than  10^ 
tons,  they  were  too  short  in  the  tested  portion,  which  was  i  inch  square 
and  only  i  y2  inches  long,  for  the  action  of  simple  tensile  resistance. 


No.  i. 


No.  2. 


Nos.  3,  4. 


No.  5. 


Nos.  6,  18,  19. 


Nos.  7,  20,  21. 


No.  8. 


No.  9. 


No.  TO. 


No.  it. 


NOS.    12,  22,  23. 


No.  13. 


No   14.  No.  15. 

Scale  of  Nos.  i  to  17 — One-tenth  full  size. 


No.  16. 


No. 


No.  24. 


No.  25.  No.  26. 

Scale  of  Nos.  24  to  27 — One-twentieth  full  size. 


No.  27. 


Figs.  239-258.— Sections  of   Cast-Iron  Beams  tested  for  Transverse  Strength 
by  Mr.  Hodgkinson,  Mr.  Berkley,  and  others.     Table  No.  228. 


CAST-IRON   FLANGED   BEAMS. 


649 


Table  No.  228. — STRENGTH  OF  CAST-IRON  FLANGED  BEAMS. 

PART  i. — ULTIMATE  TRANSVERSE  STRENGTH. 

i.  MODEL  BEAMS. 


Thickness. 

Sectional  Area  of 
Flanges. 

Breaking  Weight  at 
the  middle. 

Refer- 

Depth 

ence 

Number. 

Span. 

at 
middle. 

d 

Web. 

Lower 
flange. 

Lower 
flange. 
a 

Upper 
flange,  ratio 
to  lower. 

Calculated. 
W 

Actual. 
W 

feet. 

inches. 

inches. 

inches. 

sq.  inches. 

ratio. 

tons. 

tons. 

Mr.  HODGKINSON'S  BEAMS.     Assumed  tensile  strength,  7  tons  per  square  inch. 

i 

4-5 

5/^ 

.29 

•39 

•69 

I  tO  I 

2.47 

2.98 

2 

M 

.30 

•55 

•98 

I  tO  2 

3-27 

3.29 

3 

}> 

.32 

•57 

1.20 

I  to  4 

3.83 

3.69 

4 

it 

•33 

•56 

1.  2O 

i  to  4 

3-87 

3.64 

5 

J5 

•3°5 

.51 

?-57 

i  to  4^ 

4.68 

4.79 

6 

,, 

•38 

•53 

2.20 

i  to  4 

6.45 

6.46 

7 

J} 

•34 

•56 

2.89 

i  to  5^ 

7.85 

7-47 

8 

,, 

•33 

•57 

2.31 

i  to  3.2 

6.49 

6.71 

9 

|f 

•35 

•537 

2.92 

I  to  4.3 

8.04 

7-54 

10 

,, 

•34 

•54 

3-57 

I  to  5.6 

9.56 

8.68 

ii 

5> 

.266 

.66 

4.40 

i  to  6 

10.98 

11.65 

12 

•335 

•65 

4-31 

i  to  7 

11.00 

10.40 

13 

M 

3-32 

i  to  6.  7 

9.02 

9.40 

H 

7-'o 

6-93 

•38 

•75 

4-54 

i  to  6 

10.26 

9.90 

15 

4.10 

.40 

•74 

4-44 

i  to  6 

5-41 

6.05 

16 

9.0 

IOX 

•25 

•77 

4.72 

i  to  8.3 

13-28 

12.80 

17 

4.5 

5/? 

.40 

.46 

1.04 

none 

3-83 

3-93 

Averages  for  17  beams  

i  to  A..  6 

7.07 

7.01 

Mr.  BERKLEY'S  BEAMS.     Assumed  tensile  strength,  10.5  tons  per  square  inch. 

18 

4-5 

$yi 

.38 

•53 

2.20 

I  to  4 

9.67 

10.00 

19 

J} 

9.67 

10.00 

20 

,, 

•34 

.56 

2.'89 

i  to  3/2 

11.85 

".75 

21 

,, 

11.85 

".85 

22 

,, 

•335 

.65 

4.31 

i  to  7 

16.47 

14-25 

23 

» 

tt 

it 

M 

17.08 

18.00 

Averages  for  6  beams  

i  to  <;  "» 

12.76 

12  64 

•  L^  j.  j 

•  wt  y  vy 

**.  V4t- 

2.  LARGE  BEAMS.  —  Assumed  tensile  strength,  6.5  tons  per  square  inch. 

(Reported  by  Mr.  Hodgkinson.) 

24 
3  beams 

j  18.0 

17 

.625 

1-25 

10.31 

i  to  4.  6 

24-93 

25-0 

25 

11.67 

9 

I-5 

I   r 

12.  OO 

i  to  1.33 

21.24 

2O.  O 

261 

27-4 

30.5 

2.08 

2.07 

25.05 

I  tO  2.  1 

94.64 

76.6  + 

272 

23.1 

36.1 

3-36 

3-12 

74.60 

none 

330.0 

153.0  + 

Mr.  CUBITT'S  BEAMS. 

28 

15.0 

7.15 

1.04 

•59 

7.98 

i  to  3.6 

7-75 

7.00 

29 

, 

7.17 

1.  10 

•59 

8.ii 

i  to  3.6 

7.96 

7-13 

30 

, 

10.75 

•93 

.04 

5.25 

i  to  2.3 

11.02 

II.5O 

31 

5 

10.75 

1-05 

•05 

5.42 

i  to  2.3 

11.71 

12.00 

32s 

, 

12.75 

•73 

.88 

4-47 

I  tO  2.  7 

"•95 

10.25 

33 

, 

12.8 

•95 

.09 

5-59 

I  tO  2.  25 

14.89 

15-75 

344 

} 

14.0 

.91 

.00 

6.50 

none 

i8.39 

12.38 

355 

, 

17.25 

.68 

.84 

4.96 

i  to  2.  2 

19-39 

16.00 

36fi 

7-5 

7.15 

1.06 

•59 

8.03 

i  to  3.4 

15.63 

15-63 

376 

M 

10-75 

.92 

.02 

5.16 

i  to  2.25 

21.76 

23.87 

Averages  for  n  of  these  large  beams  

i  to  3.  17 

17.00 

17.08 

Total  averages  for  34 

beams 

i  to  4.  30 

11.31 

11.26 

650 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


Table  No.  228  (continued}. 

PART  2. — DEFLECTION  AND  ELASTIC  STRENGTH. 

i.  MODEL  BEAMS. 


Limits  of  Elastic  Strength. 

Ratio  of 

Reference 
Number. 

Form  of  Beams. 

Elastic 
Strength  to 
Breaking 
Weight. 

Coefficient 
of 
Elasticity. 

Deflection 
at  middle. 

Load. 

Inches  per 
ton  of  load. 

tons. 

inches. 

tons. 

inches. 

per  cent. 

E. 

Mr.  HODGKINSON'S  BEAMS. 

2 

3 

4 

6 

Elliptical. 

•45 

5.670 

.079 

88 

5232 

I 

Do. 

•49 

7.244 

.068 

9i 

4980 

9 

Uniform  depth. 

•33 

6.872 

.048 

9i 

4372 

10 

Do. 

•36 

7-454 

.048 

86 

3420 

ii 

12 

Do. 

.48 

10.254 

.047 

99 

3324 

13 
14 

Elliptical. 
Uniform  depth. 

.46 
.60 

8.503 
9.204 

•11 

90 
93 

5438 
4284 

15 

Do. 
Do. 

.70 

•55 

3.787 
11.450 

^048 

63 

89 

5520 
5208 

17 

Segmental. 

.42 

3.700 

.114 

94 

5600 

Averages  for  10  beams...    .         .... 

88 

47<;8 

Mr.  BERKLEY'S  BEAMS. 

H"/  0 

18 
19 

Uniform  depth. 
Do. 

•  245 
.271 

7.00 
8.00 

•035 
•034 

lo 

7400 
7634 

20 

Do. 

•387 

IO.OO 

•039 

85 

5486 

21 
22 

Do. 
Do. 

.264 

9.00 
8.00 

•037 
•033 

76 
56 

5760 

5274 

23 

Do. 

•303 

10.00 

.030 

55 

5i54 

Averages  for  6  beams  .  .    . 

70 

6118 

/ 

2.  LARGE  BEAMS. 

(Reported  by  Mr.  Hodgkinson.) 

24 
3  beams 

Segmental. 

1.  00 

20.0 

.050 

80 

4632 

8 

Segmental. 

•33 
1.29 

10.0 

76.6 

•033 
.017 

5o 

P 
4236 

27 

— 

.68 

153-0 

.0044 

? 

? 

Mr.  CUBITT'S  BEAMS. 

28 

Uniform  section. 

1-54 

6.0 

•  257 

86 

4760 

29 

Do. 

1.215 

5-° 

•243 

70 

4706 

30 

Do. 

•645 

7-o 

.092 

6l 

5340 

3i 

Do. 

1.04 

II.  0 

•095 

92 

4900 

32 

Do. 

.60 

9.0 

.067 

88 

5566 

33 

Do. 

.90 

15.0 

.060 

95 

5032 

34 

Uniform  depth. 

.41 

6.0 

.070 

49 

4840 

35 

Do. 

.76 

16.0 

.048 

100 

5226 

36 

Uniform  section. 

•3i 

10.  0 

.031 

64 

4884 

37 

Do. 

.261 

20.  o 

.013 

84 

4762 

Averages  for  12  beams  

77 

4906 

Total  averages  for  28  bean 

IS 

79 

5"3 

CAST-IRON   FLANGED   BEAMS.  651 


1  No.  26— 
z  No.  27  — 
as  the  appara 
3  No.  32  — 
4  No.  34— 
5  No.  35- 
6  No.  37— 

Notes  to  Table  No.  228. 

'broke  apparently  in  consequence  of  an  accidental  shake." 
'with  this  load,  153  tons  in  the  middle,  the  experiment  was  discontinued, 
:us  was  overstrained." 
'bottom  flange  unsound." 
'bottom  flange  unsound." 
'bottom  flange  unsound." 
'  nearly  but  not  quite  sound." 

The  contents  of  the  table  exhibit  a  surprisingly  close  conformity 
throughout  between  the  calculated  and  the  actual  ultimate  strengths  of  such 
beams  as  were  sound,  and  were  fairly  tested  and  broken.  The  total 
averages  for  34  cast-iron  beams  show  that,  with  a  ratio  of  upper  to  lower 
flange  of  i  to  4.30,  the  breaking  weight,  as  calculated,  was  11.31  tons,  and 
as  tested,  11.26  tons. 

It  further  appears  that  the  ultimate  strength  of  a  cast-iron  beam  is  scarcely 
affected  by  the  proportionate  size  of  the  upper  flange;  and  that  the  formula 
(  i  ),  page  647,  may  be  adopted  for  the  calculation  of  the  strength  of  flanged 
cast-iron  beams  of  any  ordinary  section.  It  is  sufficient  to  employ  the  factor, 
7  tons,  for  castings  of  less  than  ^  inch  in  thickness,  and  6.5  tons  for  those 
which  have  a  thickness  of  i  inch  and  upwards.  Taking  the  span  in  feet  :  — 

Ultimate  Transverse  Strength  of  Cast-iron  Flanged  Beams. 


For  the  thinner  castings  ......  W  =  d"  ^a  +  2  a//)  .............  (  2  ) 

O 

For  the  thicker  castings  ......  w  =  «T(6.5'»+i.9'O  ............  (  3  ) 

VV  =  the  breaking  weight  in  tons  at  the  middle;  a  the  sectional  area  of  the 
lower  flange,  and  a"  the  sectional  area  of  the  web,  taken  at  the  reputed 
depth,  both  in  square  inches;  /  the  span  in  feet.  The  reputed  depth 
is  the  total  depth  minus  the  thickness  of  the  lower  flange. 

The  following  are  the  constants  for  other  factors  of  tensile  strength  :  — 

Tensile  strength  per  square  inch.  Constants  in  formula  2  or  3. 

for  a.  for  a". 

6      tons  ........  .........................  6        ............   1.7 

&A    »      .................................  6-75   ............  2.0 

lYz    ,,      .................................   7-5     ............   2-2 

8  „      .................................  8        ............   2.3 

9  »      .................................  9        ............   2-6 

10        „      .................................   10      ............   2.9 

Approximate  Rule  for  the  Strength  of  Cast-iron  Flanged  Beams. 

Referring  to  formula  (  2  )  for  a  tensile  strength  of  7  tons  :  —  To  the 
sectional  area  of  the  lower  flange  add  a  fourth  of  the  sectional  area  of 
the  web,  calculated  on  the  total  depth,  both  in  inches;  multiply  the  sum 
by  the  total  depth  in  inches,  and  by  2^  ;  and  divide  the  product  by  the 
span  in  feet.  The  quotient  is  the  breaking  weight  at  the  middle,  in  tons. 

For  any  other  tensile  strength,  use  it  as  the  multiplier  instead  of  2^, 
and  divide  the  product  by  3,  and  by  the  span.  The  quotient  is  the 
breaking  weight. 


652  STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


ELASTIC  STRENGTH  AND  DEFLECTION  OF  CAST-IRON  FLANGED  BEAMS. 

From  the  observations  of  the  experimentalists  on  the  deflection  of  the 
beams  noted  in  the  second  part  of  table  No.  228,  it  is  shown  that  the  elastic 
strength — that  is,  the  limit  of  load  for  uniform  increments  of  deflection, 
irrespective  of  set — as  given  in  the  table,  averages  about  80  per  cent,  of  the 
ultimate  strengths.  The  value  of  the  coefficients  of  elasticity,  E,  were  cal- 
culated tentatively  by  means  of  the  formula  (  13  ),  page  531,  for  beams  of 
constant  depth  and  uniform  strength  loaded  at  middle;  but,  since  all  the 
beams  excepting  six  of  Cubitt's  beams  were  proportioned  for  carrying 
uniform  loads,  the  tentative  values  found  for  these  beams  were  modified 
according  to  their  special  forms,  on  the  principles  already  adopted  in  the 
general  section  on  the  deflection  of  beams,  pages  529,  530,  to  give  the  proper 
values  of  E.  The  general  average  value  for  E  is  5113,  and  the  numerical 
coefficient  of  the  formula  (  13 ),  page  531,  being  multiplied  into  this  value, 
gives  the  resultant  coefficient  for  beams  of  cast  iron.  The  general  formula 
is  as  follows ;  c  being  the  coefficient : — 

Deflection  of  Cast-iron  Flanged  Beams. 

D_  W/3 

(cE)<t"*(4a+  1.155  «"2)'" 
Values  of  (c  E),  to  be  employed  in  this  formula. 


Length  in        Length  in 
inches.               feet. 

i.  Constant  depth,  uniform  strength,  load  at  middle.  ... 

20,700  ... 

12 

2.             Do. 

do. 

uniform  load  

41,400  ... 

24 

3.  Constant  breadth, 

do. 

load  at  middle.  .  .  . 

10,35°  ••• 

6 

4.             Do. 

do. 

uniform  load  

27,600  ... 

16 

5.  Uniform  section, 

load  at  middle.  .  .  . 

20,700  ... 

12 

6.             Do. 

uniform  load  

33,120  ... 

19 

D  =  the  deflection,  b  —  the  breadth,  d"  —  the  reputed  depth,  or  the  ex- 
treme depth  minus  the  thickness  of  the  lower  flange,  all  in  inches;  /=the 
span,  in  inches  or  feet;  #-the  sectional  area  of  the  lower  flange  at  the 
middle,  and  a"  -  that  of  the  web  at  the  middle,  reckoned  on  the  reputed 
depth ;  W  =  the  load  in  tons. 

Approximate  Rule. 

To  the  sectional  area  of  the  lower  flange,  add  one-fourth  of  the  sectional 
area  of  the  web,  calculated  on  the  whole  depth,  both  in  inches;  multiply 
the  sum  by  the  square  of  the  depth  in  inches,  and  by  the  proper  coefficient 
in  the  following  list;  making  a  product  A.  Multiply  the  load  at  the  middle 
in  tons,  by  the  cube  of  the  span  in  feet;  and  divide  this  product  by  the 
product  A.  The  quotient  is  the  deflection  in  inches. 


No.  i.  Coefficient 48 

No.  2.          „         96 

No.  3.          „         24 


No.  4.  Coefficient 64 

No.  5.  „         48 

No.  6.  „         76 


WROUGHT-IRON   FLANGED   BEAMS   OR  JOISTS. 


653 


WROUGHT-IRON    FLANGED    BEAMS   OR   JOISTS. 
SOLID-ROLLED  WROUGHT-!RON  JOISTS. 

The  usual  section  of  solid-rolled  wrought-iron  beams  or  joists  is  shown 
by  Figs.  259,  260,  261,  and  262,  page  654.  For  practical  facility  of  rolling, 
the  flanges  rarely  ever  exceed  6  inches  in  breadth;  and  the  breadth  of 
flange  varies  from  a  third  of,  to  an  equality  with,  the  depth, — the  latter  ratio, 
of  course,  only  occurring  for  small  sizes.  The  flanges,  also,  have  a  taper 
section  on  each  side  of  the  web. 

In  joists  of  ordinary  proportions,  the  thickness  of  the  web  is  from  J/26th 
to  Yi3tn  of  tne  depth  of  the  beam,  being  thicker  as  the  relative  breadth  of 
the  flanges  is  increased;  but,  by  setting  the  rolls  wider  apart,  the  thickness 
may  be  increased  a  half  or  two-thirds  more.  The  mean  thickness  of  the 
flanges  is  from  */l9th  to  Vioth  of  the  depth.  The  slope  or  taper  of  the 
flanges  in  section  is  usually  about  i  in  7,  on  each  side  of  the  web.  The 
beams  can  be  rolled  to  lengths  of  30  feet;  but  they  cost  less  per  ton  when 
the  lengths  do  not  exceed  20  feet. 

Table  No.  229  shows  the  average  proportions  of  the  thickness  of  the 
web,  and  the  mean  thickness  of  the  flange,  for  various  proportional  breadths 
of  flanges,  the  depth  being  taken  as  i. 

Table  No.  229.— PROPORTIONAL  DIMENSIONS  OF  SOLID 
WROUGHT-IRON  JOISTS. 


Depth  of 
Joist. 

Breadth 
of 
Flange. 

Thickness 
of  Web. 

Mean 
Thickness 
of  Flanges. 

Depth  of 
Joist. 

Breadth 
of 
Flange. 

Thickness 
of  Web. 

Mean 
Thickness 
of  Flanges. 

•3 

1/26 

I/I9 

•75 

I/l6.3 

I/II.5 

•35 

1/24.5 

1/18 

.8 

I/I5-3 

i/i.i 

•4 

I/23 

1/17 

.85 

I/I4.6 

1/10.5 

•45 

1/22 

1/16.2 

•9 

I/I4 

I/IO 

•5 

I/2I 

I/I5-3 

•95 

I/I37 

1/9-7 

I 

•55 

1/20 

J/I4-3 

1.  00 

I/I3-5 

1/9-5 

I 

.6 

I/I9 

I/I3-5 

I 

.65 

1/18 

I/I2.8 

I 

•7 

1/17 

1/12 

The  dimensions  of  a  variety  of  solid-rolled  joists  actually  manufactured, 
having  the  minimum,  or  what  may  be  called  the  normal,  thickness  of  web, 
are  given  in  table  No.  230,  next  page.  The  reputed  weights  per  lineal  foot 
are  given  in  the  fifth  column.  The  ultimate  strengths  for  a  span  of  10  feet, 
in  the  sixth  column,  are  calculated  by  the  first  approximate  rule  with  formula 
(  7  )>  Pa£e  656;  and  the  safe  distributed  load  in  the  last  column  is  taken  as 
one-third  of  the  breaking  weight  at  the  middle,  according  to  a  factor  of  6. 

To  find,  from  table  No.  230,  the  ultimate  strength  of,  or  the  safe  per- 
manent load  for,  a  joist,  for  any  other  span  than  10  feet,  multiply  the  tabular 
weight  for  the  beam  of  the  given  section  by  10,  and  divide  the  product  by 
the  given  span. 

Inversely,  to  find  the  span  for  a  joist  of  a  given  section,  with  a  given  weight, 
multiply  the  tabular  weight  by  10,  and  divide  the  product  by  the  given  weight 


654 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


Table  No    230. — SOLID-ROLLED  WROUGHT-IRON  JOISTS  : — DIMENSIONS, 
WEIGHT,  AND  STRENGTH.     SPAN,  10  FEET. 


Depth  of 
Beam. 

Breadth 
of  Flanges. 

THICKNESS. 

Reputed 
Weight  per 
lineal  foot. 

Ultimate 
Strength, 
Loaded  at  the 
Middle. 

Safe  Perma- 
nent Load, 
Uniformly 
Distributed. 

Of  Web. 

Of  Flanges. 

inches. 

16 

inches. 

ctf 

inch. 

...3/4     - 

9/16 

...9/16... 
9/16 
...7/16... 
3/4 
...7/16... 
7/16 
...3/8    -. 
7/16 
...7/16... 
3/8 
...3/8    ... 
3/8 
"J/i6  .. 

inch. 
...   I3/I6  ... 
I3/I6 

.    7/8    . 

Ibs. 
...      62      ... 

tons. 
..   84 

cwts. 

;6o 

H 
IA. 

...     5  78 

6 
cU 

60 
...      60      ... 

68 
...  67 

453 

44.7 

12 
12 

'    6 
c 

15/16 
n/i6  ... 

56 
42 

61 

..  41? 

407 

qoo 

10 
10 
10 

9>A  ... 
9# 

4^ 
...    4#    ... 
4^ 
...    4/2    ... 

3# 

cv£ 

5/8 

Q/l6 

36 
T.2 

34 
...  26       ... 

%s 

227 

...    173 
167 
177 

9/16 
...  11/16... 

1/2 

0/16 

32 
7Q       , 

24 
2Q 

18.6 

21 

124 
I/LO 

8 
8 

...     ^  /8 

5 

4 

9/16 

1/2     . 

29 
...      21 

20 
..    i  c.4. 

133 
IO7 

8 
8       ... 
7 

7 

2X 
2^ 

3/8 
7/16  ... 

15 
I  C 

9-5 

...      Q.  1     , 

...          IV^J 

^ 
62 

3^ 
^M 

5/16 
...5/16... 
5/16 
...9/32... 
5/16 
...5/16... 
5/16 
...7/16... 
3/8 
...3/8    ... 
5/16 
...  1/4    ... 
i/4 
...  1/4    .- 
3/i6 
...  1/4    ... 
3/i6 

1/2 

7/16  . 

19 
IQ     , 

1  1.6 

.  1  0.4. 

77 
6q 

7 
7 

...      j/8      ... 
2^ 

2i 

7/16 
3/8    . 

14 
14     ... 

7-6 
...    6.6    ... 
7-8 
...     ,8    ... 

...  15.1     ... 
..      86    . 

51 
...      44 

52 
•-.      39 
40 

...      101 

35 

cy 

6X 
6X   ... 
6X 
6      ... 

5/2 

5       ... 

4#  ••• 
4 
4       ... 

3^ 
3       ••• 
3 

3X 

...     2X      .- 
2 

...    5        ... 

2 

...    4^    .- 
3 

...     2 

3 

2 
I# 

...    3 

2 

*. 

13/32 
..    3/8    .- 
7/16 
..    9/16., 
7/16 

..      1/2     ... 

7/16 
..    5/16... 
3/8 
..    5/16... 
7/32 
..    5/16... 
7/32 

...     18     ... 
ii 
...     30     ... 

10 
27 

13 

...      8     ... 

12 

...      8     ... 

$1A 

...       10      ... 

^ 

6 

3^8 
...    2.45  ... 
i.  ii 
...    2.5    ... 

1.22 

40 
21 
25 

...      16 

7-4 
...      17 
8.1 

TRANSVERSE  STRENGTH  OF  WROUGHT-IRON  JOISTS. 
A  number  of  solid-rolled   joists  of  uniform   symmetrical  section  were 


Figs.  259-262. — Sections  of  Solid  Wrought-iron  Flanged  Beams,  or  Joists. 

tested  by  Mr.  Kirkaldy  for  Mr.  Moser,  the  sections  of  four  of  which  have 


WROUGHT-IRON  FLANGED  BEAMS  OR  JOISTS. 


655 


been  ascertained  approximately,  and  are  here  annexed,  Figs.  259-262,  with 
the  following  particulars :  — 


JOISTS. 

Weight 
per 
foot. 

Depth. 

Breadth. 

Web. 

Thick- 
ness. 

Flange. 

Mean 
thickness- 

Sectional  Area. 

Web  at 
reputed 
depth. 

One 
Flange. 

Total. 

A  

Ibs. 

43-56 
37.54 

26.22 
19.16 

inches. 

11.75 
9.85 
7.90 
7.07 

inches. 
5.70 
4.60 
376 
3.00 

inch. 
.60 
.50 
.50 
.50 

inch. 

.643 
.804 
.619 
.485 

sq.  ins. 
6-537 
4.523 
3.640 

3.293 

sq.  ins. 
3.665 
3.700 
2.329 

1-455 

sq.  ins. 

13.34 
11.50 
8.033 
5.870 

B  

D  . 

E  

The  elastic  and  ultimate  transverse  strengths  of  these  beams  are  reduced 
from  the  observations  of  Mr.  Kirkaldy,  and  are  given  in  the  table  No.  231. 
In  the  last  column,  the  probable  real  ultimate  strengths  of  the  beams  are 
added;  they  are  computed  at  twice  the  elastic  strength,  in  correspondence 
with  the  well  established  ratio  of  the  tensile  elastic  and  ultimate  strengths 
of  wrought  iron.  The  ultimate  strength,  column  4,  was  calculated  by 
formula  (19),  page  511,  in  which  the  ultimate  tensile  strength,  s,  is  taken  as 
20  tons : — 

Table  No.  231. — TRANSVERSE   STRENGTH  OF    SOLID-ROLLED    WROUGHT- 
IRON  JOISTS. 

(Results  of  Experiment.) 


JOISTS. 

Span. 

Elastic 
Strength. 

Ultimate,  or  Breaking 
Weight. 

Cause 
of 
Failure. 

Probable 
Real 
Breaking 

Weight. 

Observed. 

Calculated. 

Observed. 

A  

: 

feet. 
20 
10 
20 
10 
20 
10 
10 

5 

tons. 
10.714 
21.428 
8.482 
17.857 
4.0l8 
8.705 
5402 
11.607 

tons. 
20.390 
40.780 
15.060 
30.120 
8.203 
16.406 
8.150 
21.124 

tons. 
14.310 
32.450 
11.445 
26.530 
6.371 
15.112 
8.150 
19.520 

Buckling 
)) 

j> 
)> 
?> 
» 

5J 
JJ 

tons. 
21.428 
42.856 
16.964 

35-7I4 
8.036 
17.410 
10.804 
23.214 

D 

E  

11.027 

20.331 

16.736 

— 

22.053 

As  the  cause  of  failure  was  buckling,  it  is  clear  that  the  beams  would 
have  supported  greater  weights  if  they  had  been  supported  laterally.  That 
the  want  of  such  support  was  the  cause  of  the  weakness,  is  evidenced  by  the 
fact  that  the  observed  breaking  weights  more  nearly  approach  the  calculated 
weights  for  the  shorter  than  for  the  longer  spans.  The  probable  real  break- 
ing weights  average  more  than  the  weights,  as  calculated  from  the  experi- 
mental data.  Adapting  the  formula  (19),  page  511,  by  assuming  the  value 
of  s  —  20  tons,  then — 


656  STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 

Ultimate  Transverse  Strength  of  Solid  Wrought-iron  Joists  of  Uniform 
Symmetrical  Section. 


In  tons, 


55*')  ..........................  (s) 


>...  .(6) 


... 

0.03  / 

W  =  the  breaking  weight  at  the  middle. 
a  -  the  sectional  area  of  the  lower  flange,  in  square  inches. 
a"  —  the  sectional  area  of  the  web,  taken  at  the  reputed  depth,  in  square 
inches. 

^  =  the  reputed  depth  in  inches;  that  is,  the  total  depth  minus  the 
average  thickness  of  one  flange. 
/=the  span  in  feet. 

Approximate  Rules  for  the  Strength  of  Solid  Wrought-iron  Joists  of 
Ordinary  Proportions. 

Reduce  the  second  coefficient  in  the  numerator  of  the  above  formulas, 
to  i,  and  increase  the  depth  to  the  total  depth,  d,  of  the  beam. 

ist  Approximate  Rule.  In  Tons.  —  To  the  sectional  area  of  one  flange 
add  one-fourth  of  the  sectional  area  of  the  web,  calculated  on  the  total  depth, 
both  in  inches;  multiply  the  sum  by  the  depth  in  inches  and  by  7,  and 
divide  by  the  span  in  feet.  The  quotient  is  the  breaking  weight  at  the 
middle. 

In  Hundredweights.  —  Substitute  133  for  the  multiplier  7  in  the  preceding 
calculation. 

The  second  last  column  in  table  No.  230  was  calculated  by  this  rule. 
The  formulas  are  — 


In  tons,  W  =  ...............................  (7) 


2d  Approximate  Rule.  In  Tons.  —  Multiply  the  breadth  of  the  joist  by 
the  square  of  the  depth  in  inches,  and  by  0.6;  and  divide  the  product  by 
the  span  in  feet.  The  quotient,  plus  i,  is  the  breaking  weight  at  the 
middle. 

In  Hundredweights.  —  Substitute  12  for  the  multiplier  0.6  in  the  preceding 
calculation.  The  quotient,  plus  20,  is  the  breaking  weight. 

The  formulas  are  :  — 


In  tons,  W  =  +i  ...................................   (9) 

,,r     12  bd2  /     x 

In  cwts.,  W  =  —  —  -,-  +  20  .................................   (10) 

W  =  the  breaking  weight  at  the  middle,  b  —  the  breadth,  and  d  =  the 
depth  in  inches;  /=the  span  in  feet. 

Note.  —  The  ist  approximate  rule  is  better  than  the  2d  rule. 


WROUGHT-IRON   FLANGED  BEAMS  OR  JOISTS. 


6S7 


DEFLECTION  AND  ELASTIC  STRENGTH  OF  SOLID  WROUGHT-IRON  FLANGED 
BEAMS  OR  JOISTS  OF  UNIFORM  SYMMETRICAL  SECTION. 

With  the  particulars  already  given  for  the  beams  A,  B,  D,  and  E,  page 
655,  and  the  subjoined  deflections  under  given  loads  at  the  middle,  within 
the  elastic  limits,  the  values  of  the  coefficient  of  elasticity,  E,  calculated 
from  the  general  formula  (13),  page  531,  by  inversion,  are  as  follows: — 


BEAM. 

Span. 

Load  at  the 
Middle. 

Deflection. 

E. 

feet. 

tons. 

inches. 

20 

8.929 

.848 

13,196 

10 

8.929 

.132 

10,588 

B.  ,- 

20 

7-143 

I.I50 

13,100 

10 

14.286 

•330 

11,414 

20 

3-571 

1.420 

12,120 

. 

10 

8.929 

.440 

12,232 

IO 

5-357 

.405 

13,692 

5 

8.929 

.158 

7,314 

Average  coefficient  of  elasticity,  E, 
excluding  the  last,  as  exceptio 

for  7  beams, 
nal 

12,334 

To    adapt    the    general    formula    just    named,    the    constant    becomes 
4x  12,334  =  49,336;  or  (49336-^123=)  28.5,  when  the  span  is  in  feet:  — 

Deflection  of  Solid  Rolled  Wrought-iron  Flanged  Beams  of  Uniform 
Symmetrical  Section,  loaded  at  the  Middle. 


D  = 


W/3 


28.5 


# +1.1550") 


D  =  the  deflection  in  inches,  W  =  the  load  in  tons,  /  the  span  in  feet,  d" 
the  reputed  depth  in  inches,  being  the  whole  depth  minus  the  thickness  of  the 
lower  flange;  a  the  sectional  area  of  the  lower  flange,  and  a"  the  sectional 
area  of  the  web  reckoned  on  the  reputed  depth,  both  in  square  inches. 

Note. — If  the  same  weight  be  uniformly  distributed,  the  divisor  45.7  is 
to  be  used. 

Approximate  Rule. 

Load  at  the  Middle. — To  the  sectional  area  of  one  flange  add  one-fourth 
of  the  sectional  area  of  the  web,  calculated  on  the  total  depth,  both  in 
inches;  multiply  the  sum  by  the  square  of  the  depth  in  inches,  and  by 
114,  making  a  product  A.  Multiply  the  load  in  tons  by  the  cube  of  the 
span  in  feet;  and  divide  this  product  by  the  product  A.  The  quotient  is 
the  deflection  in  inches. 

Load  Uniformly  Distributed. — Use  the  multiplier  183  in  the  calculation, 
instead  of  114. 

STRENGTH  OF  RIVETTED  WROUGHT-IRON  JOISTS. 

Compared  with  solid-rolled  joists,  the  strength  of  rivetted  joists  is  less, 
and  the  deflection  is  greater.  A  series  of  rivetted  plate-joists  of  uniform 

42 


658 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


section  were  constructed  and  tested  for  deflection  by  the  late  Mr.  Thomas 
Davies,  in  I856.1     The  sections  of  these  beams  are  shown  by  Figs.  263-268; 


No.  7. 


Figs.  263-268. — Sections  of  Rivetted  Wrought-iron  Joists. 


they  consist  of  plate-webs  and  flanges,  united  by  angle-irons,  of  which  the 
scantlings  are  given  on  the  figures, — 

No.  of  joist, i.  2.  3.  4.  5.  7. 

Weight  of  joist,...    4.25  ...    6.5  ...  20.5  ...  14.75  •••  J3-62  ...  13.62  cwts. 
Span  for  trial, n.66  ...  16.5  ...  28.5  ...  28.5    ...  22.5    ...  22.5  feet. 

The  loads  rested  on  spaces  at  the  middle  of  the  beams,  2 1  inches  wide. 

The  elastic  strengths  and  deflections  of  the  joists,  as  deduced  from  the 
record  of  the  results,  were  as  follows : — 


Elastic  strength,....  n%  ...  n%  ...  10^   ...      7     ...   13    ...   13  tons. 
Deflection, 437  ...  .625  ...  2.000  ...  1.620  ...  .875  ...  .875  inches. 

All  the  beams  except  No.  2  were  unsymmetrical,  and  an  approximate  rule 
for  strength  and  deflection  may  be  constructed,  by  making  a  calculation 
for  each  beam  in  the  position  in  which  it  was  tested,  and  in  an  inverted 
position,  in  terms  of  the  flange  and  angle-irons  which  are  undermost,  in 
each  position  respectively;  and  finding  the  mean  results.  In  this  way  the 

1  See  a  paper  read  at  a  meeting  of  the  Edinburgh  Architectural  Institute,  February  18, 
1856. 


WROUGHT-IRON   FLANGED   BEAMS   OR  JOISTS. 


059 


breaking  weights,  columns  2  and  3  in  the  following  tablet,  were  arrived  at 
for  e'ach  joist,  applying  the  formula  (  5  ),  page  656,  for  solid-rolled  joists. 
Two-thirds  of  these  mean  calculated  breaking  weights  are  given  in  the  4th 
column ;  and  they  are  probably  the  real  breaking  weights,  since  they  average 
exactly  twice  the  observed  elastic  strength  given  in  the  last  column. 


JOISTS. 

Calculated 
Breaking 
Weight. 

Mean 
Breaking 
Weight. 

Two-thirds 
of  the 
Mean. 

Observed 
Elastic 
Strength. 

No   i,                 

tons. 
34.  5  2 

tons. 

tons. 

tons. 

Do.    inverted,  

38.65 

36.^9 

24.4 

flj? 

No  2,  

31.80 

21.2 

nK 

No   3, 

37.36 

28.Q2 

33.14 

22.1 

I0|/- 

No  4  . 

13.74 

Do.    inverted,  

27.03 

20.40 

13-6 

7 

No  5,  .. 

33.47  ) 

(  13 

No.  7  (No.  5  inverted),  

5M9  ( 

42.53 

28.4 

<    j 
)  13 

Averages. 

22  O 

no 

By  an  appropriate  alteration  of  the  coefficient  in  the  rule  for  solid-rolled 
joists,  therefore,  the  following  rule  is  obtained: — 

Approximate  Rules  for  the  Strength  of  Rivetted  Wrought-iron  Joists. 

In  tons. — Find  the  sectional  areas  of  the  upper  and  the  lower  flanges 
with  their  angle-irons  respectively;-  to  half  the  sum  of  these  areas  add  one- 
fourth  of  the  sectional  area  of  the  web,  calculated  on  the  total  depth,  all  in 
inches;  multiply  this  last  sum  by  the  depth  in  inches,  and  by  4^3;  and 
divide  by  the  span  in  feet.  The  quotient  is  the  breaking  weight  at  the 
middle. 

In  hundredweights. —Substitute  94  for  the  multiplier  4^  in  the  preceding 
operation. 

The  formulas  are: — 


In  tons, W  = 


Incwts, 


(12) 
(13) 


in  which  a'  is  half  the  sum  of  the  upper  and  lower  sectional  areas,  a"  the 
sectional  area  of  the  web,  d  the  depth,  /  the  span,  and  W  the  load  at  the 
middle. 

Note  to  the  rule. — If  the  beam  is  symmetrical  in  section,  the  section  for 
one  flange  only  is  calculated. 

Similarly,  let  the  deflections  for  the  elastic  strengths,  for  each  beam 
in  its  first  position,  and  as  inverted,  be  calculated  by  the  formula  (u), 
page  657,  for  solid-rolled  joists.  They  are  given  in  the  2d  column  of  the 


66o 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


following  tablet,  and  the  mean  for  each  is  given  in  the  3d  column.  The 
actual  deflections,  in  the  4th  column,  are  greater  than  those  in  the  3d 
column,  in  the  ratios  indicated  in  the  last  column. 


JOISTS. 

Calculated 
Deflections. 

Mean 
Calculated 
Deflections. 

Actual 
Deflection. 

Ratio  of  Actual 
to  Calculated 
Deflections. 

No   i 

inches. 
.227 
.203 

I.I48 

1.483 

inches. 

.215 

.418 

I.3I6 

1.  606 

.654 

inches. 

437 
.625 

2.000 
1.620 

|  -875 
(  .875 

ratio. 

i  to  2.033 
i  to  1.495 

i  to  1.520 

i  to  1.009 

i  to  1.338 
i  to  1.338 

i  to  1.479 

Do.  inverted,  

No.  2,  

No.  i,., 

Do  inverted 

No  4. 

2.130 
1.083 

Do  inverted 

No.  5,  . 

•730) 
•578J 

g  No.  7,  . 

No  7  (No  5  inverted) 

Average  ratio,  not  includin 

There  is  considerable  variation  in  the  ratio  of  the  calculated  to  the  actual 
deflections;  the  average  is  i  to  i^.  Modify  accordingly  the  coefficient 
in  the  approximate  rule  for  solid-rolled  joists,  page  657 : — 

Approximate  Rule  for  the  Deflection  of  Rivetted  Wr ought-Iron  Joists. 

Load  at  the  middle. — Find  the  sectional  areas  of  the  upper  and  the 
lower  flanges  with  their  angle-irons  respectively;  to  half  the  sum  of  these 
areas  add  one-fourth  of  the  sectional  area  of  the  web,  calculated  on  the 
total  depth,  all  in  inches ;  multiply  this  last  sum  by  the  square  of  the  depth 
in  inches,  and  by  75,  making  a  product  A.  Multiply  the  load  in  tons  by 
the  cube  of  the  span  in  feet;  and  divide  this  product  by  the  product  A. 
The  quotient  is  the  deflection,  in  inches. 

Load  uniformly  distributed. — Use  the  multiplier  120  in  the  calculation, 
instead  of  75. 

Note  to  the  Rule. — If  there  be  no  flanges,  the  angle-irons  alone  are  to  be 
taken  as  representing  flange-area. 

Remarks. — i.  The  experimental  elastic  strengths,  as  well  as  the  deflec- 
tions, of  Nos.  5  and  7  beams,  which  were  in  fact  the  same  beam  in  reverse 
positions,  are  identical.  The  identity  here  observed  is  confirmatory  of  the 
general  principle  of  the  elasticity  of  beams,  enunciated  at  page  517. 

2.  It  follows,  from  experimental  tests,  that  the  strength  of  solid-rolled 
joists  is  to  that  of  rivetted  joists,  of  equal  weights,  as  i^  to  i;  and  that 
their  deflections,  under  equal  loads,  are  as  i  to  i^. 


BUCKLED   IRON   PLATES. 

Buckled  plates,  so  named  by  Mr.  Mallett,  the  inventor,  are  bulged 
plates,  which  are  curved  or  arched,  with  a  very  small  rise  or  curvature, 
springing  from  the  edges  of  the  plate,  a  narrow  strip  of  which,  all  round, 


RAILWAY  RAILS. 


66 1 


is  retained  in  the  original  plane  of  the  plate.  Buckled  plates  are  very  rigid, 
and  are  capable  of  sustaining  heavy  loads.  When  bolted  down,  or  rivetted 
all  round  the  edges,  they  offer  twice  the  resistance  that  they  do  if  simply 
supported;  and  if  two  opposite  sides  be  wholly  unsupported,  the  resistance 
is  only  s/8  ths.  Less  than  2  inches  of  rise  is  sufficient  for  a  ^-inch  plate, 
4  feet  square.  A  ^-inch  Staffordshire  plate,  3  feet  square,  with  a  2-inch 
flat  border,  buckled  with  a  rise  of  i  ^  inches,  is  crippled  with  a  load  of 
9  tons  distributed  over  half  the  surface ;  if  rivetted  down,  1 8  tons  are  required 
to  cripple  it.  Plates  of  soft  puddled  steel  bear  twice  these  loads  before 
being  crippled.  The  strength  appears  to  increase  as  the  square  of  the 
thickness.  The  factor  of  safety  adopted  by  Mr.  Mallet  is  4  for  steady 
loads,  and  6  for  moving  loads. 


RAILWAY  RAILS. 

RAILS  OF  SYMMETRICAL  SECTION. 

These  are  beams  of  limited  depth  and  considerable  flange-area,  and 
the  strength  should  be  calculated  by  formula  (22),  page  512,  repeated 
below;  for  the  application  of  which  the  section  of  a  double-headed  rail 
is  to  be  divided  for  the  calculation,  according  to  the  annexed  diagram, 
Fig.  269.  a,  a,  a,  a,  are  the  flange  portions,  c  d  the  web,  d  the  depth 
of  the  rail,  and  d"  the  vertical  distance  apart  of  the  centres  of  the 
flanges.  That  the  sectional  area  of  the  flange  may  be  correctly  ascertained, 


Nl      17 


Fig.  269. — For  Transverse  Strength  of  Rail  of 
Symmetrical  Section. 


Fie.  270. — Squaring  the  Section  of  a  Double- 
headed  Rail. 


the  surface  should  be  divided  into  thin  strips  parallel  to  the  neutral  axis, 
as  in  the  diagram,  the  area  of  each  of  which  should  be  calculated.  If  the 
outer  contour  of  the  flange  is  circular,  as  is  usually  the  case,  the  resultant 
centre  of  the  flange  a  a  may  be  taken  as  passing  through  the  centre  of  the 
circle.  If  the  flanges  are  otherwise  formed,  the  position  of  their  centre  of 
gravity,  ascertained  by  the  rule,  page  514,  may  be  taken.  Approximate 
results  may  be  obtained  by  squaring  the  section  of  the  flanges,  by  the  eye, 
and  calculating  from  the  centres  of  the  rectangles,  as  in  Fig.  270. 


662 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


W/ 


(2) 


(4  a'  -     +1.155  /'</*) 


W  =  the  breaking  weight  at  the  middle,  in  tons. 

0'  =  the  net  sectional  area  of  one  flange,  in  inches  (excluding  the  central 
portion  pertaining  to  the  web). 

d  —  the  total  depth  of  the  rail,  in  inches. 
d"  =  the  vertical  distance  apart  of  the  centres  of  the  flanges. 
t'  =  the  thickness  of  the  web. 

/=the  length  of  span  between  the  supports,  in  inches. 
s  =  the  ultimate  tensile  strength,  in  tons  per  square  inch. 

RULE  i. — To  find  the  Ultimate  Transverse  Strength  of  a  rail  of  symmet- 
rical section.  Multiply  the  sectional  area  of  the  flange  portion  of  one 
head  by  the  square  of  the  vertical  distance  apart  of  the  centres  of  the 
heads,  and  by  4;  and  divide  by  the  depth  of  the  rail  [B].  Multiply  the 
thickness  of  the  web  by  the  square  of  the  depth  of  the  rail,  and  by  1.155 
[C].  Multiply  the  sum  of  the  quotient  B  and  the  product  C  by  the  ulti- 
mate tensile  strength,  and  divide  by  the  span.  The  last  quotient  is  the 
breaking  weight  at  the  middle. 

RULE  2. — To  find  the  Ultimate  Tensile  Strength  of  a  rail  of  symmetrical 
section.  Multiply  the  breaking  weight  at  the  middle  by  the  span,  and 
divide  the  product  by  the  sum  of  the  quotient  B  and  the  product  C 
described  in  Rule  i.  The  last  quotient  is  the  ultimate  tensile  strength. 

Mr.  R.  Price  Williams,  in  a  paper  of  exceptional  value,1  gives  a  num- 
ber of  tests  of  the  ultimate  transverse  strength  of  iron  and  steel  rails,  made 
for  him  by  Mr.  Kirkaldy.  From  this  paper,  the 
following  data,  collected  in  table  No.  232,  are 
derived  for  several  double-headed  rails  of  sym- 
metrical section. 

The  tensile  strengths  of  these  rails,  in  the  last 
column,  are  calculated  by  Rule  2  above.  There  is 
no  information  as  to  the  observed  tensile  strength 
of  the  rails;  but,  in  the  course  of  discussion,  Mr. 
Berkley  gives  the  tensile  strength  of  steel  rails 
tested  by  him,  varying  from  40  to  50  tons  per 
square  inch;  and  the  mean  of  these  strengths  is 
the  same  as  the  average  of  the  strengths  for  steel 
rails  calculated  in  the  last  column. 

Mr.  Baker   gives  a  full-size  section,  with  the 
Fig.  271.— Section  of  Steel  Rail  breaking   weight,  for   double-headed   steel   rails, 
Rliiwl^lUit^Srd5111^   manufactured  by  Sir  John  Brown  &  Co.  for  the 
Great  Indian  Peninsula  Railway,  weighing  68  Ibs. 
per  yard.2     The  section,  Fig.  271,  has  an  area  of  6.88  square  inches,  of 

1  "On  the  Maintenance  and  Renewal  of  Permanent  Way,"  in  the  Proceedings  of  the 
Institution  of  Civil  Engineers,  1865-66,  vol.  xxv.,  page  353. 

2  The  Strength  cf  Beams,  page  86. 


RAILWAY  RAILS. 


663 


which  the  flange-area  at  the  bottom  is  1.667  inches.  The  web  is  .70  inch 
thick,  the  total  depth  5  inches,  and  the  vertical  distance  of  the  centres  of 
gravity  of  the  heads  3.72  inches.  The  ultimate  tensile  strength  varied  from 

Table  No.  232. — TRANSVERSE  STRENGTH  OF  IRON  AND  STEEL  DOUBLE- 
HEADED  RAILS.     1866. 

Span  60  inches.     Load  applied  at  the  middle. 
(Deduced  from  Mr.  Price  Williams'  data). 


DEPTH. 

SECTIONAL  AREA. 

DESCRIPTION. 

Weight 
per  Yard 
esti- 
mated). 

Thickness 
of  Web. 

Total. 

Centres  of 
Flanges. 

One 

Flange. 

Total. 

IRON  RAILS. 

Ibs. 

inches. 

inches. 

inches. 

sq.  inches. 

sq.  inches. 

L    L.  &N.  W.  Ry  

82 

5-40 

4.20 

.82 

I.9II 

8.25 

N             Do 

82 

e  40 

4..OO 

.82 

1.  071 

8.29 

M    Ebbw  Vale  Co 

82 

2.QO 

.78 

8.17 

p           Do.           

68 

C.O4 

.68 

1.70 

6.83 

STEEL  RAILS. 

78 

C.4.O 

4.2O 

•  7? 

1.81 

7-67 

B    Brown  &  Co 

7Q 

522 

•2.72 

1.902 

7.72 

C     Bessemer 

74. 

3.90 

-74 

1.701 

7-25 

D    Cammell  

72.5 

5-02 

Pa 

•73 

1.722 

7.11 

Calculated 

DESCRIPTION. 

Breaking 
Weight. 

ELASTIC  STRENGTH. 

Elastic 
Deflection. 

Nature  of  Failure. 

Tensile 
Strength 
per  square 
inch. 

IRON  RAILS. 

tons. 

tons. 

per  cent. 

inches. 

tons. 

L    L.  &  N.  W.  R. 

23.000 

11.604 

5°-4 

.198 

f  cracked,  &  \ 
\     canted      j 

26.24 

N            Do. 

21.107 

9.823 

46-5 

.184 

canted 

25.07 

M    Ebbw  Vale  

22.371 

10.716 

47-9 

.235 

snapped 

27.71 

P            Do. 

I5-500 

6.698 

43-2 

.148 

torn 

24.06 

STEEL  RAILS. 

A    Crewe      .     ... 

-zc.701 

16.070 

44-9 

252 

snapped 

43-  9  ! 

B    Brown  &  Co.  ... 

35.432 

15.181 

42.9 

.264 

<  cracked,  &  \ 
I     canted      j 

45-75 

C    Bessemer  
D    Cammell  

33.446 
31-935 

16.967 
16.070 

50-7 
50.3 

•  300 
.322 

canted 
snapped 

46.67 
46.43 

37  to  50  tons  in  nine  specimens,  averaging  45  tons  per  square  inch.  The 
breaking  weight  at  the  middle,  on  a  span  of  43.5  inches,  averaged,  in 
forty-five  experiments,  38.7  tons.  Applying  Rule  i  above,  the  calculated 
breaking  weight  is, 

[(4  X3i712  x  !.667)  +  (  1.155  x  .7ox5')] 
5 
(18.45  +  20.21)  x  45  +  43.5  =  40  tons. 


66, 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


This  weight  is  1.3  tons  more  than  the  average  given  by  experiment;  and  it 
is  as  near  as  can  reasonably  be  expected,  considering  the  variable  elements 
of  the  data.  But,  even  this  small  excess  may  be  explained  away;  for  a 
sectional  area  of  6.88  square  inches,  at  10.2  Ibs.  per  square  inch,  gives  a 
weight  of  70  Ibs.  per  yard,  whilst  the  actual  weight  was  68  Ibs.  It  is,  there- 
fore, probable  that  the  section  was  less  than  6.88  square  inches,  and  that 
the  strength,  if  it  were  calculated  from  the  exact  section,  would  be  less 

SQ 

than  40  tons  in  the  ratio  of  70  to  68,  or  about  (40  x  —  = )  38.8  tons,  which 

70 

is  virtually  identical  with  the  experimental  breaking  weight.1 

STRENGTH  OF  DOUBLE-HEADED  BESSEMER  STEEL  RAILS  RELATIVELY  TO 
THE  PROPORTION  OF  CONSTITUENT  CARBON. 

Mr.  Price  Williams  gives,  in  an  appendix  to  his  paper,  already  men- 
tioned, a  table  showing  the  strength  and  deflection  and  set  of  double- 
headed  steel  rails,  of  86  Ibs.  per  yard,  5^  inches  deep,  manufactured  by 
Sir  John  Brown  &  Co.  for  the  Great  Indian  Peninsula  Railway.  The 
sectional  area  was  probably  8.43  square  inches.  The  experiments  were 
made,  under  Mr.  Berkley's  direction,  with  rails  containing  various  propor- 
tions of  carbon.  The  following  table  is  reduced  from  Mr.  Price  Williams' 
table,  and  it  shows  a  notable  correspondence  between  the  percentages  of 
constituent  carbon  and  the  breaking  load  applied  at  the  middle;  for  whilst 
the  carbons  increase  from  .40  to  .55  per  cent.,  the  ultimate  loads  increase 
from  40  to  52^  tons. 

Table  No.  233. — TRANSVERSE  STRENGTH  OF  DOUBLE-HEADED  BESSEMER 
STEEL  RAILS.     1864. 

Span  43.5  inches.     Load  applied  at  the  middle. 
(To  show  the  influence  of  the  constituent  carbon  on  the  strength.) 


Constituent 
Carbon. 

Ultimate  Strength. 

Elastic  Strength. 
(Deduced  from  the  experimental  data.) 

Load. 

Deflection. 

Set. 

Load. 

Deflection. 

Set 

per  cent. 
.40 
.46 

49 
.50 

•55 

tons. 
40 
40 

50 
52.5 
52.5 

inches. 

3-94 
2.64 
4.18 
4.68 
4.40 

inches. 
3-74 
2-34 

3-77 
4.28 
4.02 

tons. 

15 

20 
22.5 
22.5 
25 

per  cent. 

37-5 
So 
45 
43 
48 

inches. 
.10 
.14 
.I65 
.130 
.165 

inches. 
.01 
.05 

•03 
.01 

.04 

Thirty  Bessemer- steel  rails,  manufactured  at  Barrow-in-Furness,  were 
analyzed  and  tested  in  different  ways  for  strength.  The  tensile  strength 
increased  generally  with  the  proportion  of  carbon  in  the  steel,  as  may  be 
seen  from  the  following  abstract  for  thirty  rails,  from  a  table  given  by 
Mr.  J.  T.  Smith:2— 

1  Mr.  Baker  arrives  at  a  breaking  weight  of  38.9  tons  by  means  of  the  ingenious  diagram- 
matic reduction  already  noticed. 

z  "  On  Bessemer  Steel  Rails,"  in  the  Proceedings  of  the  Institution  of  Civil  Engineers^ 
1874-75,  v°l-  xlii.,  PaSe  74- 


RAILWAY  RAILS. 


66$ 


Carbon. 

per  cent 

.28 


SOFT  RAILS. 

Tensile  Strength 

per  square  inch. 

tons. 

30-90 

29    32.60 

30    32.94 

31     32.67 

32     33-04 


Carbon. 


HARD  RAILS. 

Tensile  Strength 
per  square  inch. 
tons. 

37-01 


per  cent. 

•36 

•39  ..................  4i.4i 

•40  ..................  37-68 

•43  ..................  39-10 

.44  ..................  41.02 

•45  ..................  44-00 

•50  ..................  45-79 

•57  ..................  50-42 


Averages  .30  32.43  .44 

RAILS  OF  UNSYMMETRICAL  SECTION. 


42.05 


The  general  rule  at  page  517,  is  applicable  for  the  calculation  of  the  trans- 
verse strength  of  bridge-rails  and  flange-rails,  which  are  the  only  varieties  of 
rails  that  are  not  symmetrical  in  section.  That  rule  embodies  the  final  cal- 
culation formulated  in  equation  (  25  ),  page  516,  in  terms  of  the  total  tensile 
resistance  of  the  section,  and  the  vertical  distance  apart  of  the  centres  of 
tension  and  compression.  From  that  equation,  it  follows  that  W  I—  4  S  </3; 
and,  as  S  is,  by  the  sixth  step  of  the  rule,  page  517,  equal  to  1.73  s  x  (sum  of 
ist  products,  tensional)^,,  in  which  h^  is  the  height  of  the  neutral  axis 
above  the  base,  by  substitution,  W/=  4  x  i. 73  ^  (sum  of  ist  products)  x^3- 

«, 
and,  putting  A  =  the  sum  of  the  ist  products, — 


W_ 


_6.g2  sd3  A 


(3) 


6.92  d3  A 


(4) 


W  =  the  breaking  weight  at  the  centre,  in  tons. 
^  =  the  ultimate  tensile  strength  of  the  metal,  in  tons  per  square  inch. 

d3  =  the  vertical  distance  apart  of  the  centres  of  tension  and  compression, 
in  inches. 

hi  =  the  height  of  the  neutral  axis  above  the  base  of  the  section,  in  inches. 
/•the  span,  in  inches. 

A  =  the  sum  of  the  products  obtained  by  multiplying  the  areas  of  the 
strips  of  the  reduced  section  under  tensional  stress,  by  their  mean  distances 
respectively  from  the  neutral  axis,  in  inches,  as  described  in  step  4  of  the 
rule,  page  517. 

RULE  3. — To  find  the  Ultimate  Transverse  Strength  of  a  Rail  of  Unsym- 
metrical  Section,  i.  Divide  the  section  into  strips,  which  may  be  of  equal 
thickness,  parallel  to  the  base.  2.  Reduce  the  width  of  the  flange  portions 
in  the  ratio  of  1.73  to  i.  3.  Find  the  position  of  the  centre  of  gravity  of 
the  section  as  thus  reduced  (by  the  rule,  page  514);  it  is  that  of  the 
neutral  axis  of  the  total  section.  4.  Multiply  the  areas  of  the  strips  of  the 
reduced  section,  below  the  neutral  axis,  by  their  respective  mean  distances 
from  it;  and  also  by  the  squares  of  these  distances;  and  divide  the  sum  of 
these  second  products  by  the  sum  [A]  of  the  first  products;  the  quotient 


666 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


is  the  distance  of  the  position  of  the  resultant  centre  of  resistance  below 
the  neutral  axis.  5.  Make  the  same  calculation  ( 4 )  to  find  the  position  of 
the  resultant  centre  above  the  neutral  axis.  6.  The  sum  of  the  two  dis- 
tances thus  found,  is  the  vertical  distance  apart  of  the  centres  of  tension  and 
compression.  7.  Multiply  the  sum  A  by  the  distance  apart  of  the  centres 
of  stress,  and  by  the  ultimate  tensile  strength  in  tons  per  square  inch,  and 
by  6.92;  and  divide  the  product  by  the  height  of  the  neutral  axis  above 
the  base  of  the  section,  and  by  the  span.  The  quotient  is  the  breaking 
weight  in  tons  at  the  middle. 

RULE  4. — To  find  the  Ultimate  Tensile  Strength  of  a  Rail  of  Unsymmetrical 
Section,  when  the  Transverse  Strength  is  given.  Perform  the  same  prelim- 
inary operations  as  for  Rule  3 — Nos.  i,  2,  3,  4,  5,  and  6;  then,  7,  multiply 
the  breaking  weight  in  tons  at  the  middle  by  the  length  of  the  span,  and 
by  the  height  of  the  neutral  axis  above  the  base  of  the  section;  and  divide 
the  product  by  the  sum  A  (referred  to  in  Rule  3),  and  by  the  distance 
apart  of  the  centres  of  stress,  and  by  6.92.  The  quotient  is  the  ultimate 
tensile  strength  in  tons  per  square  inch. 

Note. — The  dimensions  are  in  inches,  and  the  pressures  and  weights  in  tons. 

Steel  Flange-Rails. — The  steel  rails  designed  by  Mr.  John  Fowler,  and 

manufactured  by  the  Dowlais  Iron  and  Steel  Company,  for  the  Metropolitan 

Railway  (Fig.  272),  are  4^  inches  high  and  6^  inches  wide  at  the  base; 

they  have  a  sectional  area  of 
8.24  square  inches,  and  weigh 
84  Ibs.  per  yard.  Several  of 
these  rails  were  tested  by  Mr. 
Kirkaldy :  in  which  the  web  was 
.65  inch  thick,  the  head  2.5 
inches  wide,  and  the  flange 
6.4  inches  wide.  The  thickness 
of  the  flanges  was  .65  inch  near 
the  web,  and  .37  inch  near 
the  edge.  The  ultimate  tensile 
strength  is  said  by  Mr.  Baker  to 
be  35  tons  per  square  inch. 

To  apply  the  rule  for  the  trans- 
verse strength,  produce  the  sides 
of  the  web  to  the  top  and  the 
bottom  of  the  section,  at  c'  c"  and 

d'  d"  ;  and  reduce  the  width  of  the  flange  portions,  a  a  and  b  b,  in  the  ratio  of 
1.73  to  i,  following  the  contour-lines  a'  a'  and  b'b'.  Divide  the  section  into 
strips,  say  .20  inch  in  width,  and  find  the  centre  of  gravity  of  the  reduced 
section;  the  line  ii,  passing  through  it,  is  the  neutral  axis,  2.51  inches  below 
the  top,  and  1.99  inches  above  the  bottom.  The  resultant  centres  of  resist- 
ance are  1.595  inches  above  and  1.779  inches  below  the  neutral  axis;  and 
their  distance  apart  is  (1.595  +  1.779  =  )  3.374  inches. 

To  find  the  total  stress  in  tension,  in  the  lower  part  of  the  section,  the 
sum  of  the  first  products,  4.324  (which  is  the  same  for  tension  and  com- 
pression), is  multiplied  by  1.73  times  35  tons,  the  ultimate  tensile  strength, 
and  divided  by  1.99  inches,  the  distance  of  the  neutral  axis  from  the  base: — 

4-324  x  '35  x  I>73'  =  131.56  tons,  total  tensile  resistance. 
1.99 


Fig.  272.— Section  of  Steel  Flange-Rail  for  the 
Metropolitan  Railway. 


RAILWAY  RAILS. 


667 


The  breaking  weight  at  the  middle,  on  a  span  of  60  inches,  is,  then, 

,17     131.56  x 
_:L_D  — 


DO 


=  29.59 


This,  it  may  be  noted,  is  an  application  of  formula  (  25  ),  page  516. 

Let  the  Metropolitan  rail  be  calculated  for  its  transverse  strength  when 
upside  down:  the  product  4.324  x  (35  x  1.73)  is  divided  by  2.51  inches, 
the  distance  of  the  upper  surface,  now  downwards,  of  the  head  of  the  rail 
from  the  neutral  axis  :  — 

4-324x(35  x  I>73)=I04t30  tons>  totai  tensile  resistance  upside  d 

and  the  breaking  weight  at  the  middle,  on  a  span  of  60  inches,  is 
W  =  *°4.3°  x  3.374  x  4  =         6  tons. 

DO 

To  compare  these  calculated  results  with  the  results  of  Mr.  Kirkaldy's 
experimental  tests,  these  are,  with  Mr.  Fowler's  permission,  here  ab- 
stracted :  — 

Table  No.  234.  —  TRANSVERSE  STRENGTH  OF  STEEL  FLANGE-RAILS  FOR 
THE  METROPOLITAN  RAILWAY.     1867. 

Span  60  inches.     Load  applied  at  the  middle- 
(Reduced  from  Mr.  Kirkaldy's  Reports.) 


Ultimate  Strength. 

Elastic  Strength. 

Elastic 
Deflec- 
tion 
per  ton. 

Load. 

Deflec- 
tion. 

Load. 

Deflec- 
tion. 

Set. 

S  olid  rail,  normal  position— 
ist  specimen  

tons. 

30-393 
29.632 
29.043 
28.457 
28.733 

inches. 

942 
8.69 
10.54 
9.18 
14.78 

tons. 

I2.50O 
II.607 
11.607 
12.500 
11.607 

per  cnt. 

inches. 

.255 
.232 
.238 
.250 
.232 

inches. 

.022 
.028 
.030 
.021 
.030 

inches. 

2d        do 

^d        do 

4th       do  

5th       do  

Averages  

29.270 

10.52 

11.964 

41 

.2414 

.026 

.0202 

Solid  rail  inverted. 

22.OI4 

542 

I5.I80 

69 

.290 
•235 

.040 

.0191 

Normal    position,   holes  y 
punched  in  the  flanges,  f 
Average  of  six  speci-  ( 

I5.6l8 

.669 

11.904 

76 

.023 

.0197 

Normal    position,   holes  ") 
drilled.     Mean  of  two  \ 
specimens            ) 

23.934 

4.51 

12.500 

52 

.267 

.025 

.0214 

Note. — The  holes,  punched  or  drilled,  were  i.io   inches   in  diameter, 
.75  inch  from  the  edge  of  the  flange. 


668 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


Fig.  273. — Section  of  Iron 
Flange-Rail.     Scale,  %th. 


The  breaking  weight  of  the  rail  varied,  in  six  specimens,  from  28.46  tons 
to  30.39  tons;  average,  29.27  tons.  The  calculated  breaking  weight  is 
29.59  tons,  or  about  y$  ton  in  excess  of  the  average. 

Inverted,  one  specimen  broke  with  22.01  tons;  the  calculated  breaking 
weight  is  23.46  tons,  or  nearly  i^  tons  more. 
The  elastic  strength  in  this  position  was  greater 
than  in  the  normal  position. 

Influence  of  Holes  in  the  Flange.  —  When  punched, 
the  effect  was  to  reduce  the  breaking  weight  nearly 
a  half.  When  drilled,  the  ultimate  strength  was 
only  reduced  about  a  sixth.  But  the  elastic 
strength  remained,  in  both  cases,  unimpaired; 
and  the  elastic  deflection  per  ton  was  practically 
identical  in  all  cases  —  averaging  about  .20  inch 
per  ton. 

Wrought-Iron  Flange-  Rails.  —  The  annexed  sec- 
tion, Fig.  273,  shows  a  wrought-iron  flange-rail, 
5  inches  high,  weighing  75  Ibs.  per  yard.  Ten  specimens  of  rail  of  this 
section,  of  Cleveland  manufacture,  were  tested  for  transverse,  tensile,  and 
compressive  strength  by  Mr.  Kirkaldy.  The  samples  for  the  tensile  and 
compressive  tests  were  cut  from  the  middle  of  the  head  and  of  the  flange. 

Tensile.  Compressive. 

HEAD:  —  Elastic  strength  per  square  inch  ......  13.21  tons.     18.13  tons- 

Ultimate  strength      „         „       ......  20.93     »        67.00     „ 

FLANGE:  —  Elastic  strength  per  square  inch....  13.62     „ 
Ultimate  strength      „         ,,       ....21.83     » 

Ultimate  transverse  strength,  span  3  feet  ......  33.60  tons. 

The  centre  of  gravity  of  the  reduced  section,  that  is,  the  neutral  axis  of 
the  entire  section,  shown  in  the  Fig.  273,  is  2^  inches  above  the  base  of  the 
section,  or  the  height  is  one-half  the  total  height  of  the  rail.  The  resultant 
centre  of  tensile  stress  is  1.974  inches  below  the  neutral  axis,  and  that  of 
compressive  stress  is  1.683  inches  above  it,  as  indicated.  The  vertical 
distance  apart  of  these  resultant  centres  is  (1.683  +  I<974=:)  3-^57  inches, 
and  by  Rule  4, 


*%.  "2    ( 


- 
6.92x3.657x5.058 


=23.62  tons  per  square  inch, 


the  ultimate  tensile  strength  of  the  wrought-iron  flange-rail,  in  its  lower  or 
flange  portion. 

DEFLECTION  OF  RAILS. 

Double-headed  Rails. — Formulas  for  the  deflection  of  double-headed  rails 
are  deduced  by  equating  the  values  of  s,  the  tensile  strength  per  square 
inch,  given  by  formula  ( 2 ),  page  662,  and  by  formula  ( 2 ),  page  528;  thus : — 

=  - — ;  whence, 


<*-7 


RAILWAY  RAILS.  669 

From  this  equation,  the  following  values  of  D  and  E  are  deduced  :  — 

W/3 


_ 

4E  (4tfV/2+ 
W/3 


4D 


The  values  of  E,  by  formula  (  6  ),  for  the  rails  tested  by  Mr.  Price 
Williams,  as  detailed  in  table  No.  232,  page  663,  are  as  follows:  — 

Iron  Rails,  double-headed.  Steel  Rails,  double-headed. 

E.  E. 

L  ............  11,146  ............  A  ............  13,038 

N  ............  10,571  ............  B  ............  13,588 

M  ............  9,683  ............  C  ............  12,982 

P  ............  12,457  ............  D  ............  i3>°°7 

Averages...  10,964  ................................  I3»I54 

That  is,  the  iron  rails  were  extended,  say,  '/".ooo  of  their  length  per  ton  of 
tensile  stress  per  square  inch  of  section  ;  and  the  steel  rails  were  extended, 
say,  Vis.ooo  of  their  length  per  ton  per  square  inch.  It  has  already  been 
deduced  from  direct  experiments  on  the  elongation  of  bars  (see  pages  623, 
624),  that  the  extension  of  iron  was  from  I/IO>0oo  to  I/I3>00o  part  of  the  length, 
and  that  of  steel  was  I/I3>00o  part  of  the  length,  per  ton  per  square  inch. 
Thus,  the  deductions  from  experiment  on  transverse  resistance,  are  strongly 
corroborated  by  the  results  of  experiment  on  direct  tensile  resistance. 

Substituting  these  values  of  E,  in  round  numbers,  in  formula  (  5  ),  the 
following  formulas  for  the  deflection  of  iron  and  steel  rails,  like  those  tested 
by  Mr.  Price  Williams,  are  derived  :  — 

Deflection  of  Double-headed  Rails,  within  the  Elastic  Limit,  Loaded  at 

the  Middle. 


IRON...          .  D  =  -  _       __  .  (  7  ) 

44,000  (4  a  d  2+  1.155 

STEEL  ..........  D= 


D  =  the  deflection  at  the  middle,  in  inches. 
W=the  load  at  the  middle,  in  tons. 

tf'  =  the  net  sectional  area  of  one  flange,  in  inches  (excluding  the  central 
portion  pertaining  to  the  web). 

d=  the  total  depth  of  the  rail,  in  inches. 

dT  =  the  vertical  distance  apart  of  the  centres  of  the  flanges,  in  inches. 
/'  =  the  thickness  of  the  web,  in  inches. 
/=the  length  of  span  between  the  supports,  in  inches. 

Flange-Rails.  —  By  a  similar  process,  equating  the  values  of  s,  given  by 
formula  (  4  ),  page  665,  and  by  formula  (  2  ),  page  528,  formulas  are  deduc- 
ible  for  the  deflection  of  flange-rails  within  elastic  limits  :  — 


670  STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 

W//^    =4^ED;  and  W/3  ^,  =  6.92x4x^^3  ED  A; 
6.92  a3  A          /2 

whence  the  following  values  of  D  and  E  :  — 


To  find  the  value  of  E  by  the  formula  (  10  ),  for  Mr.  Fowler's  steel  rail, 
investigated  at  page  666,  for  which  the  value  of  the  quantity  A  is  6.983:  — 


1 1.964  tons  x  60  inches3  x  1.99  inches 
:  4.5  inches  x  3.374  inches  x  .2414  inch: 


: . . . =  7264. 

27.68  x  4.5  inches  x  3.374  inches  x  .2414  inchx  6.983 


That  is  to  say,  the  flange  of  Mr.  Fowler's  steel  rail  is  extended  I/7264  part 
of  its  length  per  ton  of  tensile  stress  per  square  inch.  This  fraction  is 
considerably  greater  than  the  fraction  that  was  found  for  the  double-headed 
steel  rails  tested  by  Mr.  Price  Williams,  averaging  I/I3>000  part.  The  greater 
extensibility,  nearly  twice  as  much,  is  in  accordance  with  the  relative  tensile 
strengths  of  the  steels  of  which  the  different  rails  were  made  —  35  tons  per 
inch  for  the  flange-rail,  and  45  tons  for  the  double-headed  rails. 

Substituting  in  formula  (  9  ),  the  value  of  E,  just  found,  the  formula  is 
reduced  to  the  following  form  for  the  deflection  of  steel  flange-rails  like 
Mr.  Fowler's:  — 

Deflection  of  Steel  Flange-  Rails,  within  the  Elastic  Limit,  Loaded  at 

the  Middle. 


D=  i  /  „  \ 

200,000  dd3  A 

To  find,  in  the  absence  of  data,  the  probable  numerical  constant  for  the 
deflection  of  iron  flange-rails,  let  the  constant  in  this  formula  be  reduced  in 
the  ratio  of  52,000  to  44,000,  the  correlative  constants  for  steel  and  iron,  in 

formulas  (  7  )  and  (  8  );  or  to  2oo,ooox  —  =  170,000:  — 

Deflection  of  Iron  Flange-Rails,  within  the  Elastic  Limit,  Loaded  at 

the  Middle. 

D- 


..................... 

170,000  aa3A 

D  =  the  deflection  at  the  middle,  in  inches. 

W  =  the  load  at  the  middle,  in  tons. 

^x  =  the  height  of  the  neutral  axis  of  the  reduced  section,  above  the  base 
of  the  section,  in  inches. 

d=  the  total  height  of  the  rail,  in  inches. 

</3  =  the  vertical  distance  apart  of  the  centres  of  tension  and  compression, 
in  inches. 

/=the  span,  in  inches. 

A  =  the  sum  of  the  products  obtained  according  to  Rule  3,  page  665. 


STEEL  SPRINGS.  671 

STEEL  SPRINGS. 

The  author,  in  1854-55,  investigated  the  elastic  strength  of  laminated 
springs,  in  his  work  on  Railway  Machinery?-  and  he  deduced  the  following 
formulas  for  their  elasticity  and  working  strength  :  — 


<»> 


E  =  the  elasticity,  or  deflection,  in  sixteenths  of  an  inch  per  ton  of  load. 

S  =  the  working  strength,  or  load,  in  tons. 

/=the  span  when  loaded,  in  inches. 

b  =  the  breadth  of  plates  in  inches,  supposed  uniform. 

/  =  the  thickness  of  plates  in  sixteenths  of  an  inch. 

n  -  the  number  of  plates. 

RULES  FOR  THE  ELASTICITY  OF  LAMINATED  SPRINGS. 

RULE  i.  —  To  find  the  elasticity  of  a  laminated  spring.  Multiply  the 
breadth  in  inches  by  the  cube  of  the  thickness  of  each  plate  in  sixteenths 
of  an  inch,  and  by  the  number  of  plates  ;  multiply  the  cube  of  the  span  in 
inches  by  1.66.  Divide  the  second  product  by  the  first.  The  quotient  is 
the  elasticity  in  sixteenths  of  an  inch  per  ton  of  load. 

RULE  2.  —  To  find  the  span  due  to  a  given  elasticity,  and  number  and  size 
of  plates.  Multiply  the  elasticity  by  the  breadth  in  inches,  and  by  the  cube 
of  the  thickness  in  sixteenths,  and  by  the  number  of  plates;  and  divide 
by  1.66.  Find  the  cube  root  of  the  quotient.  The  result  is  the  span  in 
inches. 

RULE  3.  —  To  find  the  number  of  plates  due  to  a  given  elasticity,  span,  and 
size  of  plate.  Multiply  the  cube  of  the  span  in  inches  by  1.66.  Multiply 
the  elasticity  by  the  breadth  of  plate  in  inches,  and  by  the  cube  of  the 
thickness  in  inches.  Divide  the  first  product  by  the  second.  The  quotient 
is  the  number  of  plates. 

Note  i.  —  The  span  and  the  elasticity  are  those  due  to  the  spj/Jng  when 
weighted. 

Note  2.  —  When  extra  thick  back  and  short  plates  are  used,  they  must  be 
replaced  by  an  equivalent  number  of  plates  of  the  ruling  thickness,  prior 
to  the  application  of  Rules  i  and  2.  This  is  found  by  multiplying  the 
number  of  extra-thick  plates  by  the  cube  of  their  thickness,  and  dividing  by 
the  cube  of  the  ruling  thickness.  Conversely,  the  number  of  plates  of  the 
ruling  thickness  given  by  Rule  3,  required  to  be  removed  and  replaced  by 
a  given  number  of  extra-thick  plates,  are  found  by  the  same  calculation. 

Note  3.  —  It  is  assumed  that  the  plates  are  similarly  and  regularly  formed, 
and  that  they  are  of  uniform  width,  and  but  slightly  tapered  at  the  ends. 

RULES  FOR  THE  WORKING  STRENGTH  OF  SPRINGS. 

RULE  4.  —  To  find  the  working  strength  of  a  laminated  spring.  Multiply  the 
breadth  of  plates  in  inches  by  the  square  of  the  thickness  in  sixteenths,  and 

1  Railway  Machinery,  1855,  page  242.     Also,  Railway  Locomotives,  1860. 


672  STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 

by  the  number  of  plates;  multiply  the  working  span  in  inches  by  11.3. 
Divide  the  first  product  by  the  second.  The  quotient  is  the  working 
strength  in  tons  of  load. 

RULE  5.  —  To  find  the  working  span  due  to  a  given  working  strength,  and 
number  and  size  of  plates.  Multiply  the  breadth  of  plate  in  inches  by  the 
square  of  the  thickness  in  sixteenths,  and  by  the  number  of  plates  ;  multi- 
ply the  working  strength  in  tons  by  11.3.  Divide  the  first  product  by  the 
second.  The  quotient  is  the  working  span  in  inches. 

RULE  6.  —  To  find  the  number  of  plates  due  to  a  given  working  strength, 
span,  and  size  of  plate.  Multiply  the  working  strength  in  tons  by  the  span 
in  inches,  and  by  11.3;  multiply  the  breadth  of  plate  in  inches  by  the 
square  of  the  thickness  in  sixteenths.  Divide  the  first  product  by  the 
second.  The  quotient  is  the  number  of  plates. 

RULE  7.  —  To  find  the  required  original  compass  of  the  spring.  Multiply 
the  elasticity  in  sixteenths  per  ton  by  the  working  strength  in  tons,  and  add 
the  product  to  the  desired  working  compass.  The  sum  is  the  whole 
original  compass,  to  which  an  allowance  of  from  ^  to  3/8  inch  should 
be  added,  for  the  permanent  setting  of  the  spring. 

Note  i.  —  The  span  is  that  due  to  the  form  of  the  spring  when  weighted. 

Note  2.  —  Extra  thick  back  or  short  plates  must  be  replaced  by  an 
equivalent  number  of  plates  of  the  ruling  thickness,  before  applying  the 
Rules  4  and  5.  To  find  this,  multiply  the  number  of  extra-thick  plates  by 
the  square  of  their  thickness,  and  divide  by  the  square  of  the  ruling  thick- 
ness. Conversely,  the  number  of  plates  of  the  ruling  thickness  given  by 
Rule  6,  required  to  be  removed  and  replaced  by  a  given  number  of  extra- 
thick  plates,  are  found  by  the  same  calculation. 

Helical  Springs.  —  Most  of  the  data  on  the  strength  of  helical  springs  are 
indefinite  and  contradictory.  It  may  be  assumed  that  the  elasticity,  or 
deflection  per  unit  of  load,  is  as  the  fourth  power  of  the  diameter  or  of  the 
side  of  the  bar,  if  round  or  square,  of  which  the  spring  is  constructed  ;  as  the 
cube  of  the  mean  diameter  of  the  coil  or  helix,  as  the  number  of  free  coils 
of  the  springs,  and  as  the  load  applied.  In  the  "  Report  on  Safety  Valves,"1 
the  following  formula  is  propounded  :  — 


E  =  Compression  or  Extension  of  one  coil,  in  inches. 

d=  diameter  from  centre  to  centre  of  steel  bar  composing  the  spring,  in 
inches. 

w  -  the  weight  applied,  in  pounds. 

D  =  the  diameter,  or  the  side  of  square,  of  the  steel  bar  of  which  the  spring 
is  made,  in  i6ths  of  an  inch. 

C  =  a  constant  which,  from  experiments  made,  may  be  taken  as  2  2  for 
round  steel,  and  30  for  square  steel. 

The  deflection  for  one  coil  is  to  be  multiplied  by  the  number  of  free  coils, 
to  obtain  the  total  deflection  for  a  given  spring. 

1  Transactions  of  the  Institution  of  Engineers  and  Ship-builders  in  Scotland,   1874-75, 
page  39. 


ROPES. 


6/3 


It  is  also  stated  in  the  Report  that  the  relation  between  the  safe  load, 
size  of  steel,  and  diameter  of  coil,  has  been  deduced  from  the  works  of 
Professor  Rankine;  and  that  it  may  be  taken  for  practical  purposes  as 
follows : — 


D= 


4.29 


,  for  round  steel,  .................  (2) 


ROPES— HEMP  AND  WIRE. 
HEMPEN  ROPES. 

By  the  old  ropemakers'  rule  the  breaking  strength  in  hundredweights 
was  equal  to  four  times  the  square  of  the  girth  of  the  rope  in  inches.  This 
is  equivalent  to  Gregory's  rule,  by  which  the  breaking  strength  in  tons  was 
equal  to  one-fifth  of  the  square  of  the  girth  in  inches.  The  square  of  half 
the  girth  represented  the  weight  in  pounds  per  fathom.  The  following  table 
of  the  strength  of  cordage,  is  reduced  from  Mr.  Glynn's1  data.  The  ropes 
recorded  in  the  second  part  of  the  table  are  machine-made  ropes.  Made 
by  the  warm  register,  the  rope  is  stronger  and  more  durable  than  by  the 
cold  register,  as  it  is  more  thoroughly  penetrated  by  the  tar.  But  it  is 
less  pliable,  and  cold-register  rope  is  now  generally  used  for  cranes,  and 
block  and  tackle. 

Table  No.  235. — BREAKING  STRENGTH  OF  TARRED  HEMP  ROPES. 
(Reduced  from  Mr.  Glynn's  data.) 


Size  of  Rope. 

Made  by  the  Old  Method. 

Made  by  the  Register. 

Girth. 

Diameter. 

Common 
Staple 
Hemp. 

Best 
Petersburg 
Hemp. 

Cold 
Register. 

Warm 
Register. 

inches. 

inches. 

tons. 

tons. 

tons. 

tons. 

3 

•95 

2.22 

2.70 

3-30 

3-85 

3X 

4 

.11 
.27 

3-33 
3-92 

3-87 
4.67 

5.00 
5.85 

1:1! 

4X 

•43 

4.60 

5-55 

7.29 

8.68 

5 

•59 

5-95 

7.08 

9.15 

10.71 

$y* 

-75 

6.90 

8.3I 

11.07 

13.00 

6 

.91 

8.10 

9.65 

10.94 

14.80 

6% 

2.07 

9.16 

10.54 

15.46 

18.10 

7 

2.24 

10.24 

12.26 

18.00 

21.00 

7/2 

2-39 

11.15 

1373 

20.60 

24.10 

8 

2.54 

12.00 

14.30 

23-43 

2742 

Specimens  of  white  2-inch  rope,  exhibited  at  Kew  Gardens,  bore  the 
following  breaking  weights : — 


1  On  the  Construction  of  Cranes  and  Machinery,  page  94. 


43 


6/4 


STRENGTH  OF   ELEMENTARY   CONSTRUCTIONS. 


2-inch  Neapolitan, 2.75  tons,  breaking  weight. 

„       Konigsberg, 1.97     „ 

„       French, 2.17     „ 

„       St.  Petersburg, 2.17     „  „ 

„       Italian, 2.32     „  „ 

Specimens  of  rope  supplied  by  the  National  Association  of  Rope  and 
Twine  Spinners,  were  tested  by  Mr.  Kirkaldy. 


Rope. 

Circumfer- 
ence. 

Weight  per 
Fathom. 

Ultimate  Strength. 

Total. 

3erLb.-w'ght 
per  Fathom. 

Russian  rope,        48  t 
Machine  yarn,      50 
Hand-spun  yarn,  51 

breads  

inches. 
5.26 
5.07 

5-39 

pounds. 

5-74 

5-35 
6.04 

tons. 

4-95 
5.14 
8.16 

tons. 
.863 
.961 
1.350 

Extension  in  50  inches  Length.     Stress  per  Pound-weight  per  Fathom. 

500  Ibs. 

1000  Ibs. 

1500  Ibs. 

2000  Ibs. 

2500  Ibs. 

3000  Ibs. 

Russian  rope,  

inches. 
3.38 
3-25 
3.26 

inches. 
5.29 

4-53 
4.46 

inches. 
6.59 
5.56 
5.29 

inches. 

6.56 

5.9I 

inches. 

6^5 

inches. 
6.63 

Machine  yarn 

Hand-spun  yarn,  

The  bearing  capacity  of  a  hemp  rope  is  proportional  to  its  thickness,  the 
number  of  its  strands,  the  slackness  with  which  they  are  twisted,  and  the 
quality  of  the  hemp.  Karl  von  Ott  states  that  ropes  0.866  inch  in  diameter, 
whose  threads  had  been  shortened  by  twisting  l/5  tn>  Xtn>  and  ^d  of 
their  original  length,  broke  respectively  with  loads  of  6834  Ibs.,  5335  Ibs., 
and  4519  Ibs.  He  adds  that  the  ultimate  strength  of  ropes,  according  as 
they  are  wetted,  or  tarred,  or  dry,  usually  varies  between  7000  Ibs.  and 
11,400  Ibs.  per  square  inch,  and  that  a  working  strength  of  one-sixth,  say 
1422  Ibs.,  or  0.63  ton  per  square  inch  may  be  adopted. 

WIRE  ROPES. 

The  strength  of  wire  ropes  of  iron  and  steel,  manufactured  by  Messrs. 
R.  S.  Newall  &  Co.,  is  given  in  table  No.  236,  together  with  that  of  hemp 
rope,  for  comparison.  From  the  table,  the  following  data  are  derived : — 

1.  The  Breaking  Strength  is — 

About  i  ton  per  Ib.  weight  per  fathom  for  round  hemp  rope. 

2  »  »  »  »  iron      » 

3  to  3^  „  „  „  steel     „ 

2.  The  Working  Load  is — 

3  cwts.  per  Ib.  weight  per  fathom  for  round  hemp  rope. 
6        „  „  „  „  iron      „ 

10        „  „  „  „  steel     „ 

3.  The  Working  Load  in  Cwts.  is — 

s/eths  of  the  square  of  the  circumference  in  inches,  for  round  hemp  rope. 
About  5  times  „  „  „  „  „  iron       „ 

9  times  „  „  „  „  „          steel      „ 


ROPES. 


675 


Table  No.  236. — STRENGTH  OF  ROPES — HEMP,  IRON,  STEEL. 

(Messrs.  R.  S.  Newall  £  Co.) 
i.   ROUND  ROPES — for  Inclined  Planes,  Mines,  Collieries,  Ships'  Standing  Rigging,  &c. 


HEMP. 

IRON. 

STEEL. 

TENSILE  STRENGTH. 

Circum- 
ference. 

Weight  per 
Fathom. 

Circum- 
ference. 

Weight  per 
Fathom. 

Circum- 
ference. 

Weight  per 
Fathom. 

Working 
Load. 

Ultimate 
Strength. 

inches. 

pounds. 

inches. 

pounds. 

inches. 

pounds. 

cwts. 

tons. 

..    2V   .. 

2 

...    I 

...I 

6 

2    . 

I 

I 

9 

3 

•2  \/ 

4    ... 

T  fyh. 

...    2 

...    12    ... 

...     4  ... 

I* 

ilA 

ilA 

15 

5 

•    4l/2    •• 

C     . 

-, 

18  ... 

...     6  ... 

2 

3% 

1/8 

2 

21 

7 

j-    T/ 

7 

2  ^At 

A 

2  */ 

2/1 

8 

2% 

27 

9 

...  6     ... 

Q    • 

2^/9. 

,.     C 

J  i7A  . 

•     ^ 

,.    ^O    .. 

...     IO    ... 

2^ 

33 

ii 

..  6y2  .. 

...    IO    ... 

...  6      . 

.   .   2 

-  I/ 

^6 

...    12    . 

2X 

4 

39 

13 

...  7 

...     12     ... 

...    27A    .. 

...  7 

...   2X    •.• 

A  y^7 

...  42  ... 

3 

71A 

1      45 

15 

7  5^ 

.    14    . 

...  8      ... 

23/( 

..5                -   & 

...  16  .  . 

3% 

5i 

17 

...  8      ... 

...   16   ... 

...  ^ft  ... 

...  9     ... 

...   2#... 

...  5X  ••• 

...  54  ... 

...   18  ... 

3/^2 

IO 

2/8 

6 

60 

20 

8  y^ 

...   18   ... 

.3  C/C 

...ii 

6// 

66 

...    22    ... 

3X 

12 

4 

72 

24 

...  9>£  ... 

...    22     ... 

...  3JA  ... 

...13        ... 

...  3X  - 

...  8      ... 

...  78  ... 

...    26    ... 

IO 

...    26 

4 

14 

84 

28 

...  4X  ... 

...15        ... 

...  3ft  ... 

...  9     ... 

...  90  ... 

...    30    ... 

II 

...    30 

4ft 

16 

96 

32 

... 

...  4^  . 

...18      ... 

...  3^  ... 

...10 

...108  ... 

...    36    ... 

12 

34 

4/s 

20 

3% 

12 

120 

40 

2.  FLAT  ROPES.  —  For  Pits,  Hoists,  &c.  &c. 

4      x  i  J-^ 

...    20  ... 

2/£  X    l/z      -    -       T  T 

AA 

2O 

5      x  iX 

24 

2/2   X    „ 

13 

23 

5>£  x  i^ 

...    26  ... 

23/    -^     yf 

...     I  I    . 

60 

..    27    .. 

28 

3      x  „ 

16 

2    x  y2 

10 

64 

28 

6      x  i^ 

...    30  ... 

3/4  x  „ 

...    18  ... 

2%  X  ^ 

...     II     ... 

...      72... 

...    32    ... 

7      x  i^ 

36 

3^x  „ 

20 

»     x  „ 

12 

80 

36 

8X  x  2*^ 

...   40  ... 

...     22    ... 

2>^  x  X 

...     13    ... 

...    88... 

...    40    ... 

8/"2  x  2X  1       45 

4      x 

25 

2X  x  H 

15 

100 

45 

9     x  2^ 

...    50  ... 

4X  x  */£ 

...     28    ... 

3      x  „ 

...    16  ... 

...  112  ... 

...  50  ... 

9^  x  2ft 

55 

4/2  x  „ 

32 

3/4  x  ,, 

18 

128 

56 

10      X  2>^ 

...   60  ... 

4^  x  » 

...    34  ... 

3/2  x  ,, 

...   20  ... 

...  60  ... 

6;6 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


Table  No.  237. — STRENGTH  OF  CABLE  FENCING  STRANDS  AND 
SOLID  FENCING  WIRE. 

(Reduced  from  Messrs.  Francis  Morton  &  Co.'s  "Standard  Quality"  Table.) 


Size  of  Fencing  Strand. 

Solid  Wire  of 
Equal  Diameter. 

Length  of  One  Ton. 

Ultimate  Strength  per 
Square  Inch. 

Strand. 

Wire. 

Strand. 

Wire 
(annealed). 

4fc     No.*/, 

No. 
O 

inch. 
.326 

yards. 

3,000 

yards. 
2,700 

tons. 
2.419 

tons. 
2.000 

*     No.5/0 

I 

j 

.300 

3,800 

3,200 

1.828 

1.683 

4t  N°-4/° 

2 

.274 

5,600 

3,850 

1.562 

1.402 

fNo.  3/o 

3 

.250 

6,000 

4,650 

1.340 

1.169 

No.  oo 

4 

.229 

6,200 

5,500 

1.160 

.988 

wjjxh      No.  o 

5 

.209 

7,800 

6,600 

.893 

.817 

%      No.  i 

6 

.191 

9,800 

7,900 

.714 

.682 

H        No.  2 

7 

.174 

1  1  ,000 

9,550 

.627 

.566 

Jb     Na  2A 

8 

.159 

15,000 

11,400 

•49  1 

•473 

Note. — The  number,  size,  and  strength  of  the  Iron  Wire  quoted  in  this  Table  are  the 
same  as  those  of  Ryland's  Warrington  Wires,  table  No.  82,  page  247. 

Table  No.  238. — TENSILE  STRENGTH  OF  AMERICAN  IRON  WIRE  ROPE 

AND  HEMP  ROPE. 

(Mr.  ].  A.  Roebling.) 


Trade 
Number. 

Circumfer- 
ence of 
Wire 
Rope. 

Circumfer- 
ence of 
Hemp 
Rope  of 
equal 
Strength. 

Ultimate 
Strength. 

Trade 
Number. 

Circumfer- 
ence of 
Wire 
Rope. 

Circumfer- 
ence of 
Hemp 
Rope  of 
equal 
Strength. 

Ultimate 
Strength. 

No. 

inches. 

inches. 

tons. 

(English.) 

No. 

inches. 

inches. 

tons. 
(English.) 

FINE  WIRE. 

H 

3.26 

8X 

18.2 

I 

6.62 

i5# 

67.3 

15 

2.98 

7X 

14.5 

2 

6.20 

I4# 

59 

16 

2.68 

6X 

"-3 

3 

5-44 

13 

49 

17 

2.40 

S/2 

8 

4 

4.90 

12 

39-6 

18 

2.12 

5  x 

6.9 

5 

4.50 

I0# 

32 

»9 

•9 

4^ 

5-3 

6 

3-91 

9X 

24.7 

20 

•63 

4 

3-72 

7 

3.36 

8 

18.4 

21 

•53 

3-3 

2.57 

8 

2.98 

7 

14.5 

22 

•3i 

2.8 

i-93 

9 

2.56 

6 

10.4 

23 

•23 

2.46 

i-5 

10 

2.45 

5 

7.8 

24 

.11 

2.2 

1.16 

COARSE  WIRE. 

25 

•94 

2.04 

•94 

II 

4-45 

I0# 

33 

26 

.88 

i-75 

•74 

12 

4.00 

10 

27-3 

27 

.78 

1.50 

•5i 

13 

3-63 

9/2 

22.7 

CHAINS. 


6/7 


French  Wire  Rope. — For  mining  purposes,  each  strand  consists  of  a  core 
of  hemp  and  12  wires;  and  the  rope  has  5  or  6  strands  on  a  central  hemp 
core.  Flat  ropes  are  formed  by  laying  3  or  4  ropes  side  by  side,  and 
binding  or  lacing  them  with  annealed  wire;  but  flat  ropes  are  seldom 
employed. 

Table  No.  239. — FRENCH  IRON  WIRE  ROPES  FOR  MINING  SERVICE. 

(Manufactured  by  MM.  Harmegnies,  Dumont,  &  Co.,  Anzin.) 
Working  depth,  400  metres,  or  440  yards. 


FLAT  ROPES. 

ROUND  ROPES. 

Number 
of 
Strands. 

Width. 

Thickness. 

Weight 
Yard. 

Working 
Load. 

Number. 

Diameter. 

Weight 
Y?rd. 

Working 
Load. 

strands. 

inches. 

inch. 

Ibs. 

tons. 

No. 

inches. 

Ibs. 

tons. 

8 

5-1 

.87 

16 

5 

10 

1.30 

6.5 

3 

8 

4-7 

•79 

13 

4-5 

II 

1.  10 

5 

2-5 

6 

3-9 

•83 

12 

4 

12 

.98 

3-8 

2 

8 

4-3 

.67 

II 

3-5 

13 

•83 

3 

i-5 

6 

3-5 

•79 

10 

3 

H 

•71 

2.6 

i 

6 

3-2 

.67 

9 

2-5 

15 

.63 

2 

•75 

6 

3-2 

•63 

8 

2 

16 

•59 

i-5 

•5 

6 

2.8 

•59 

7 

1.8 

17 

•Si 

i 

.25 

9 

2.4 

•55 

6 

i-5 

Note  to  Table. — i.  Steel  wire  ropes  may  be  a  third  less  in  weight  than 
iron  wire  rope  for  the  working  load.  2.  Hemp  ropes  should  be  a  third 
heavier  than  iron  wire  rope  for  the  same  working  load. 

Steel  Wire  Ropes. — Ropes  consisting  of  26  steel  wires,  No.  14  W.  G.,  or 
.085  inch  in  diameter,  are  made  for  steam  ploughing  purposes.  The  weight 
of  the  rope  is  about  2  Ibs.  per  yard, — less  than  i  ton  per  1000  yards. 
Each  wire,  it  is  said,  bears  a  tensile  stress  of  from  2000  Ibs.  to  i  ton; 
and,  at  this  rate,  the  rope  should  have  a  tensile  resistance  equal  to  24  or 
26  tons. 


CHAINS. 

Chains  are  constructed  either  with  open  links,  Figs.  274  and  275,  or  with 
stud-links,  Figs.  276,  277,  278,  and  279. 

The  standard  proportions  of  the  links  of  chains,  in  terms  of  the  diameter 
of  the  bar  iron  from  which  they  are  made,  are  as  follows : — 

Extreme  Length.  Extreme  Width. 

Stud-link 6      diameters 3.6  diameters. 

Close-link 5  „        3.5 

Open-link 6  „        3.5 

Middle-link 5.5         „        3.5 

End-links 6.5         „        4.1          „ 

End-links  are  the  links  which  terminate  each  i5-fathom  length  of  chain; 
they  are  made  of  thicker  iron,  generally  1.2  diameters  of  the  common  links. 


6/8 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


Ordinary  Stud-link  Chain-cable. — The  admiralty  test  for  the  tensile 
strength  of  ordinary  stud-link  chain-cables,  is  at  the  rate  of  630  Ibs.  per 
circular  ^6 -inch  section  of  one  side  of  a  link:  equivalent  to  22.92  tons  per 
square  inch  of  one  side,  or  to  1 1.46  tons  per  square  inch  of  both  sides  taken 
together, — just  within  the  elastic  limit. 


No.  4. 


No.  6. 


Figs.  274-279. — Links  of  Chain-Cables.  No.  i,  Circular  Link.  No.  2,  Oval  Link.  No.  3, 
Oval  Stud-Link,  with  pointed  stud.  No.  4,  Oval  Stud-Link,  with  broad-headed  stud. 
No.  5,  Obtuse-angled  Stud-Link.  No.  6,  Parallel-sided  Stud-Link. 

The  weight  of  a  link  in  similar  cables,  increases  as  the  cube  of  any  lineal 
dimension,  say  the  thickness;  and  the  weight  per  yard  increases  as  the 
square  of  the  thickness  of  chain.  Whence  the  formula — 

Weight  per  yard  of  Common  Stud-link  Chain-cable. 
W  =  26.9  */2;  or,  in  round  numbers,  27  d* (  i  ) 

W  =  the  weight  per  yard  in  pounds;  d=  the  thickness  of  the  chain,  or  the 
bar  from  which  it  is  made,  in  inches. 

The  weight  of  a  bar  of  iron  a  yard  long  is  10  Ibs.  per  square  inch  of  sec- 
tion, or  7.854  Ibs.  per  circular  inch;  that  is,  a  i-inch  round  bar  weighs  7.85 
Ibs.  per  yard,  whilst  a  stud-chain  cable  of  i-inch  iron,  weighs  26.9  Ibs.  per 
yard,  or  3.42  times  the  weight  of  a  i-inch  bar.  Generally,  therefore,  a  stud 
chain-cable  weighs  3.42  times  as  much  as  a  bar  of  the  same  size  and  length. 

The  table  No.  240  contains  the  dimensions,  weights,  and  strengths  of 
ordinary  stud-link  chain-cables.  Column  4  shows  the  weight  of  100  fathoms 
of  cable  in  8  lengths;  including  4  swivels  and  8  joining  shackles.  The 
sixth  column  gives  the  ultimate  strength  by  actual  tests  made  at  Woolwich, 
in  1842-43,  averaging,  as  shown  in  the  last  column,  16  tons  per  square 
inch,  or  two-thirds  of  the  strength  of  the  original  bar,  assumed  at  24  tons 
per  square  inch. 

The  safe  working-stress  is  5.73  tons  per  square  inch  of  both  sides 
together,  or  half  of  the  proof-stress. 

Open-link  Chains,  Figs.  274  and  275. — The  chain,  Fig.  275,  is  sometimes 
called  a  close-link  chain,  to  distinguish  it  from  the  circular-link  chain,  Fig. 
274.  The  ultimate  strength,  generally,  is  the  same  as  that  of  stud-link 
chains;  but  the  elastic  limit  is  less  than  that  of  the  others,  and  the  proof- 
stress  for  close-link  chains  is  just  two-thirds  of  that  for  stud-link  chains,  or 


LEATHER   BELTING. 


679 


Table  No.  240. — ORDINARY  STUD-LINK  CHAIN-CABLE 
WEIGHT  AND  STRENGTH. 


Dimensions  of  Link. 

Weight  of  100 
Fathoms. 

Average 
Ultimate 
Strength. 

Admiralty 
Proof-stress 
adopted  by 
Lloyds'. 

Ultimate 
Strength 
per  square 
inch  of 
Both  Sides 
of  Link. 

Diameter 
of  each 
Side. 

Length 
of  One 
Link. 

Width 
of  One 
Link. 

Total. 

Per 

Fathom 

(6  Feet). 

inches. 

inches. 

inches. 

cwts. 

Ibs. 

tons. 

tons. 

per  cent. 

tons. 

7/l6 

2^j 

i-575 

9.20 

"•3 

— 

3% 

— 

— 

# 

3 

1.8 

12 

13-4 

— 

4K 

— 

— 

9/i6 

3^ 

2.025 

15.2 

17.2 

— 

5/^ 

— 

— 

ft 

3^ 

2.25 

18.75 

21 

9-58 

7 

73 

I5.6 

"/* 

4/^ 

2.475 

22.7 

25-4 

— 

%/4 

— 

— 

4/^ 

2.7 

27 

30.2 

J3-51 

io/^ 

75 

15-3 

7/% 

sX 

3-i5 

36.75 

41.2 

20.38 

13* 

67 

16.9 

I 

6 

3-6 

48 

53-8 

24-25 

18 

74 

15-4 

\y% 

6^ 

4-05 

60.75 

29.54 

22^ 

77 

14.9 

ify 

7^ 

4-5 

75 

84 

— 

28>^ 

— 

— 

i^ 

8/^ 

4-95 

90.75 

101.6 

— 

34 

— 

— 

!/•£ 

9 

5-4 

108 

121 

59-58 

40^ 

68 

16.9 

l$i 

5.85 

126.75 

142 

47^ 

— 

— 

tl 

ioK 

6-3 

6-75 

i47 
168.75 

164.6 
I89 

74.12 
92.88 

55^ 
63X 

H 

lil 

2 

12 

7.2 

192 

215 

99-54 

72  1 

72 

i5.8 

2j^ 

12^ 

216.75 

242.8 

— 

2X 

1-3^2 

8.  i 

243 

276.2 

— 

91/^ 

— 

— 

14* 

8.55 

270.75 

303.2 

— 

lOll/2 

— 

— 

2^2 

15 

9.0 

300 

336 

— 

\\2.l/2 

— 

— 

2$ 

\&yz 

9.9 

363 

406.6 

— 

136^ 

— 

— 

Average 

s                                             .  .         

72 

15-9 

i. — The  Safe  Working-stress  is  taken  at  half  the  Proof-stress. 
2. — The  Proof-stress  and  Safe  Working-stress  for  close-link  chains  are  respectively  two- 
thirds  of  those  of  stud-link  chains. 

7.64  tons  per  square  inch  of  section  of  both  sides,  or  410  Ibs.  per  circular 
j^-inch  of  section  of  one  side.  The  safe  working-stress  is  half  the  proof- 
stress,  or  3.82  tons  per  square  inch  of  section. 

The  weight  of  close-link  chain  is  about  three  times  the  weight  of  the  bar 
from  which  it  is  made,  for  equal  lengths. 

Karl  von  Ott,  comparing  the  weight,  cost,  and  strength  of  the  three 
materials,  hemp,  iron  wire,  and  chain  iron,  concludes  that  the  proportion 
between  the  cost  of  hemp  rope,  wire  rope,  and  chain,  is  as  2:1:3;  and 
that,  therefore,  for  equal  resistances,  wire  rope  is  only  of  half  the  cost  of 
hemp  rope,  and  a  third  of  the  cost  of  chains. 


LEATHER  BELTING. 

According  to  the  experiments  of  Messrs.  Briggs  and  Towne,  the  tensile 
strength  of  single  leather  belts,  .219  inch  thick,  was, 

Per  Per  square 

inch  wide.  inch  of  Section. 

Through  the  lace-holes, 210  Ibs 960  Ibs. 

Through  the  rivet-holes, 382    „       1740    „ 

Through  the  solid  parts, 675    „       3080    „ 


68o 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


Messrs.  Norris  &  Co.'s  beltings,  as  tested  by  Mr.  Kirkaldy,  gave  the 
following  results  for  ultimate  tensile  strength : — 

Table  No.  241. — TENSILE  STRENGTH  OF  LEATHER  BELTING. 


SIZE. 

English 
Belting. 

Helvetia 
Belting. 

DOUBLE. 

i  ^  inches 

Ibs. 
I4,86l 

Ibs. 
17  622 

7 

6,IQ^ 

1  1,  080 

6                               

5,60^ 

IO,4.?6 

4.,  36  5 

6  2O7 

2         „         

2,942 

4,237 

SINGLE. 
jo 

8846 

ii  888 

4.,O6O 

5.426 

» 
A                                              

^,24.8 

•2,04.8 

ilA 

-J  QO7 

•3    -277 

Spill's  machinery  belting  is  manufactured  from  flax-yarn,  saturated  with 
a  compound  substance  said  to  be  incapable  of  decomposition.  According 
to  the  annexed  results  of  tests  it  is  stronger  than  leather  belts : * — 


No.  i, 

No.  2,  5 

No.  3,  10 

Leather  belt,  4 


Tensile  Strength, 
per  inch  Wide. 

5  inches  wide, I254  Ibs. 


1489  „ 


525 


Untanned  leather  belts  are  said  to  be  half  as  strong  again  as  tanned 
leather  belts.  Mr.  John  Mason,  of  Bulkley,  Barbadoes,  uses  belts  cut  from 
raw  cowhide,  simply  dried  in  the  sun.  They  last  longer,  he  says,  than 
leather  belts,  and  are  made  at  a  fourth  of  the  cost  of  the  latter.2 

India-rubber  belts,  made  of  American  cotton  canvas,  cemented  in  layers 
by  vulcanized  india-rubber,  and  covered  by  a  compound  of  rubber,  have  been 
proved  to  possess  considerably  greater  frictional  adhesion  than  leather  belts. 


STRENGTH   OF   BOLTS   AND   NUTS. 

Mr.  Brunei's  Experiments? — Mr.  Brunei  tested  the  tensile  strength  of 
screwed  bolts  and  nuts  of  Shropshire  iron,  from  ^  inch  to  ij^  inches  in 


8 


Fig.  280.— Screw  Bolt  and  Nut. 

diameter,  applying  the  stress  between  the  head  and  the  nut,  when  16  inches 

1  Exhibited  Machinery  of  1862,  page  423.  2  Engineering,  June  19,  1874. 

3  The  particulars  of  these  experiments  are  derived  from  the  Author's  work  on  Railway 
Locomotives,  1860. 


BOLTS  AND   NUTS.  68  1 

apart,  and  placed  as  in  Fig.  280.  The  length  of  the  screwed  part  was  3^ 
inches.  In  most  instances,  the  bolt  snapped  at  the  base  of  the  screwed 
part. 

Diameter  Total  Breaking  Weight.         Breaking  Weight  per  sq.  inch. 

inches.  tons.  tons. 


10  ..................  32 

I2  .................  •  27 

15%  ..................  25 

20  ..................  25 

21  ..................  21 

29  ........  ..........  23 


To  find  to  what  extent  the  screwing  of  a  bolt  diminishes  its  tensile 
strength,  Mr.  Brunei  tested  four  i^-inch  bolts  and  nuts  to  the  annexed 
form,  Fig.  281,  on  which  the  screwed  part  was  enlarged  to  i^  inches  in 
diameter.  The  bolts  were  broken  in  the  shank,  and  the  average  breaking 
weight  was  equal  to  25.2  tons  per  square  inch,  showing  an  addition  of 
2.2  tons  per  inch,  as  compared  with  the  screwed  shank,  Fig.  280.  Inversely, 
it  may  be  inferred  that  the  strength  of  i^-inch  bolts  was  reduced  2.2  tons, 
or  8  per  cent,  by  screwing. 


Fig.  aSi.-— Screwed  Enlarged  Bolt  and  Nut. 

The  heads  of  the  i  ^-inch  bolts  were  i  %  inches  thick,  and  they  stood 
fast  during  all  the  trials.  The  depth  of  the  nuts  of  these  bolts  varied  from 
i  %  inch  to  24  mcn- 

Nuts  i  inch  deep,  or  8/I0ths  of  the  diameter,  stood  well. 
Do.    y%       „          or  7/I0ths     „  „          thread  strained. 

Do.    24       »          or  6/ioths     „  „          thread  stripped. 

The  thread,  it  appears,  was  stripped  when  the  depth  of  the  nut  was  only 
3/5  ths  of  the  diameter.  Nevertheless,  in  ordinary  good  practice,  a  depth 
of  half  the  diameter  has  been  found  sufficient  for  both  the  head  and  the  nut. 
But  it  may  well  be  better  to  make  them  deeper,  to  allow  for  contingencies. 

Working  Stress  for  Screwed  Bolts. — A  working  stress  of  i  ^  tons  per 
square  inch  has  been  assigned  for  screwed  bolts.  In  France,  it  has  been 
taken  as  high  as  324  tons  Per  square  inch. 

Whitworttts  System  of  Standard  Sizes  of  Bolts  and  Nuts. — The  thickness 
of  the  bolt  head  is  ^6  ths  of  the  diameter,  and  that  of  the  nut  is  equal  to  the 
diameter.  The  angle  of  the  triangular  thread  is,  in  this  system,  55°.  The 
top  and  the  bottom  of  the  thread  are  rounded  off,  and  the  reduction  so 
made  of  the  exact  height  of  the  triangle  is  one-third ;  that  is,  one-sixth  from 
the  top,  and  one-sixth  from  the  bottom.  The  actual  height  of  the  thread 
becomes  rather  more  than  3/s  ths,  and  less  than  2/3  ds,  or  about  63  per  cent., 
of  the  pitch.  See  table  No.  242,  next  page. 

For  screws  with  square  threads,  the  number  of  threads  per  inch  is  one- 
half  of  the  number  for  triangular  threads. 


2/8 
3X 


4 
4X 


2.509 
...    2.634 


'     3/2 

3X 
3X- 
3 

3     • 


BOLTS   AND   NUTS. 


683 


The  American  standard  pitches  are  nearly  identical  with  the  Whitworth 
standards. 

American  Standard  Sizes  of  Bolts  and  Nuts. 

(United  American  Railway  Master  Car-builders'  Association,  in  Convention  at  Richmond, 
I  Va.,  June  15,  1871.) 

ROUGH  BOLTS. — The  breadth  across  the  flats  of  the  bolt-head  and  the 
nut  =  i  YZ  diameters  +  Y%  inch. 

The  thickness  of  the  head  =  ^  diameter  4-  J/i6  inch. 
The  thickness  of  the  nut  =  i  diameter. 

FINISHED  BOLTS. — The  breadth  across  the  flats  of  the  bolt-head  and  the 
nut  =  i  y2  diameters  +  x/i6  inch. 

The  thickness  of  the  head  and  of  the  nut  =  i  diameter  -  X/X6  inch. 


Diameter. 

Number  of 
Threads 
per  Inch. 

Diameter. 

Number  of 
Threads 
per  Inch. 

Diameter. 

Number  of 
Threads 
per  Inch. 

inches. 

threads. 

inches. 

threads. 

inches. 

threads. 

X 

20 

IM 

6 

3^ 

3 

5/i6 

18 

1/2 

6 

4 

3 

% 

16 

Itt 

S/2 

4X 

27/& 

7/i6 

14 

Iff 

s 

4X 

V/4 

/2 

13 

tg 

s 

4X 

2% 

9/i  6 

12 

2 

4/2 

5 

".        2/2 

H 

II 

2X 

4/2 

5X 

Z/2 

ti 

10 

2/2 

4 

S/2 

2*/S 

% 

Q 

2^ 

4 

Sti 

23/S 

i 

8 

3 

3/2 

6 

2X 

!# 

7 

3X 

3/2 

W 

7 

3/2 

3/4 

Table  No.  243. — WHITWORTH'S  STANDARD  PITCHES  FOR 
SCREWED  IRON  PIPING. 


Diameter  of 
Piping. 

Number  of 
Threads 
per  inch. 

Diameter  of 
Piping. 

Number  of 
Threads 
per  inch. 

Diameter  of 
Piping. 

Number  of 
Threads 
per  inch. 

inches. 

threads. 

inches. 

threads. 

inches. 

threads. 

% 

28 

H 

14 

1/2 

II 

X 

19 

X 

H 

1% 

II 

3A 

19 

i 

II 

2 

II 

/2 

H 

iX 

II 

above  2 

8 

M.  Armengaud  gives  a  table  of  the  dimensions  of  bolts  and  nuts,  based  on 
the  average  practice  in  France.  It  is  here  translated  into  English  measures, 
for  threads  of  triangular  and  of  square  section.  The  thickness  of  the  nut 
for  triangular  threads  is  equal  to  the  diameter  of  the  bolt,  as  in  Whitworth's 
system.  The  depth  of  the  square  thread  is  nearly  equal  to  half  the  pitch, 
or  to  the  thickness  of  the  thread. 


684 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


Table  No.  244. — ARMENGAUD'S  FRENCH  STANDARD  BOLTS  AND  NUTS. 

With  Hexagonal  Heads  and  Nuts. 
i.  TRIANGULAR  THREAD — (Equilateral  Triangle). 


SCREW. 

HEAD  AND  NUT. 

Diameter  of  Bolt 
and  Screw. 

Diameter 
at  Bottom 
of  Thread. 

Number  of 
Threads 
per  inch. 

Thickness 
of 
Head. 

Thickness 
of  Nut. 

Breadth 
across  the 
Flats. 

Working 
Tensile 
Stress. 

millimetres. 

inches. 

inches. 

threads. 

inches. 

inches. 

inches. 

Ibs. 

5 

.20 

•13 

18.1 

.24 

.20 

•55 

44 

7.5 

•30 

.22 

16 

•30 

•30 

.68 

99 

10 

•39 

•31 

14.1 

•38 

•39 

.88 

178 

12.5 

.49 

•39 

12.7 

•44 

•49 

1.04 

277 

15 

•59 

.48 

11.5 

•52 

•59 

1.20 

400 

17-5 

.69 

.58 

10.6 

.58 

.69 

1.40 

545 

20 

•79 

.66 

9.8 

.66 

•79 

1.50 

7i3 

22.5 

.89 

.76 

9.1 

•72 

.89 

1.68 

902 

tons. 

25 

.98 

.84 

8.5 

.80 

.98 

1.84 

.50 

30 

1.18 

1.02 

7-5 

•94 

i.iS 

2.16 

•73 

35 

1.38 

1.20 

6-7 

i.  08 

1.38 

2.48 

•99 

40 

1.58 

1.40 

6.0 

1.22 

1.58 

2.80 

1.30 

45 

1.77 

1.56 

5-5 

1.36 

1.77 

3-20 

1.64 

50 

1.97 

1.74 

5-i 

1.50 

1.97 

3-44 

2.03 

55 

2.17 

1.92 

4-7 

1.64 

2.17 

3-76 

2-45 

60 

2.36 

2.08 

44 

1-74 

2.36 

4.08 

2.92 

65 

2.56 

2.26 

4.1 

1.92 

2.56 

4.40 

3-42 

70 

2.76 

2.44 

3-8 

2.06 

2.76 

4.70 

3-97 

75 

2-95 

2.60 

3-5 

2.20 

2.95 

5.00 

4.56 

80 

3-15 

2.78 

3-4 

2.34 

3-15 

5-35 

5.12 

2.  SQUARE  THREAD. 

Depth  of 

Thread. 

LUllb. 

20 

•79 

.072 

6.57 

— 

1.82 

•32 

— 

25 

.98 

.O8l 

5.97 

— 

2.01 

•51 

— 

30 

1.18 

•093 

5.40 

— 

2.22 

•73 

— 

35 

1.38 

.10 

4-93 

— 

2.41 

•99 

— 

40 

i-57 

.106 

4-53 

— 

2.63 

1.30 

— 

45 

1-77 

.114 

4.20 

— 

2.85 

1.64 

— 

50 

1.97 

.128 

3-91 

— 

3-07 

2.03 

— 

55 

2.17 

•13 

3.65 

— 

3-30 

2-45 

— 

60 

2.36 

.14 

3-43 

— 

3-50 

2.92 

— 

65 

2.56 

•15 

3-23 

— 

3-70 

342 

— 

70 

2.76 

.158 

3-o6 

— 

3-92 

3-97 

— 

75 

2-95 

.166 

2.92 

— 

4.13 

4.56 

— 

80 

3-15 

.174 

2.76 

— 

4.36 

5.18 

— 

,     85 

3-35 

.183 

2.63 

— 

4.58 

5.85 

— 

90 

3-54 

.192 

2.51 

— 

478 

6.56 

— 

95 

3-74 

.200 

2.41 

— 

5.00 

7oO 

— 

100 

3-94 

.209 

2.31 

— 

5.22 

8.10 

— 

105 

4-i3 

.220 

2.22 

— 

5-43 

8-93 

— 

1  10 

4-33 

.226 

2.13 

— 

5.66 

9.80 

— 

115 

4-53 

.230 

2.06 

— 

5.87 

10.71 

— 

120 

4.72 

2.24 

2.00 

— 

6.08 

11.66 

— 

SCREWED   STAY-BOLTS  AND   STAYED   SURFACES. 


685 


SCREWED   STAY-BOLTS  AND   STAYED   SURFACES. 

Screwed  Stay-Bolts. — Sir  William  Fairbairn  tested  the  strength  of  24 -inch 
stay-bolts,  with  enlarged  ends,  screwed  into  ^5-inch  plates  of  copper  and 


Figs.  282,  283.— Flat  Stayed  Plates. 

of  iron,  some  of  them  being  rivetted  or  headed  in  addition,  as  in  the 
Figs.  282  and  283. 

BOLTS.  PLATES.  Breaking  Weight. 

1 .  Copper  into  copper,  screwed  and  rivetted, 7.2  tons. 

2.  Iron       into  copper,      do.  do 10.7 

3.  Iron       into  copper,  screwed  only 8.1    „ 

4.  Iron       into  iron,  screwed  and  rivetted 12.5    „ 

Notes.— ist  Test.  The  bolt 
broke  through  the  shank. 

2d  Test.  The  rivet-head  was 
broken  off,  and  the  bolt  was 
drawn  out  of  the  plate,  strip- 
ping the  thread. 

$d  Test.  The  bolt  stripped 
the  thread  of  the  plate. 

4th  Test.  The  bolt  broke 
through  the  shank;  screw  and 
plate  uninjured. 

Flat  Stayed  Plates.—  Sir  Wil- 
liam Fairbairn  tested  two  flat 
boxes,  Fig.  284,  22  inches 
square,  having  top  and  bottom 
plates  of  ^-inch  copper  and  3/fr- 
inch  iron  respectively,  inclosing 
a  2*4 -inch  water-space;  stayed 
with  ^/jg-inch  iron  stays,  having 
enlarged  ends  screwed  and 
rivetted  into  the  plates,  to  re- 
present the  Conditions  Of  the  Fig.  284. -Flat  Stayed  Plates. 

firebox  of  a  locomotive.     The  stays  were  placed  at  intervals  of  5  inches 


p 

11               ll                EJ 

O( 

81 

Oi 

81 

01 
Oi 
01 
0! 
0! 
Oi 
<3:l 
OC 

30OOOOOOOOOC 

)0 

o 
o 
o 
o 
o 
o 
o 
o 

0 

o 
o 

o 

0 

o 

0 
0 

o 

o 

0 
0 

)OOOOOOOOOOC 

686 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


in  the  first  box,  and  4  inches  in  the  second.  Under  the  pressure  of  water, 
the  sides  of  the  first  box  commenced  to  bulge  or  swell  between  the  stays 
at  a  pressure  of  455  Ibs.  per  square  inch,  and  the  box  was  burst  at  815  Ibs., 
by  the  drawing  of  the  head  of  a  stay  bolt  through  the  copper  plate,  as 
shown. 

In  the  second  box,  the  bulging  commenced  at  a  pressure  of  5 1 5  Ibs.  per 
square  inch;  it  amounted  to  Ji  inch  at  1600  Ibs;  and  at  1625  Ibs.  one  of 
the  stays  was  drawn  through  the  iron  plate,  stripping  the  thread. 

At  the  5-inch  intervals,  rupture  took  place  when  the  stress  on  each  stay- 
bolt  was  9  tons;  at  the  4-inch  intervals,  the  ultimate  stress  was  n^  tons. 

Flat  Plates  of  Marine  Boilers. — Two  experimental  flat  boxes  were  tested 
at  Plymouth  Dockyard,  by  Mr.  Phillips.1  They  were  constructed  respec- 
tively of  7/i6-inch  and  of  ^-inch  iron  plates,  stayed  at  intervals  of 
15^  inches  by  15^  inches,  with  i^-inch  screwed  stay-bolts  rivetted  over 
at  the  ends,  giving  a  superficies  of  240  square  inches  for  each  bolt. 
Tested  by  hydrostatic  pressure,  the  plates  were  bulged  between  the  stay- 
bolts,  and  were  finally  pushed  off,  or  drawn  away  from  the  bolts,  under  the 
following  pressures : — 


PLATES. 

Sectional  Area  of  Stay-bolts. 

Bursting  Pressure 
per  Square  Inch 
of  Surface. 

Total  Pressure 
for  each  Stay-bolt. 

In  Body. 

At  Thread. 

inch. 

7/i6 

y* 

square  inch. 
I.48 
1.48 

square  inch. 

(say)     1.2 
(say)     1.2 

Ibs. 
105 
140 

tons. 
11.25 
H73 

When  nuts  were  applied  to  the  ends  of  the  stay-bolts  through  the  7/l6-inch 
plate,  they  bore  a  pressure  amounting  to  165  Ibs.  per  square  inch,  on  the 
plate;  when  the  box  gave  way  at  a  rivetted  joint. 

Rules  for  Flat  Stayed  Surfaces. — Mr.  Wm.  Bury2  propounds  the  following 
rules  for  the  staying  of  the  flat  surfaces  of  marine  boilers: — ist.  The 
diameter  of  the  screwed  stays  over  the  threads,  should  never  exceed  three 
times  the  thickness  of  the  plates.  2d.  The  working  steam-pressure  allowed 
per  square  inch  of  section  of  the  stay-bolts,  at  the  threads,  is  5000  Ibs. 
3d.  The  formula  for  the  working  pressure  in  pounds  per  square  inch,  with 
the  above-named  proportions,  is, — 

112  (thickness  of  plate  in  sixteenths  inch)  _ ,, 

area  of  stayed  surface  for  each  stay,  in  square  inches  =  ' 

which  appears  to  agree  with  safe  practice.     Mr.  Bury  gives  the  following 
data,  by  this  rule : — 


1  See  Engineering,  September  I,  1876,  page  185. 
'''Engineering,  September  15,  1876,  page  236. 


HOLLOW  CYLINDERS. 


687 


Table  No.  245. — PROPORTIONS  OF  FLAT  STAYED  SURFACES  OF  BOILERS. 
For  a  working  pressure  of  $o  Ibs.  per  square  inch. 


Diameter  of 

Sectional  Area 

Area  of  Surface  for 

Distance  of  Centres 

Stay-bolts. 

at  Threads. 

each  Stay-bolt. 

of  Stay-bolts. 

inch. 

square  inch. 

square  inches. 

inches. 

i/i 

0.8 

133 

1  1  *A. 

IX 

I.O 

166 

13 

I5/I6 

I.I 

183 

\y% 

1.2 

200 

14/4 

11/2 

i-5 

250 

15^ 

Mr.  Bury  recommends  that  nuts  should  be  applied  on  the  uptake  ends 
of  the  bolts  outside  the  plates,  where  they  are  above  the  water-line.  He 
reckons  on  a  bursting  pressure  six  times  the  working  pressure. 


HOLLOW   CYLINDERS:— TUBES,    PIPES,    BOILERS,  &c. 

RESISTANCE  TO  INTERNAL  OR  BURSTING  PRESSURE.     TRANSVERSE 

RESISTANCE. 

The  action  of  a  centrifugal  pressure  within  a  cylinder  is  illustrated  by 
Fig.  94,  page  2  74.  The  resistance  offered  by  the  sides  of  the  cylinder  to 
internal  pressure  transversely,  is  not  uniformly  exerted  throughout  the  thick- 
ness of  the  sides.  On  the  contrary,  the  resistance  varies,  and  is  a  maximum 
at  the  inner  surface  of  the  cylinder,  and  when  the  stress  on  the  inner  .sur- 
face does  not  exceed  the  limit  of  elastic  resistance,  the  tensional  stress 
diminishes  uniformly  through  the  thickness  of  the  sides,  and  is  a  minimum 
at  the  outer  surface. 

For  cast-iron,  in  which  the  strain  increases  approximately  in  proportion 
to  the  stress,  this  simple  ratio  of  decrease  holds  approximately  up  to  the 
bursting  strength,  which  is  measured  by  the  total  resistance  opposed  to 
breakage  when  the  internal  surface  is  strained  to  the  ultimate  limit  of  its 
tensile  strength.  But  in  the  stretching  of  wrought  iron  and  steel,  there  is 
a  break  in  the  uniformity  of  the  stretching,  at  the  yielding  point,  as  is  shown 
very  clearly  by  Fig.  222,  page  624;  for,  beyond  the  yielding  point,  the 
extension  proceeds  in  a  greatly  accelerated  ratio  with  the  stress. 

Take,  for  instance,  the  cast-iron  cylinder  of  a  hydraulic  press,  10  inches 
in  diameter  internally  and  20  inches  externally,  shown  in  cross  section  in 
Fig.  285.  Divide  the  thickness  of  it  into  an  indefinite  number  of  concentric 
rings  of  equal  thicknesses,  a,  b,  c,  d,  e-,  and  suppose,  only  for  the  sake  of 
argument,  that  the  first  or  innermost  ring  is  stretched  by  internal  pressure 
to  1 1  inches  in  diameter  inside.  All  the  other  rings  will  be  stretched  to 
larger  diameters,  in  such  proportions  that,  whilst  the  circumferential 
extension  is  the  same  for  all  the  rings,  the  increase  of  diameter  will  be 
inversely  as  the  original  diameter  of  each  ring,  so  that  the  outermost,  or 
20-inch  ring,  will  be  stretched  only  ^  inch  in  diameter,  or  half  the  diamet- 


688 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


rical  stretch  of  the  innermost  ring.  The  comparative  stretches  of  the  suc- 
cessive rings  are  shown  by  shadings  on  the  right  side  of  the  figure;  and 
the  shadings  at  the  same  time  show  by  their  thicknesses  the  relative  stresses 
on  the  successive  rings. 


Fig.  285. — Diagram  to  show  the  Stretching  of 
Hollow  Cylinders  by  Internal  Pressure. 


56  78  910 

as  e 


Fig.  286. — Diagram  to  show  the  Hyperbolic  Ratio 
of  the  Stress  throughout  the  Thickness,  by  In- 
ternal Pressure. 


Since  the  stress  is  inversely  as  the  radial  distance  from  the  centre,  if  its 
values  be  represented  by  ordinates  to  the  radius  treated  as  a  base-line,  cae, 
in  the  longitudinal  section,  Fig.  286,  they  will,  if  connected  at  the  ends, 
form  a  hyperbolic  curve  a'  e';  and  the  area  comprised  between  the  curve 
and  the  base-line,  is  a  measure  of  the  total  resistance  of  the  section.  Let 

r  =  the  inside  radius  c  a, 
r'  =  the  outside  radius  c  e, 

s  =  the  maximum  tensile  stress,  in  tons  per  square  inch, 

<^=the  inside  diameter  =  2  r, 
d'  —  the  outside  diameter  =  2  r', 

R  =  the  ratio  of  the  outside  diameter  to  the  inside  diameter  =  —  =  — , 

d      r 

p  —  the  internal  pressure  in  tons  per  square  inch. 

Then,  the  rectangular  area  c  c'  a'  a  is  a  measure  of,  or  is  equal  to,  rxs, 
for  a  length  of  i  inch  parallel  to  the  axis  ;  and  the  area  of  resistance  under  the 
hyperbolic  curve  a  a'  e'  e,  is  equal  to,  for  both  sides  (r  x  s)  x  hyp  log  R  x  2. 
The  internal  pressure  to  be  resisted,  for  i  inch  of  length,  is  equal  to  /  d, 
the  product  of  the  inside  diameter  and  the  hydrostatic  pressure  per  square 
inch;  and  it  is  equal  to  the  resistance;  that  is,  p  d=  (r  x  s  x  hyp  log  R  x  2). 
Or,  p  d  —  2  r  s  x  hyp  log  R  =  ds  x  hyp  log  R ;  and 

/  =  j-xhyp  log  R (  i ) 


s  = 


hyp  log  R 
hyp  log  R=£ (3) 

These  formulas  express  the  relations  of  the  internal  pressure,  and  the 
maximum  tensile  stress  on  the  metal  at  the  inner  surface,  within  the  limits 
of  elastic  strength.  They  are  given  as  rules,  below. 


HOLLOW  CYLINDERS.  689 

Bursting  Strength. — For  cast-iron  cylinders,  the  foregoing  formulas  may 
also  be  employed  in  calculations  for  the  bursting  strength,  and  the  corre- 
sponding ultimate  breaking  strength. 

To  calculate  the  bursting  strength  of  wrought  iron  and  of  steel  cylinders, 
let  the  base-line  cae,  Fig.  286,  represent,  as  before,  the  inside  radius  ca, 
and  the  outside  radius  c  e.  Draw  the  verticals  c  c'  and  a  a',  to  measure  the 
ultimate  tensile  strength  of  the  metal  per  square  inch;  conceive  the  verti- 
cals to  be  bisected  at  points  which  may  be  indicated  as  c"  and  a",  and  draw 
c"  a"  e"  parallel  to  the  base.  The  rectangle  a  a"  e"  e  would  represent  the 
resistance  of  the  section  due  to  the  elastic  strength  of  the  material,  which 
is  uniform  throughout  the  thickness  and  is  taken  as  half  the  ultimate  tensile 
strength.  Draw  intermediate  vertical  ordinates  through  the  radial  intervals 
of  the  thickness,  between  a  and  c;  and  set  off  the  lengths  of  the  upper 
segments,  above  the  middle  level  a"  e" ,  to  represent  the  values  of  the  uni- 
formly varying  tensions  in  excess  of  the  elastic  limit,  forming  a  hyperbolic 
curve,  say,  a1  e'.  The  resistance  of  the  section  thus  treated,  consists  of  two 
parts: — the  uniform  resistance  aa" e" e,  equal  to  (r'-r)^y2  s;  and  the 

varying  resistance  a"  a'  e  e",  equal  to  (rx  — )  x  hyp  log  R.    Twice  the  sum 

of  these  resistances  is  equal  to  the  internal  pressure  per  inch  of  length 
of  the  cylinder;  whence,1 

(R  +  hyplogR-i) 


2  /,\ 

(R  +  hyplogR-i)'" 

)  =        +i  .............................   (6) 


RULES  FOR  THE  STRENGTH  OF  HOLLOW  CYLINDERS,  WITHIN  THE  LIMITS 
OF  ELASTIC  STRENGTH. 

RULE  i.  To  find  the  Internal  Pressure  for  a  given  maximum  tensile  stress 
on  the  material.  Multiply  the  hyperbolic  logarithm  of  the  ratio  of  the 
external  to  the  internal  diameter,  by  the  maximum  tensile  stress  in  tons 
per  square  inch  of  the  metal.  The  product  is  the  internal  pressure  in  tons 
per  square  inch. 

RULE  2.  To  find  the  maximum  Tensile  Stress  on  the  sides  for  a  given  inter- 
nal pressure.  Divide  the  pressure  in  tons  per  square  inch  by  the  hyperbolic 
logarithm  of  the  ratio  of  the  external  to  the  internal  diameter.  The  quo- 
tient is  the  maximum  tensile  stress  on  the  metal  in  tons  per  square  inch. 

1  The  formula  (4)  is  thus  deduced  :  — 

/  d=(  (r'  -  r)  x  ±  x  2  )  +  (  2  (  r  x  L)  x  hyp  log  R);  or, 

p  d-  (d-^-x  s)  +  (--  x  s  x  hyp  log  R). 


Dividing  both  sides 


and,  substituting  R  for  — ,  the  formula  for  the  pressure  becomes, 

p  =  ,  (R  +  hypiogR-D (4) 

44 


6Q3  STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 

RULE  3.  To  find  the  Ratio  of  the  Outside  Diameter  to  the  Inside  Diameter, 
for  a  given  maximum  tensile  stress  on  the  sides,  and  a  given  internal  pressure. 
Divide  the  pressure  by  the  stress,  both  in  tons  per  square  inch.  The 
quotient  is  the  hyperbolic  logarithm  of  the  ratio  of  the  diameters,  for  which 
the  ratio  may  be  found  in  a  table  of  hyperbolic  logarithms. 

RULES  FOR  THE  BURSTING  STRENGTH  OF  HOLLOW  CYLINDERS. 

Cast  Iron. 

The  rules  and  formulas  ( i ),  ( 2 ),  and  ( 3 ),  may  be  employed  for  calculat- 
ing the  bursting  strength,  and  the  corresponding  ultimate  tensile  strength, 
of  cast-iron  hollow  cylinders. 

Wrought  Iron  and  Steel. 

RULE  4.  To  find  the  Bursting  Pressure  for  a  given  ultimate  tensile  strength 
of  the  material.  To  the  ratio  of  the  outside  to  the  inside  diameter,  add 
the  hyperbolic  logarithm  of  this  ratio,  and  from  the  sum  deduct  i.  Multi- 
ply half  the  remainder  by  the  ultimate  tensile  strength  in  tons  per  square 
inch.  The  product  is  the  bursting  pressure  in  tons  per  square  inch. 

RULE  5.  To  find  the  Ultimate  Tensile  Strength  of  the  material  for  a  given 
bursting  pressure.  To  the  ratio  of  the  outside  to  the  inside  diameter,  add 
the  hyperbolic  logarithm  of  this  ratio,  and  from  the  sum  deduct  i.  Divide 
twice  the  bursting  pressure  in  tons  per  square  inch  by  the  remainder  just 
found.  The  quotient  is  the  ultimate  tensile  strength  in  tons  per  square 
inch. 

RULE  6.  To  find  the  Ratio  of  the  Outside  to  the  Inside  Diameter,  for  a  given 
bursting  pressure  and  ultimate  tensile  strength.  Divide  twice  the  bursting 
pressure  by  the  tensile  strength,  and  add  i  to  the  quotient.  The  sum  is 
equal  to  the  ratio  plus  the  hyperbolic  logarithm  of  the  ratio.  The 
value  of  the  ratio  is  found  by  trial  and  error,  in  a  table  of  hyperbolic 
logarithms. 

Notes  to  Rules  i  to  6. — i.  The  hyperbolic  logarithm  of  a  number  is  equal 
to  the  product  of  its  common  logarithm  by  2.3026.  2.  The  pressure  and 
the  tensile  stress  may  be  expressed  in  pounds  or  in  hundredweights,  instead 
of  tons. 

ist  Example.  —  Let  the  inside  diameter  of  the  cast-iron  cylinder  of  a 
hydraulic  press  be  10  inches,  the  outside  diameter  30  inches,  and  the 
ultimate  strength  of  the  metal  7  tons  per  square  inch;  to  find  the  bursting 

pressure.     The  ratio  of  the  diameters  is  (— =)  3,   of  which  the  hyper- 

10 

bolic  logarithm  is  1.0986  (see  table  No.  2,  page  61).     By  rule  i,  the  burst- 
ing pressure  is  (1.0986  x  7  =)  7.69  tons  per  square  inch. 

Average  Stress  on  the  Metal. — As  the  total  transverse  resistance  per  inch 
of  length  of  cylinder  is  equal  to  p  d,  which  is  the  product  of  the  inside 
diameter  by  the  bursting  pressure  per  square  inch,  the  average  stress  on  the 

metal  is  equal  to   rf    .  ;  that  is  to  say,  it  is  equal,  in  tons  per  square  inch, 
d-d 

to  the  product  of  the  inside  diameter  by  the  bursting  pressure  in  tons  per 
square  inch,  divided  by  the  difference  of  the  inside  and  outside  diameters. 


HOLLOW   CYLINDERS.  691 

In  the  foregoing  example,  the  average  stress,  which  bursts  the  cylinder,  is 

equal  to  /I0  x  7-  9  =  ^  3.845  tons  per  square  inch  of  section  —  little  more 

30-  10 
than  half  the  direct  tensile  resistance  of  the  metal. 

2d  Example.  —  A  steam-boiler,  7  feet  in  diameter  inside,  of  7/l6-inch 
wrought-iron  plates,  was  burst  at  a  longitudinal  double-rivetted  joint  by  a 
pressure  of  310  Ibs.  per  square  inch.  The  outside  and  inside  diameters, 
d  and  d',  were  84.875  inches  and  84  inches  respectively,  the  ratio  of  which 
is  1.0104.  By  formula  (  5  )  the  ultimate  tensile  strength  was 

_   3'°*'  62°  —      62°     =29,886  Ibs., 

i.  0104  +  hyp  log  1.0104—  i      i.  0104  +  .  010345  —  i     .020745 

or  13.34  tons  per  square  inch  of  the  section  of  the  solid  plate. 

$d  Example.  —  A  cast-iron  pipe,  10  inches  in  diameter  inside,  is  ^  inch 
in  thickness.  What  is  the  bursting  strength  when  the  ultimate  tensile 
strength  of  the  material  is  equal  to  7  tons  per  square  inch?  The  ratio  of 
the  outside  to  the  inside  diameter  is  as  11.5  to  10,  or  as  1.15  to  i;  and,  by 
formula  (  i  ), 

7  x  hyp  log  1.15  =  7  x  .1398  =  .9786  ton, 

or  2192  Ibs.  per  square  inch,  is  the  bursting  pressure. 

APPROXIMATE  RULES  FOR  TRANSVERSE  RESISTANCE  TO  BURSTING 

PRESSURE. 

When  the  diameter  is  very  considerable,  compared  to  the  thickness,  the 
transverse  resistance  to  bursting  pressure  may  be  taken  approximately  as 
directly  proportional  to  the  thickness  of  the  metal,  and  inversely  propor- 
tional to  the  diameter.  The  total  pressure  on  a  i-inch  length  of  section  of 
both  sides  together,  is  equal  to  the  product  of  the  diameter  by  the  pressure 
per  square  inch. 

Let  d—\h^  diameter,  in  inches;  /  =  the  thickness  of  metal  at  each  side, 
in  inches  ;  s  =  the  ultimate  tensile  strength  of  the  metal,  in  tons  per  square 
inch  ;  and  p  -  the  pressure  in  pounds  per  square  inch.  Then  dp  is  the  total 
pressure  on  a  i-inch  length  of  both  sides  together;  2  /  is  the  sectional  area 
of  both  sides;  and  2/.$-x  2240  =  dp,  or, 


RULE  7.  —  The  bursting  pressure  in  pounds  per  square  inch  of  surface  is 
equal  to  4480  times  the  product  of  the  thickness  by  the  ultimate  tensile 
strength  per  square  inch,  divided  by  the  diameter. 

RULE  8.  —  The  thickness  of  metal  required  at  each  side  is  equal  to  the 
product  of  the  diameter  and  the  pressure  per  square  inch,  divided  by  the 
ultimate  tensile  strength  in  tons  per  square  inch,  and  by  4480. 

RULE  9.  —  The  ultimate  tensile  strength  in  tons  per  square  inch  of  section 
of  metal  is  equal  to  the  product  of  the  diameter  by  the  bursting  pressure  in 
pounds  per  square  inch;  divided  by  the  thickness  of  metal  and  by  4480. 

Note.  —  When  the  material  is  made  of  jointed  plates,  the  tensile  stioigth 
of  the  whole  plate  is  to  be  multiplied  by  the  coefficient  of  strength  of  the 
joint,  to  give  the  reduced  strength  to  be  employed  as  the  value  of  s  in  the 
calculation. 


692 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


LONGITUDINAL  RESISTANCE  TO  BURSTING  PRESSURE. 

When  the  ends  of  a  cylinder  are  closed,  and  make  one  piece  with  the 
cylindrical  portion,  the  total  longitudinal  resistance  to  internal  pressure  is 


b         b  d       d 

Fig.  287. — Diagram  for  the  Resistance  of  a  Flat-headed  Cylinder  to  Bursting  Pressure. 

directly  proportional  to  the  thickness  of  the  metal,  and  to  the  diameter; 
whilst  the  total  bursting  force,  acting  on  the  ends,  is  proportional  to  the 

Table  No.  246. — EXPERIMENTAL  RESISTANCE  OF  SOLID-DRAWN  TUBES  TO 
BURSTING  PRESSURE  AND  COLLAPSING  PRESSURE. 

(Deduced  from  Messrs.  Russell's  data. ) 
WROUGHT-!RON  TUBES. 


External 
Diameter. 

Thickness. 

'  Internal 
Diameter. 

Bursting  Pressure. 

Collapsing  Pressure. 

Difference 
of  Burst- 
ing and 
Collapsing 
Pressures. 

Per 
square  inch 
of 
Surface. 

Per  square 
inch  of 
Section  of 

Metal. 

Per 

square 
inch  of 
Surface. 

Per  square 
inch  of 
Section  of 
Metal. 

inches. 

31X 
3l/s 

i* 

2^ 

2X 
2 

I* 

Avera 

B.W.G. 
10 
10 
II 
II 
II 
1  1 
12 
12 

ges,  omi 

inch. 
•134 
•134 
.120 
.120 
.120 
.120 
.109 
.109 

tting  the 

inches. 
2.982 
2.857 
2.760 
2.510 
2.260 
2.OIO 
1.782 
1-532 

last  tube 

Ibs. 
4800 
4500 
4500 
5200 
5000 
5900 
5900 
5600 

tons. 
23.84 
21.42 
23.10 
24.28 
21.02 
22.06 
21-53 
17-57 

Ibs. 
3300 

3150 
3500 
3500 
3600 
4500 
4900 
4OOO 

tons. 
17.86 
16.40 

19.53 
17.89 
16.74 
18.82 
20.07 
H.33 

tons. 
5.98 
5.02 

3-57 
6-39 
4.28 

3-24 
1.46 
3-24 

— 

22.40 

— 

18.20 

— 

I'A 

2 

iH 

13 
13 
15 

17 

.095 
.095 
.072 
.058 

HOMOGEN 
2.810 
2.060 
1.856 
1.409 

EOUS  ME" 
3600 
7600 
4000 
4600 

PAL  TUBE 

23-77 
36.78 
23.02 
24.94 

3150          22.20 
4600          24.32 
3500          21.70 
4000          25.02 

— 

i# 

13 

.095 

BESSEMER  STEEL  TUBES. 
1.56       j)     7800       28.92 

4600     i      18.91 

— 

2^ 
I* 
l/B 
# 

— 

5/i6 
5/i6 

5/i6 

HYI 

r# 

i 
% 

X 

)RAULIC    T 
proved  to 
1    11,000 
6,OOO 

4,000 

12,000 

'UBES.  •• 

14-73 
4.29 
2.23 
2.15 

— 

— 

— 

HOLLOW  CYLINDERS.  693 

square  of  the  diameter.  It  results  that  the  longitudinal  resistance  per 
square  inch  to  bursting  force  is  inversely  proportional  to  the  diameter. 
Let  the  circle  and  the  rectangle,  Fig.  287,  be  a  cross  section  and  a 
longitudinal  section  of  a  cylindrical  boiler;  the  area  of  the  circle  is  a 
measure  of  the  longitudinal  pressure  of  the  steam  on  the  ends  of  the  boiler. 
Set  off  the  interval  a  a  on  the  circular  section,  and  the  interval  ccon.  the 
longitudinal  section;  and  draw  the  diameters  ab,ab,  and  the  parallels  cd,cd. 
The  areas  of  pressure  to  be  resisted  are  respectively  the  two  triangular 
spaces  ab,  and  the  rectangle  cd;  and,  since  the  former  have  only  half  the 
surface  of  the  latter,  it  follows  that  the  longitudinal  stress  per  square  inch 
on  the  shell  is  only  half  the  transverse  stress,  and  that  the  amount  of  the 
longitudinal  resistance  is  proportionally  twice  as  much  as  that  of  the  trans- 
verse resistance. 

On  the  same  showing,  a  hollow  sphere  resists  twice  the  pressure  per 
square  inch,  that  a  tube  of  equal  diameter  and  equal  thickness  can  do. 

Wrought-iron  Tubes. — Messrs.  J.  Russell  &  Sons  tested  the  resistance  of 
solid-drawn  wrought-iron  tubes  to  bursting  pressure,  and  to  collapsing  pres- 
sure, on  the  results  of  which  table  No.  246  is  based. 

The  bursting  pressure  of  the  wrought-iron  tubes  in  tons  per  square  inch 
of  section  of  metal,  appears  to  be  practically  constant;  and  it  may  be 
taken  for  practical  purposes  that  the  ultimate  strength  is  measured  by  the 
tensile  strength  of  the  material. 

Resistance  of  a  Lancashire  Boiler  to  Bursting  Pressure. — A  boiler  7  feet 
in  diameter,  made  of  7/i6-hich  plates,  was  tested  by  Mr.  L.  E.  Fletcher, 
and  bore  a  pressure  of  310  Ibs.  per  square  inch,  when  it  failed  at  one  of 
the  longitudinal  seams,  which  were  double-rivetted.  Applying  rule  9, 
page  691,  the  ultimate  tensile  strength  was  equal  to  (84  mches  x  310  Ibs.  = ) 

•4375  x  440° 

13.29  tons  per  square  inch  of  section  of  the  entire  plate.  This  instance 
formed  the  subject  of  the  2d  example,  page  638.  In  this  instance,  the 
tensile  strength,  as  calculated  by  the  exact  rule  5,  page  690,  is  13.34  tons 
per  square  inch.  This  is  .05  ton,  or  about  2/5  ths  of  i  per  cent,  more 
than  is  given  by  the  approximate  rule. 

Take  the  correctly  calculated  strength,  13.34  tons,  with  the  net  section 
of  plate  between  the  rivets,  which  was  two-thirds  of  the  section  of  the  con- 
tinuous plate.  Then  13.34x^-20.01  tons  per  square  inch,  the  tensile 

strength  of  the  plate  between  the  rivet-holes. 

Resistance  of  a  Cylindrical  Marine  Boiler  and  a  Superheater  to  Bursting 
Pressure}- — A  cylindrical  boiler,  1 1  feet  3  inches  in  diameter  inside,  of  ^-inch 
plates,  double-rivetted,  was  burst  by  a  hydraulic  pressure  of  230  Ibs.  per  square 
inch,  equivalent,  by  rule  9,  to  J35  x  23° — s.  _  2^  tQns  per  Square  jncjj 

•  75   *448o 

of  section  of  the  solid  plate.  The  rivet-holes  were  i  x/i6  inch  in  diameter,  at 
2^4  inches  pitch,  leaving  61.36  per  cent,  of  solid  metal  between;  and  the 
ultimate  tensile  strength  of  metal  left  between  the  rivet-holes  was  9.241  x 

100  .     , 

-? — -=  15.06  tons  per  square  inch. 

*  \) 

A  superheater,  99.915   inches  in  diameter  inside,   of   9/l6-inch  plates, 
1  The  data  are  derived  from  Engineering,  July  21,  1876,  page  47. 


694  STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 

double-rivetted,  with  ^/l6-mch  rivet-holes  at  2^  inches  pitch,  was  burst  at 
the  same  time  by  a  hydraulic  pressure  of  245  Ibs.  per  square  inch,  equiva- 

lent to  99-9*5  x  2^5  =  9.655  tons  per  square  inch  of  the  solid  plate.     The 

.  .562"  x  4480 
metal  left  between  the  rivet-holes  was  67.6  per  cent,  of  the  entire  section, 

and  the  resistance  of  that  metal  was  9.655  x  -  —  -  =  14.28  tons  per  square 
inch  of  its  net  section.  '' 

Now  the  tensile  strength  of  the  plates  of  the  boiler  and  the  superheater, 
tested  by  Mr.  Kirkaldy,  averaged  20.5  tons  per  square  inch  for  the  boiler, 
and  20.2  tons  for  the  superheater.  These  instances  have  already  been 
noticed  at  page  638,  in  the  discussion  of  rivet-joints,  and  they  forcibly 
demonstrate  the  essential  weakness  of  rivetted  lap-joints  in  very  thick 
plates.  The  net  tensile  resistance  of  the  plates  between  the  holes  was 
reduced  a  fourth. 

Cast-Iron  Pipe.  —  For  a  lo-inch  pipe,  ^  inch  thick,  having  an  ultimate 
tensile  strength  of  7  tons  per  square  inch,  the  bursting  pressure  is,  by 
formula  (  7  ),  page  691, 

4.4.80  x  .7?  x  7  ,,  , 

—  '-2  —  '  =  2352  Ibs.  per  square  inch. 
10 

By  a  previous  calculation,  page  691,  with  the  exact  formula  (  i  ),  the  burst- 
ing pressure  was  found  to  be  2192  Ibs.  per  square  inch,  showing  that  the 
ordinary  approximate  formula  (  7  )  gives  160  Ibs.,  or  7^  per  cent.,  more 
than  the  correct  formula. 

RESISTANCE  OF  HOLLOW  CYLINDERS  TO  EXTERNAL  COLLAPSING 

PRESSURE. 

Solid-drawn  Tubes,  —  By  the  action  of  a  centripetal  force  on  the  outside 
of  a  hollow  cylinder,  compressive  stress  is  produced,  tending  to  collapse  the 
cylinder.  In  table  No.  246,  the  resistance  of  wrought-iron  tubes  to  col- 
lapse is  given.  It  is  less  than  the  resistance  to  bursting,  and  the  difference 
between  the  bursting  and  the  collapsing  pressures  increases  with  the  dia- 
meter, as  shown  in  the  last  column.  When  plotted  and  arranged  into  a 
curve,  or,  as  in  this  case,  a  straight  line,  the  value  of  the  difference,  in 
terms  of  the  diameter,  is  2^3  (d-  i),  which  probably  holds  for  diameters 
up  to  6  inches.  When  the  diameter  d  is  only  one  inch,  d-  i  =o,  and  the 
difference  vanishes.  The  average  bursting  pressure  being  22.40  tons  per 
square  inch  of  section  of  metal,  the  collapsing  pressure  is  22.40-2^ 
(</-i);  or  — 

Collapsing  Pressure  per  square  inch  of  Longitudinal  Section  of  Metal  for 
Solid-drawn  Iron  Tubes  up  to  6  inches  in  diameter. 


in  which  d  is  the  external  diameter  in  inches,  and/'  is  the  collapsing  pres- 
sure in  tons  per  square  inch  of  longitudinal  section.  The  thickness  is 
taken  as  from  x/iSth  to  I/25tli  of  the  diameter. 

The  corresponding  pressure  per  square  inch  on  the  external  surface  of 
the  tube,  is  equal  to  the  total  collapsing  pressure  for  i  inch  in  length  of 


the  tube,  divided  by  the  diameter  ;  or  it  is    -     _-         .  an(^  reducing, 


HOLLOW   CYLINDERS. 


695 


Collapsing  Superficial  Pressure  per  square  inch  (ist  formula). 

4480  P't.  , 
d 


(5) 


p  being  the  superficial  pressure  in  pounds  per  square  inch,  and  /  and  d  the 
thickness  and  the  diameter  in  inches. 

The  collapsing  pressure  may  be  expressed  in  terms  of  the  diameter,  by 
substituting  in  the  above  formula  (  5  )  the  value  of  /'  in  (  4  ),  when  it 


becomes,  by  reduction,  /  =  /  x  ( 


*  *  2°°° 


-  11947),  or,  in  round  numbers,  — 


Collapsing  Superficial  Pressure  per  square  inch  (2d  formula). 


(6) 


The  table  No.  247  shows  the  bursting  and  collapsing  pressures  of  solid 
wrought-iron  tubes  of  the  usual  diameters  and  thicknesses,  calculated  by 
means  of  the  preceding  formulas  :  — 

Table  No.  247.  —  SOLID-DRAWN  IRON  TUBES  —  CALCULATED  BURSTING 
AND  COLLAPSING  PRESSURES. 


External 
Diameter. 

Thickness. 

Internal 
Diameter. 

Bursting  Pressure. 

Collapsing  Pressure. 

Per 

square  inch 
of  Internal 
Surface. 

Per  square 
inch  of 
Section  of 
Metal. 

Per 
square  inch 
of  External 
Surface. 

Per  square 
inch  of 
Section  of 
Metal. 

inches. 

B.W.G. 

inch. 

inches. 

Ibs. 

tons. 

Ibs. 

tons. 

Ijf 

H 

.083 

.084 

7700 

22.4 

6500 

21.7 

iH 

14 

.083 

.209 

6900 

22.4 

5800 

21.3 

!/•£ 

.083 

•334 

6200 

22.4 

5200 

21.0 

m 

H 

.083 

•459 

5700 

22.4 

4700 

20.7 

i% 

14 

.083 

.584 

5300 

22.4 

4300 

20.3 

i  # 

H 

.083 

.709 

4900 

22.4 

4000 

2O.O 

2 

14 

.083 

.834 

4500 

22.4 

3700 

197 

2/^ 

13 

.095 

•935 

4900 

22.4 

3800 

19.3 

2^ 

13 

.095 

2.060 

4600 

22.4 

3600 

19.0 

12 

.109 

2.282 

4800 

22.4 

3600 

18.3 

2^ 

12 

.109 

2.532 

4300 

22.4 

3100 

177 

3 

II 

.120 

2.760 

4400 

22.4 

3000 

17.0 

3X 

II 

.120 

3.010 

4000 

22.4 

2700 

I6.3 

3/^ 

10 

•134 

3-232 

4200 

22.4 

2700 

157 

3^ 

10 

•134 

3482 

3900 

22.4 

2400 

15.0 

4 

10 

•134 

3-732 

3600 

22.4 

2100 

H.3 

4X 

10 

•134 

3.982 

3400 

22.4 

1900 

137 

4X 

10 

•134 

4.232 

3200 

22.4 

1700 

13.0 

4^f 

10 

•134 

4.482 

3000 

22.4 

1600 

I2.3 

5 

10 

.134 

4-732 

2800 

22.4 

1400 

II.7 

5X 

9 

.148 

4-954 

3000 

22.4 

1400 

II.O 

5^ 

9 

.148 

5.204 

2800 

22.4 

1200 

10.3 

9 

.148 

5-454 

2700 

22.4 

1  100 

9-7 

64 

9 

.148 

5.704 

2600 

22.4 

IOOO 

9.0 

696  STRENGTH   OF  ELEMENTARY  CONSTRUCTIONS. 

Large  Furnace-  Tubes.  —  The  furnace-tubes  of  Lancashire  and  Cornish 
boilers  vary  in  diameter  from  18  inches  to  4  feet,  and  they  are  usually  com- 
posed of  rings  of  plates  rivetted  together.  The  resistance  to  collapse  under 
external  steam  pressure  is  derived  partly  from  the  longitudinal  tension  to 
which  they  are  subject;  and  partly  from  their  direct  resistance  to  compres- 
sion and  collapse.  It  is  supposed  that  longitudinal  tensional  resistance  is 
brought  into  action  to  an  important  extent,  for  supporting  the  tube.  This 
is  a  mistaken  supposition.  The  records  of  collapses  of  furnace-tubes  show 
that  the  length  of  the  tube  was  the  least  influential  factor;  and  of  small  value 
unless  for  very  short  lengths  of  from  3  feet  to  6  feet  or  9  feet,  denned  by 
stiffening  rings  or  seams,  if  not  by  the  actual  length  between  the  end  plates. 
Practically,  all  the  resistance  to  collapse  of  unfortified  lengths  of  plain 
furnace-tubes  is  supplied  by  the  compressive  resistance  and  the  stiffness  of 
the  tube. 

From  the  monthly  reports  of  Mr.  Lavington  E.  Fletcher,  21  cases  of 
collapsed  iron  tubes  have  been  extracted,  comprising  flues  of  from  32  to 
48  inches  in  diameter,  ffi  inch  and  ?/l6  inch  in  thickness,  and  from  18  to 
40  feet  in  length,  under  collapsing  pressures  of  from  40  Ibs.  to  70  Ibs.  per 
square  inch.  In  some  instances,  no  doubt,  the  tubes  had  been  sensibly 
worn;  but  in  most  instances,  they  had  been  in  good  order. 

By  plotting  the  results  of  these  collapsed  tubes,  and  tracing  a  mean  curve 
through  the  plots,  the  following  formula  is  derived  :  — 

Collapsing  Pressure  of  Plain  Iron  Furnace-tubes  of  Cornish  and  Lancashire 

Steam  Boilers. 


,       v 

(3  ; 


p  =  collapsing  pressure,  in  pounds  per  square  inch. 

/  =  thickness  of  the  plates  of  the  furnace-tube,  in  parts  of  an  inch. 

</=  internal  diameter  of  the  furnace-tube,  in  inches. 

This  formula  is  directly  applicable  to  furnace-tubes  of  any  length  greater 
than  9  feet. 

From  the  results  of  hydraulic  tests  made  at  the  Leeds  Forge,  it  appears 
that  a  plain  flue-tube,  3  feet  i  inch  inside  diameter,  of  ^-inch  plate,  7  feet 
long,  bore  a  pressure  of  175  Ibs.  per  square  inch  before  giving  way  by 
collapse  ;  and  that  a  like  tube,  corrugated,  on  Fox's  system,  bore  a  pressure 
of  450  Ibs.  per  square  inch  before  giving  way. 

LEAD  PIPES. 

Mr.  Jardine  found  that  a  i^-inch  lead  pipe,  .20  inch  thick,  sustained  a 
pressure  of  1000  feet  of  water,  or  29}^  atmospheres,  without  any  alteration 
of  form.  Under  1200  feet  of  water,  or  35  atmospheres,  it  began  to  en- 
large; and  it  burst  under  1400  feet,  or  40  atmospheres,  having  swollen 
to  a  diameter  of  i^  inches.  A  2-inch  pipe,  .20  inch  thick,  sustained  a 
pressure  of  800  feet  of  water,  or  23*^  atmospheres,  with  scarcely  any 
enlargement;  but  it  burst  under  1000  feet,  or  29  atmospheres.  From  these 
results  it  appears,  by  the  aid  of  rule  2,  page  689,  that  the  elastic  strength 
of  lead  is  equal  to  15  cwts.  per  square  inch  of  sectional  area  and  that  the 
ultimate  strength  is  equal  to  i  ton  per  square  inch. 

1  Monthly  Reports  to  the  Manchester  Steam-Users'  Association,  1862-69. 


FRAMED  WORK. 


697 


FRAMED  WORK:— CRANES,  GIRDERS,  ROOFS,  &c. 

WHEN  THE  WEIGHT  OR  FORCE  is  PARALLEL  TO  ONE  OF  THE  MEMBERS 

OF  THE  FRAME. 

For  the  purpose  of  resisting  the  stress  of  heavy  loads,  the  triangle  consti- 
w 

o 

a> 


Fig.  288. 


Fig.  289.  Fig.  290. 

Illustrations  of  Stress  in  Framed  Work. 


ji .     j,     '!,• 

OU       CL     CL- 

Fig  291. 


tutes  the  fundamental  feature  of  framed  work,  as  distinguished  from  solid 
work  or  web-work.  When  a 
load  W  is  applied  direct  to  a 
vertical  pillar  ab,  Fig.  288,  the 
resistance  is  in  the  line  of  the 
stress,  and  no  framework  is 
employed.  But,  if  the  load  be 
applied  at  c,  Fig.  289,  hori- 
zontally  apart  from  <z,  the  tri- 
angular frame  a  b  c  is  con- 
structed to  carry  it.  Complete 
the  parallelogram  ad,  and  it 
is  seen  that,  if  the  vertical 
stress  of  W  be  measured  by 
ab  or  cdt  the  horizontal  tensile 
stress  in  a  c,  and  the  diagonal 
compressive  stress  in  cb,  are 
measured  by  the  lengths  of 
these  members  respectively. 
It  is  obvious  here,  as  in  other 
cases,  that  when  a  counteract- 
ing resistance  is  opposed  ob- 
liquely to  a  weight  or  other 
force,  the  resisting  stress  is 
necessarily  greater  than  the 
force;  and  that  the  diagonal 
stress  increases  with  the  over- 
hang, as  in -Fig.  290,  where, 
under  the  weights  c,  c',  and  c" 
successively  further  from  the 
origin  a,  the  diagonal  com- 


Os 


Fig.  295. 
Illustrations  of  Stress  in  Oblique  Framed  Work. 

The  hori- 


pressive  resistances  cb,  cb,  and  c"  b  are  successively  increased. 

.zontal  tensional  resistances,  measured  by  ca,  c' a,  and  c"a,  are  likewise 


698  STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 

successively  increased.  These,  in  fact,  being  perpendiculars  to  ab,  are 
the  leverages  of  the  weights  tending  to  pull  over  the  upper  end  a  of  the 
upright  ab;  and  simultaneously  to  push  out  the  lower  end  b.  The  verticals 
cd,  c'd',  and  c"d',  equal  and  parallel  to  ab,  are  successively  the  measures 
of  the  same  weight  at  the  several  positions. 

If  a  diagonal  of  constant  length,  equal,  say,  to  ab,  be  moved  into  vari- 
ous positions,  be',  be",  and  be'",  on  the  lower  end  b,  with  a  given  weight  W 
supported  at  the  end,  as  in  Fig.  291 ;  the  weight  is  to  the  thrust  in  the  dia- 
gonal, proportionally,  as  the  vertical  cd',  c" d",  or  c"'d'",  is  to  the  diagonal; 
showing  that  the  thrust  in  the  diagonal  increases  as  the  elevation  is  dimin- 
ished ;  and  that  the  horizontal  tension  in  a' c,  a"  c" ,  and  a" c",  also  increases. 

In  oblique-angled  frames,  like  Figs.  292,  293,  and  294,  the  stresses  in 
the  three  members  are  respectively  as  their  lengths.  The  horizontal 
pulling  and  thrusting  stresses  at  the  upper  and  lower  points  a  and  b  respec- 
tively, are  measured  by  the  perpendicular  c'  a  or  ee,  in  Figs.  292  to  294; 
and  they  are  the  same  as  if  the  upper  member  had  been  horizontal,  as 
at  ac'. 

WHEN  THE  WEIGHT  OR  FORCE  is  NOT  PARALLEL  TO  ANY  LEADING 
MEMBER  OF  THE  FRAME. 

The  weight  W,  Fig.  295,  is  supported  by  the  slanting  triangular  frame 
a  be.  The  vertical  cd  represents  the  weight.  By  the  parallelogram  of 
forces,  the  stresses,  tensile  and  compressive,  in  ac  and  be,  proportionally 
to  the  weight,  are  ascertained.  Draw  the  vertical  be',  then  the  triangle 
bc'e  represents  the  three  forces  in  equilibrium: — be'  for  the  weight,  and 
be  and  c' c  for  the  thrust  and  the  pull  in  the  respective  members.  This 
triangle  of  forces  is  the  same  as  if  the  members  cc'  and  eb  had,  in  fact,  been 
fixed  to  a  vertical  wall  or  member  be',  as  in  the  bracket,  Fig.  293;  and  it 
is  apparent  that  wherever  the  members  ac  and  be  be  extended  to  or 

attached,  the  stresses  for  a  given  weight 
remain  unaltered.  The  compressive  stress  in 
the  horizontal  member  ab,  is  expressed  by 
the  horizontal  bd. 

The  construction  of  the  parallelogram  may 
be  dispensed  with  by  simply  drawing  the  ver- 
tical be'  forming  up  the  triangle  of  forces  bee. 
The  horizontal  bd,  measures  the  thrust  in  the 
member  ab. 

Fig.  296. — Elementary  Iruss.  T  .        _.  ,  .  i        r 

Let  a  be,  Fig.   296,  be  a  triangular  frame, 

with  equal  limbs  ea  and  eb  resting  on  supports  at  a  and  b,  and  loaded 
at  the  apex  by  W.  Complete  the  parallelogram  ee;  then  ce  is  the  weight, 
cd  is  half  the  weight,  and  ea  and  eb  are  the  compressive  stresses  in  the 
sides.  Again,  the  stresses  ac  and  be  are  resolved  into  the  vertical  pres- 
sures ac'  and  be"  on  the  points  of  support,  each  equal  to  cd,  half  the 
load;  and  the  horizontal  tensile  stresses  ad  and  bd,  in  the  lower  member, 
equal  and  opposite  to  each  other.  One  of  these,  or  the  half  of  ab,  is  the 
measure  of  the  tension  in  this  member. 

Otherwise,  by  equality  of  moments: — yz  W  x  y2  l=dx  tension  in  ab;  in 
which  d  (=  ^  W)  is  the  depth  cd,  and  /  is  the  span  ab.  Thence,  j^  /  = 
ad,  is  the  tension  in  ab. 


FRAMED   WORK. 


699 


Further,  reducing  the  equation  of  moments, 

tension  in  ab=  — —    .* (A) 

Or,  the  tension  in  the  horizontal  member  ab  is  equal  to  the  product  of  the 
weight  by  the  span,  divided  by  4  times  the  rise. 

The  horizontal  thrust  at  the  apex  is  equal  to  the  tension  in  ab. 

The  length  of  the  inclined  members  ac  and  cb,  in  terms  of  the  span  and 

the  rise,  =  A/ (%  span) 2  + rise2;  from  which  the  stress  in  these  members 
is  given. 

FRAMED  GIRDERS — THE  WARREN-GIRDER,  LOADED  AT  THE  MIDDLE. 

Let  two  equilateral  triangles  or  bays,  like  Fig.  296,  be  framed  together  as 
in  Fig.  297,  forming  in  all  three  triangles;  and  loaded  at  the  middle.  Com- 
plete the  parallelogram  de,  and  the  weight  measured  by  de  is  resolved  into 
the  tensile  stresses  dc'  and  dc".  Each  of  these  is  resolved  into  two  com- 
pressive  stresses: — c'a  and  c"b  to  the  points  of  support,  and  c' c"  and  c" c' 
equal  and  opposed  to  each  other.  The  thrusts  at  a  and  b  are  resolved  into 
the  vertical  components  ac'"  and  bc#  each  equal  to  cd,  half  the  weight, 
resisted  and  carried  by  the  supports  at  a  and  b;  and  the  horizontal  com- 
ponents ad'  and  bd",  equal  and  opposite  to  each  other,  and  each  of  them 
exhibiting  the  tensile  stress  in  the  lower  member  ab. 


Fig.  298. 
Framed  Girders — the  Warren-Girder. 

The  compressive  stress  in  the  upper  member  c'  c"  is  double  the  tensile 
stress  in  the  lower  member  a  b. 

Suppose  a  girder  of  five  equilateral  triangles,  Fig.  298,  having  three  bays 
below  and  two  above,  loaded  at  the  central  apex  c.  Complete  the  parallelo- 
gram ce;  the  weight  ce  is  resolved  into  the  thrusts  ca  and  cb ,  which  are 
resolved  into  opposing  tensions  a'b'  and  b'  a',  and  tensions  a'  c'  and  b'  c" . 
These  tensions  are  resolved  into  opposing  thrusts  c  c  and  c"  c,  and  thrusts 
c'  a  and  c"  b  which  terminate  at  the  points  of  support  a  and  b.  These 
final  thrusts  are  resolved  into  the  vertical  components  ac"  and  bc^  each 
equal  to  half  the  weight,  and  the  horizontal  tensions  ad'  and  bd".  These 
horizontal  tensions,  which  are  half  the  tension  exerted  in  the  middle  bay, 
are  transmitted  to  the  middle  bay  a'b',  where  they  balance  each  other. 
The  middle  bay  is  thus  subjected  to  two  tensile  stresses: — the  stress  due  to 
the  thrust  of  the  weight  on  the  middle  diagonals  ca'  and  cb',  measured  by 
a'b',  the  length  of  a  bay;  and  the  transmitted  stress  excited  at  the  supports, 
measured  by  ad'  or  bd",  the  length  of  half  a  bay.  The  total  tension  on  the 


/oo 


STRENGTH   OF   ELEMENTARY   CONSTRUCTIONS. 


middle  bay  is  therefore  measured  by  i  ^  times  the  length  of  a  bay,  and  that 
on  each  side  bay  by  half  a  length,  or  a  third  of  the  tension  in  the  middle  bay. 

The  compressive  stress  in  each  of  the  upper  bays  cc',  cc",  is  measured 
by  the  length  of  a  bay,  and  is  twice  the  stress  in  the  end  lower  bay. 

The  successive  stresses  in  the  lower  and  upper  bays  consecutively,  it  is 
seen,  increase  uniformly  from  each  end  towards  the  middle,  thus : — 

Tablet  a. 


Bays  in  compression,  
Bays  in  tension,  

a  a' 

c'c 

c  c" 
a'b'        — 

b'  b 

Stresses  as 

i 

2 

T.                    2 

i 

3    i 

At  the  same  time  the  function  of  the  diagonals,  or  braces,  is  to  transmit 
the  incidence  of  the  weight  to  the  supports,  by  compression  and  tension 
alternately. 

Invert  this  girder,  as  in  Fig.  299,  and  suspend  the  weight  from  the 
inverted  apex  c.  The  stresses  in  the  several  members  are  of  the  same 
intensity,  but  reversed,  thus : — 

Tablet  b. 


Bays  in  compression, 

a  a! 

a!  V 

1 

b'  b 

Bavs  in  tension,  

c'  c 

cc" 

Stresses  as 

i 

>~> 

T. 

2 

i 

Let  the  girder,  Fig.  298,  be  doubled  in  length,  to  comprise  six  lower 
bays,  and  five  upper  bays,  as  in  Fig.  300;  and  loaded  at  the  middle.     The 


O 


w 


Fig.  300. — Warren-Girder. 

horizontal  stresses  in  the  flanges  are  accumulated  from  each  end  towards 
the  middle,  where  they  are  a  maximum,  as  in  tablet  c. 

Tablet  c. 


. 

Bays  in  compression,  .. 
Bays  in  tension,  

ad 

*r 

a'  a" 

r'V 

—  1  c'  c" 
a"  d\    — 

db" 

c'c, 

yff 

c±c<> 

Vb 

Stresses  as  

I 

2 

T. 

4 

5    1     6 

e 

4" 

3 

2 

I 

Valuation  of  the  horizontal  stress  in  terms  of  the  load. — The  unit-stress  i, 
in  the  tablet  c,  is  measured  by  a  d'9  Fig.  300,  the  horizontal  component  of  the 
oblique  thrust  csa,  of  which  csd'  is  the  vertical  component,  or  half  the 
weight.  The  value  of  the  unit-stress  relatively  to  that  of  c~d't  or  half 


FRAMED   WORK.  70 1 

the  weight,  may  be  measured  by  means  of  a  scale  of  parts.     Or,  trigono- 
metrically,  let  the  angle  at  a  be  signified  by  a,  then, — 

c^d'  :  ad1  :  :  sine  a  :  cosine  a  :  :  ^  W  :  unit-stress;  and 

T/  iir  cosine  a  /     x 

unit-stress  at  a  =  yz  W  — : (  i  ) 

sine  a 

In  the  Warren-girder,  the  angle  a  is  60°;  and 

unit-stress  a.ta  =  j4  W'-^=^A  W  x  .577;  or 
.060 

unit-stress  at  a  =  . 2885  W  (2) 

showing  that  the  horizonal  unit-stress  in  each  of  the  end  bays  is  equal  to 
the  weight  at  the  centre  multiplied  by  .2885. 

The  stresses  in  the  other  bays,  above  and  below,  are  in  simple  proportion 
to  their  numerical  order  from  the  support  at  each  end  towards  the  centre : — 

stress  on  any  bay  =  unit-stress  x  N, (  3  ) 

in  which  N  is  the  order-number  of  the  bay.     The  stress  on  the  central  bay 
is  also  expressed  by  the  equation, — 

stress  on  the  central  bay  =  unit-stress  x  —  — , (  4  ) 

in  which  n  is  the  total  number  of  bays.     Also, 

stress  on  the  middle  pair  of  bays  =  unit-stress  x (5) 

2 

In  the  example,  Fig.  300,  the  stress  on  the  central  bay  c'  c",  by  formula 
(3)  or  (4),  is,— 

(.2885  Wx6),  or  (.2885  W  XI1±-I)=  1.731  W; (a) 

and  the  stress  in  the  central  pair  of  bays  is, — 

(.2885  Wx  5),  or  (.2885  WxH-^-V  1.443  W (b) 

It  appears  that  the  stress  at  the  middle  of  the  longer  boom  is  greater 
than  the  stress  at  the  middle  of  the  shorter  boom  by  one  unit-stress. 

Valuation  by  momerits, — The  tension  in  the  central  bay  is  given  by  the 

expression  (A),  page  699,  namely  — -,  in  which  W  is  the  weight,  /  the  span, 

4f 
=  6  bays,  and  d  the  depth  =  .866  (sine  a)  proportionally,  the  length  of  a 

bay  being  i.    The  tension  is,  then, ^— -  =  1.732  W,  as  already  found  (a). 

4  x  .866 

Stress  in  the  braces. — The  stress  in  the  braces  is  to  half  the  weight,  as  thie 
length  of  a  brace  is  to  the  depth  of  the  girder,  or  as  radius  to  sine  a,  therefore, — 

Stress  in  each  brace  =  yz  W  x L_  =  — ^ —   .  .  (  6  ) 

sine  a     2  sine  a 

In  the  Warren-girder,  sine  a -.866,  and 

the  stress  in  the  brace  (Warren-girder)  is — —  =  .577  W,...(  7  ) 

2  x  .866 

which  is  twice  the  unit-stress  in  the  flange. 


702  STRENGTH    OF   ELEMENTARY   CONSTRUCTIONS. 

THE  WARREN-GIRDER  LOADED  AT  AN  INTERMEDIATE  POINT  OTHER 
THAN  THE  CENTRE. 

In  a  Warren-girder,  Fig.  301,  loaded  at  d,  as  in  an  ordinary  loaded  beam, 
the  weight  on  the  supports  at  a  and  b  are  respectively 

-  W  andyW,    (8) 

in  which  /  is  the  total  number  of  bays  in  the  longest  flange,  and  m  and  n 
the  number  of  bays  to  the  left  and  to  the  right  of  the  weight. 


Ow 

Fig.  301. — Warren-Girder,  loaded  at  any  intermediate  point. 

Stress  in  the  Braces. — The  stresses  in  the  braces  d  c'  and  d  c",  which 
immediately  support  the  weight,  are  as  n  and  m,  or  inversely  as  the  lengths 
of  the  two  segments;  and,  adapting  formula  (6), 

n         W 

Stress  in  the  braces  of  the  longer  side  =  —  x  — (o) 

/      sine  a 

j  ,  i         m        W  ,       \ 

Do.  do.  shorter  do.  =—  x  _ (  10  ) 

/      sine  a 

Sine  a^  .866,  and  in  this  example  the  stresses  are, 

2  W 

In  the  longer  side  =—  x  — —  =  .385  W,  ,. (c) 

6      .866 

In  the  shorter  side  =  -^- x-——:=  .770  W,   (d} 

6      .866 

transmitted  to  the  supports  a  and  b,  and  there  resolved  into  vertical  and 
horizontal  components. 

Second  Process  for  the  stress  in  the  braces — Unit-coefficient  of  diagonal  stress. — 
The  sum  of  the  stresses  (c]  and  (d)  is  1.155  W,  which  bears  to  the  weight 
W  the  ratio  of  the  length  of  a  brace  to  the  depth  of  the  girder;  since 
i.  :  .866  :  :  1.155  :  i.  Divide  the  coefficient  1.155  by  the  number  of 
diagonals,  2  /  or  12,  and  the  quotient  .09625  is  a  unit-coefficient  per 
diagonal.  Multiply  this  unit-coefficient  by  2  m  and  2  «,  or  the  number  of 
diagonals  in  the  longer  and  the  shorter  sides : 

.09625  x  4  diagonals  =    .385 
.09625  x  8       do.       =    .770 


The  products  are  the  coefficients  (c)  and  (d).  This  process  for  arriving  at 
the  stresses  in  the  diagonals  is  the  simplest  where  a  number  of  calculations 
are  to  be  made  for  one  girder. 


FRAMED   WORK. 


703 


Horizontal  Stress  in  the  Booms. — The  unit-stress  at  each  end  of  the  girder, 
adapting  formula  (i),  is  as  follows: — 


n'     cosine  a 
Unit-stress  at  support  a  =  -j-  W  g.ne  q 


.   (12) 


Do. 


support 


sine  a 


In  Fig.  301  the  number  of  bays  m'  and  n'  are  respectively  four  and  two 
bays,  to  the  left  and  to  the  right;  together,  six  bays  =  /.     Then, 

Unit-stress  at  support  a  -  -^-W  x  .  577  =  .  192  W; 


Do. 


support  £  = 

6 


W; 


and  the  unit-stress  at  b  is  equal  to  twice  the  unit-stress  at  a. 

The  stress  in  the  intermediate  bays,  between  each  support  and  the 
weight,  is  as  before  (formula  3), 

For  the  long  end  a  d,  (unit-stress  at  a)  x  N  ;  ............  (  14  ) 

For  the  short  end  b  d,  (       do.        at  b]  x  N;  ............  (15) 

in  which  N  is  the  order-number  of  the  bay,  on  either  side  of  the  weight, 
reckoned  from  the  point  of  support  at  the  same  side. 

The  successive  stresses  thus  calculated  are  given  in  the  following  tablet/, 
in  which  the  unit-stress  at  a  is  taken  as  i,  and  that  at  b  is  proportionally 
as  2. 

Tablet  /(Fig.  301). 


t 
In  compression  

CTC$ 

rs  r1" 

r"V 

c'r" 

c"c± 

In  tension  .         

ci  (i 

d  d1 

a"  a'" 

d"  d 

db 

tib 

The  horizontal  stresses  ) 
are  as  ..                     \ 

2 

3 

4 

5 

6 

7 

8 

6 

4 

2 

The  maximum  stress  is  in  the  bay  c'  /',  over  the  weight,  in  compression. 

The  tensile  stress  in  the  two  bays  a'"  d  and  d  b ',  contiguous  to  the 
weight,  are  as  7  and  6  respectively,  and  they  do  not  balance  each  other. 
But,  as  a  matter  of  fact,  there  is  a  balance  of  stress,  and  it  is  completed  by 
the  difference  of  the  horizontal  components  of  the  stresses  in  the  two  braces 
dc,  dc",  from  which  the  weight  is  directly  suspended,  being  respectively 
equal  to  the  unit-stresses  for  the  long  and  short  ends.  The  difference  of 
these  is  2  -  i  =  i,  or  one  unit-stress  in  the  direction  db\  and  (6+  i  =)y  is 
the  total  stress  in  the  bay  db ',  which  balances  the  opposite  stress,  also  7,  in 
the  bay  da"} 

THE  WARREN-GIRDER  UNIFORMLY  LOADED. 

A  uniform  load  on  a  Warren-girder  is,  in  fact,  a  load  equally  divided  and 
applied  to  the  apices  of  the  web,  as,  for  example,  in  Fig.  302,  in  which  the 

1  With  this  explanation,  it  may  be  said  with  propriety  that  the  sum  of  the  increments  of 
stress  on  the  one  side  of  the  weight  is  equal  to  the  sum  of  the  increments  of  stress  on  the 
other  side.  But,  abstractly,  it  is  an  erroneous  assumption. 


704 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


shorter  flange,  which  is  uppermost,  comprises  six  apices,  on  which  the  weights 
W,  W",  &c.,  are  placed. 

Stress  in  the  Braces. — The  stress  in  the  braces  caused  by  each  weight 
may  be  calculated  separately  in  the  manner  already  explained.     The  unit- 


W 

O 


w" 
O 


w* 
O 


AAAAA7\ 


O 


0 


0 


Fig.  302. — Warren-Girder,  uniformly  loaded — Longer  Flange  Undermost. 

coefficient  of  stress  for  one  diagonal  in  12,  as  in  Fig.  302,  is  1.155  +  12  = 
.09625.  The  stresses  caused  by  the  weight  W,  which  is  i  diagonal  from  a, 
and  1 1  diagonals  from  b,  are, 

In  the  brace  i 09625  x  n  diagonals  =  1.058  W,; 

In  the  braces  2  to  12 09625  x    i       do.       =    .096  W,. 

Calculating,  in  the  same  way,  the  stresses  caused  by  the  other  weights, 
the  constituents  of  stress  on  each  diagonal  are  obtained,  the  coefficients  of 
which  are  given  in  the  following  table,  No.  249,  in  which  compressive  stress 
is  distinguished  as  + ,  and  tensile  stress  as  -  .  The  resulting  coefficient  of 
stress  in  each  brace  is  given  in  the  second  last  column : — 


Table  No.  249. — COEFFICIENTS  OF  STRESS  IN  THE  BRACES  OF  A  WARREN- 
GIRDER  UNIFORMLY  LOADED,  WITH  THE  LONGER  FLANGE  UNDER- 
MOST. Fig.  302. 


Ratio  of 
Segments 
of  Girder. 

W 
I  tO  II. 

W" 
3  to  9. 

W" 
5  to  7. 

W4 

7  to  5. 

ws 

9  to  3. 

W6 

II  tO  I. 

Resultant 
Stress 
in  each 
Diagonal. 

Units  of 
Result- 
ant 
Stress. 

braces. 
I 
2 

3 

4 

1 

+  1.058  ) 
+  .096] 

-  .096 

+  .096 

-  .096 

+   .096 

+  .866 
-.866 
+  .866  ) 
+  .289  j 
-.289 
+  .289 

+  .674 
-.674 
+  .674 
-.674 
+  .674  ) 
+  48i( 

+  .481 
-.481 
+  .481 
-.481 
+  .481 
-.481 

ON  ON  ON  ON  ON  ON 
OO  OO  OO  OO  OO  OO 

r)  N  M  <s  n  N 

+  1  +  1  +  1 

+    .096 
-    .096 
+    .096 
-    .096 
+    .096 
-    .096 

in  parts  of 
W'. 
+  3.464 
-2.310 
+  2.310 
-I.I55 
+  I.I55 
±0.000 

6 

4 
4 

2 
2 
0 

8 
9 

10 

ii 

12 

-  .096 

+   .096 
-   .096 
+  .096 

-  .096 

+   .096 

-.289 
+  .289 
-.289 
+  .289 
-.289 
+  .289 

-.481 
+  .481 
-.481 
+  .481 
-.481 
+  .481 

+  .481  ) 
+  .674  J 
-.674 
+  .674 

-.674 
+  .674 

+  .289 
-.289 
+  .289) 

+  .866  ( 
-.866 
+  .£66 

+    .096 
-    .096 
+    .096 
-    .096 
+    .096) 
+  1.058  J 

±0.000 

+  1.155 
-1.155 

+  2.310 

-2.310 

+  3.464 

0 
2 

2 

4 
4 
6 

When  the  longer  flange  is  uppermost,  as  in  Fig.  303,  and  the  weights  are 
applied  to  the  upper  apices,  one  weight  is  supposed  to  be  divided  into 


FRAMED  WORK. 


70S 


halves,  of  which  one  half  is  placed  directly  over  each  support,  leaving 
5  weights  supported  by  the  girder.     The  stresses  in  the  braces,  of  which 


ww 

VWP 

Fig.  303. — Warren-Girder,  uniformly  loaded — Longer  Flange  Uppermost. 

there  are  12,  as  in  Fig.  302,  being  calculated  in  terms  of  the  unit-coefficient 
.09625,  the  constituent  and  resulting  stresses  are  given  in  table  No.  250: — 

Table  No.  250. — COEFFICIENTS  OF  STRESS  IN  THE  BRACES  OF  A  WARREN- 
GIRDER  UNIFORMLY  LOADED,  WITH  THE  LONGER  FLANGE  UPPER- 
MOST. Fig.  303. 


Ratio. 

W 

2  to  10. 

W" 
4  to  8. 

W" 
6  to  6. 

W4 

8  to  4. 

W6 

10  tO  2. 

Resultant 
Stress  in 
each  Brace. 

Units  of 
Resultant 
Stress. 

braces. 
T 
I 

2 

3 

4 

I 

-9^ 
+  .963 

+  .193) 
-•193 
+  .193 
-.193 

-.770 
+  .770 
-.770 
+  .770) 
+  •385! 
-.385 

-•577 
+  •577 
-•577 
+  .577 
-•577 
+  .577  | 

-.385 
+  .385 
-385 
+  .385 
-.385 
+  .385 

-.193 
+  .193 
-.193 
+  .193 
-.193 
+  •193 

in  parts  of 
W'. 
-2.888 

+  2.888 

-  1.732 
+  1.732 
-  .577 

+  .577 

5 
5 
3 
3 
i 
i 

7 
8 

9 

10 

ii 

12 

+  -I93 
-.193 
+  •193 
-.193 
+  -I93 
-.193 

+  .385 
-.385 
+  .385 
-385 
+  .385 
-.385 

+  .577J 
-.577 
+  •577 
-.577 
+  •577 
-.577 

-.385 
+  .385} 
+  .770) 
-.770 
+  .770 
-.770 

-.193 
+  .193 
-.193 

+  '963$ 
-.963 

+  .577 
-  .577 
+  1.732 
-1.732 

+  2.888 
-2.888 

i 
i 
3 
3 
5 
5 

The  unit  of  resultant  stress  in  the  braces  in  these  tables,  Nos.  249  and 
250,  is  taken  as  .577  W,  being  the  stress  caused  in  a  brace  by  a  half-weight 
(formula  (  7  ),  page  701);  and  the  respective  values  of  the  stress  are  ex- 
pressed in  units  of  that  value  in  the  last  column  of  the  tables. 

In  the  girder,  Fig.  302,  having  the  longer  flange  undermost,  the  stress  on 
the  middle  pair  of  braces  is  —  o,  and  on  the  successive  pairs  towards  the 
supports  each  way,  the  stress  increases  in  arithmetical  progression,  thus  :  — 


On  braces  ...............  i 

The  units  of  stress  are  6 


...4&5  ...6&7  ...  8&9  ...  lo&n  ...  12 
...     2      ...     o     ...     2      ...      4       ...  6 


In  the  girder,  Fig.  303,  having  the  longer  flange  uppermost,  the  stress  in 
the  braces  increases  also  in  arithmetical  progression,  but  by  a  different 
distribution,  being  as  i  in  the  two  middle  pairs,  thus  :  — 

On  braces  ...............  I&2...3&4...  5&6...  7&8...9&io...  n&i2 

The  units  of  stress  are     5      ...     3     ...     i      ...     i      ...     3        ...       5 

45 


706  STRENGTH   OF  ELEMENTARY  CONSTRUCTIONS. 

The  maximum  stress  in  the  braces  exists  at  the  extremities,  and  amounts 
to  6  units  in  the  first  girder,  and  5  units  in  the  second ;  these  are  the  stresses 
due  to  half  the  load  on  each  girder,  or  6  half-weights  and  5  half-weights 
respectively.  The  stress  in  any  intermediate  brace  is  that  due  to  the  num- 
ber of  half-weights  between  it  and  the  centre  of  the  girder.  Putting  n" 
equal  to  the  number  of  half-weights  between  the  brace  and  the  centre  of 
the  Warren-girder, 

Stress  in  a  given  brace  =  . 577x2  #*W 

-1.155  «"w/  (16) 

In  its  general  form,  for  any  angle  of  brace,  made  with  the  flange,  the  formula 
is  a  modification  of  formula  (  6  ),  page  701 : — 

n  "  W 

Stress  in  a  given  brace  =  — (  17  ) 

sine  a 

The  braces  which  meet  at  an  unloaded  apex  are  equally  stressed : — one 
by  compression,  the  other  by  tension. 

Stress  in  the  Flanges. — The  flanges  receive  increments  of  stress  at  each 
apex,  advancing  from  the  supports  to  the  centre,  where  the  total  stress  is  a 
maximum.  The  increment  of  stress  at  any  apex  is  equal  to  the  horizontal 
component  of  the  resultant  of  the  two  resultant  diagonal  stresses  at  the 
apex. 

Radius  :  cosine  a  :  :  resultant  diagonal  stress  :  horizontal  component, 
and  therefore, 

Horizontal  component  =  diagonal  stress  x  cosine  a; (  18) 

that  is  to  say,  each  unit  of  resultant  diagonal  stress,  or  .577  W,  causes  a  unit 
of  horizontal  stress,  or  .2885  W  (formula  (  2  ),  page  701). 

The  process  of  deducing  the  horizontal  stresses  from  the  diagonal 
stresses,  and  summing  them  up,  is  shown  in  the  following  analyses : — 


WARREN-GIRDER  UNIFORMLY  LOADED — ANALYSIS  OF  STRESS  IN  FLANGES. 
**  «ft  ?.«,:!  !*~:»clJ  \ 

Longer  Flange  Undermost,  Fig.  302 — Half  of  Girder. 

1.  No.  of  braces,  and  No.  of  bays,       123456 

2.  Units  of  resultant  stress  in  braces,      +6  -4       +4        -2  +2       o 
3, 4.  Resultant  stress  of  braces  )       —  6  +  4     4  +  4     4+2  2  +  2       2 

at  apices, for —  10          8           6  42 

5.  Horizontal      components     of  \ 

these,  or  increments  of  hori-  >      —  10          8           6  4          2 
zontal  stress  in  bays,  units,.,  j 

6.  Accumulated    increments    of)  Q  Q 

stress  in  bays,  units, }  24  28        ^ 

6  l6        24         3°  34        36 


FRAMED  WORK.  707 

Longer  Flange  Uppermost,  Fig.  303  —  Half  of  Girder. 

1.  No.  of  braces,  and  No.  of  bays,       12345  6 

2.  Units  of  resultant  stress  in  braces,      -5       +5       -3       +3       -i        +i 

3.  4.  Resultant  stress  of  braces  )       —     5  +  5    5  +  3    3  +  3    3+1     i  +  i 

at  apices,  .......................  J  or  —       10         8          6          4  2 

5.  Horizontal     components     of  \ 

these,  or  increments  of  hori-  >      —       10         8          6          4  2 

zontal  stress  in  bays,  units,..  ) 

6.  Accumulated    increments    of  I       _  jg  ^ 

stress  in  bays,  units,  ..........  j 

7.  Total  horizontal  stress  in  bays,  ) 

units,  ............................  j  -*  -^ 

The  resultant  stress  at  the  central  apex,  between  braces  6  and  7,  is,  in 
both  cases,  equal  to  o;  and  therefore  there  is  no  increment  of  horizontal 
stress  at  the  centre. 

The  increment  of  horizontal  stress  in  the  central  bay,  No.  6,  is,  in  both 
cases,  equal  to  2  units,  and  the  increments  increase  by  2  units,  from  bay  to 
bay,  up  to  bay  No.  2,  where  the  increment  amounts  to  10  units. 

So  that,  inversely,  the  increments  of  horizontal  stress  in  the  flange  diminish 
in  arithmetical  progression  as  they  approach  the  centre. 

Let  n  =  the  number  of  braces,  or  the  total  number  of  bays,  in  half  the 
girder; 

N  =  the  order-number  of  a  given  bay,  counted  from  the  end  of  the  girder, 
above  and  below; 

W'  =  the  weight  on  one  apex,  for  which  the  unit  of  horizontal  stress, 
transmitted  through  one  of  a  pair  of  diagonals,  is  .2885  W; 

The  equations,  for  the  horizontal  stress  in  the  given  bay,  based  upon  the 
foregoing  analysis,  are  as  follows  : 

i  st.  When  the  longer  flange  is  undermost:  — 
Horizontal  stress  in  a  given  bay  =  .2885  W'(«  +  (N  -  i)  (2«-N))  ......  (19) 

2d.  When  the  longer  flange  is  uppermost:  — 
Horizontal  stress  in  a  given  bay  =  .2885  W'((;/—  i)-f  (N  -  i)  (2*2  —  N))  (20) 

These  formulas  are  very  easy  of  application.  The  reasoning  by  which 
they  have  been  constructed  by  the  author  is  given  in  the  foot-note.1 

1  The  increments  of  horizontal  stress,  at  the  several  bays,  line  5,  in  the  "Analysis  of 
stress,"  are  in  arithmetical  progression,  having  the  common  difference,  2,  originating  at 
the  centre.  The  order-number  of  the  terms  of  the  progression,  counting  from  the  centre, 
is  expressed  by  (n  -  (N  -  i)  )  ;  and  (n  -  (N  -  i)  )  x  2,  is  the  value  of  the  increment  in  units. 
For  the  first  increment,  for  example,  in  bay  No.  2,  N  =  2,  and  ;/  —  6  ;  and  the  value  of 
the  increment  is  (6  -(2-  i)  )  x  2=10,  as  given  in  the  analysis.  If  N',  N",  N'",  &c., 
represent  for  the  moment  the  successive  order-numbers  of  the  bays  following  No.  2  bay, 
the  values  of  the  successive  accumulated  increments  of  stress,  line  6,  are  as  follows  :  — 

In  No.  2  bay,  (n  -  (N  -  i)  )  x  2 

No.  3    „     («  -  (N  -  i)  )  -t-  («  -  (N'  -  i)  )  x  2 
No.  4    „     («_(N 


and  so  on.     The  value  for  each  bay,  putting  N  for  the  order-number  of  the  bay,  and 
condensing  the  expression,  is  — 


708  STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 

ROLLING  LOAD  ON  THE  WARREN-GIRDER. 

Concentrated  Rolling  Load  on  the  Warren-Girder. — The  stress  caused  in 
successive  diagonals  by  a  passing  weight  is  alternately  tensile  and  com- 
pressive;  and  the  stress  is  a  maximum  when  the  load  is  on  the  apex. 

Distributed  Rolling  Load  on  the  Warren-Girder. — Suppose  that  the  rolling 
load  is  practically  of  uniform  distribution,  as  a  railway  train,  the  stresses  in 
the  diagonals  and  flanges  may  be  tabulated  and  analyzed,  as  exemplified  at 
pages  704  and  705.  If  the  train  extend  over  the  whole  of  the  girder,  the 
case  becomes  one  of  a  girder  uniformly  loaded.  The  stresses  in  partially- 
covered  girders  may  be  analyzed  in  like  manner,  and  the  changes  in  direction 
and  intensity  of  stress  determined. 

But  it  is  essential,  at  the  same  time,  that  the  stresses  caused  by  the 
permanent  weight  of  the  bridge  should  be  determined;  since  the  actual 
ultimate  stress  in  any  member  is  the  resultant  of  the  action  of  the  whole  of 
the  load,  both  permanent  and  passing.  The  maximum  stress  in  the  flanges 
takes  place  when  the  passing  load  covers  the  whole  of  the  girder. 

PARALLEL  LATTICE-GIRDER. 

Latticing  is  the  combination  of  two  or  more  systems  of  triangulation  in  the 
web  of  a  girder,  in  which  the  diagonals  cross  each  other.  The  number  of 
apices  is  proportionally  multiplied,  and  the  length  of  the  bays  is  propor- 
tionally shortened.  The  effect  is  that  the  weight  is  distributed  over  a 
greater  number  of  points  in  the  flange,  the  graduations  of  stress  in  the  flange 
are  reduced,  whilst  also  the  stress  in  the  diagonals  is  proportionally  reduced. 

There  is  a  special  advantage  in  lattice- 
work, in  affording  the  means  of  stiffen- 
ing the  braces  by  simple  connections 
at  the  intersections. 

If  the  diagonal  stresses  be  calcul- 
Fig.  3o4.-Paraiiei  Lattice-Girder.  ated,  in  the  first  instance,  as  for  a 

single  triangular  system,  let  them  be 

divided  by  the  number  of  systems  in  the  lattice ;  the  quotient  is  the  aliquot 
part  of  the  stress,  as  distributed  to  each  diagonal.  The  fundamental 
triangulation  is  shown  by  thick  lines,  Fig.  304;  and  in  this  instance,  where 
only  one  additional  system  is  interpolated,  the  stress  in  the  fundamental 
diagonals  is  reduced  to  a  half  of  what  they  would  sustain  if  they  stood 
alone. 

THE  PARALLEL  STRUT-GIRDER. 

In  the  parallel  strut-girder,  Fig.  305,  having  vertical  and  diagonal  bracing, 
supporting  a  single  weight,  W,  on  the  upper  flange  at  the  centre,  the  vertical 

(KX2x(N-i))-(Nx2x(N-i))+I+(N"l)x2x(N-i)  =  (N-i)x(2(tt-N)  +  N) 

To  this  is  to  be  prefixed  the  initial  horizontal  stress  at  bay  No.  I,  which  is  n  units,  or 
6  units,  for  Fig.  302,  and  (w-i)  units,  or  5  units,  for  Fig.  303.  Thence  the  formulas 
(19)  and  (20)  : — 

Horizontal  stress  in  a  given  bay  =  . 2885  Wi(«  +  (N- i)  (2«-N)), (19) 

when  the  longer  flange  is  undermost ;  and 

Horizontal  stress  in  a  given  bay  =  .2885  Wi(«-  l)  +  (N-  i)  (2  «-N) ),....  (20) 
when  the  longer  flange  is  uppermost. 


xxxxxxx 


FRAMED   WORK. 


709 


brace  or  strut  cd  receives  and  supports  the  whole  of  the  weight;  and  the 
compressive  stress  is  divided  and  transmitted  by  tension  through  the 
diagonals  dc  and  dc",  according  to  the  parallelogram  of  forces,  of  which 


Figs.  305  and  306. — Parallel  Strut-Girders. 

the  diagonal  de,  equal  to  twice  dc,  the  depth  of  the  girder,  represents  the 
weight.  The  depth  dc  represents  half  the  weight,  and  the  tensional  stress 
in  each  of  the  diagonals  d  c'  and  d  c"  is  represented  in  magnitude  and 
direction  by  the  diagonals. 

The  tensions  of  these  diagonals  balance  each  other  horizontally  at  their 
intersection  at  the  lower  flange  at  d,  and  thus  they  do  not  throw  any 
horizontal  stress  on  the  lower  flange. 

STRUT-GIRDER  WITH  A  CONCENTRATED  MOVING  WEIGHT. 

When  the  load  moves  over  the  girder,  each  strut  requires  to  be  braced 
by  a  pair  of  diagonals  intersecting  at  the  foot  of  the  strut,  in  the  same  way 
as  the  strut  cd,  under  the  fixed  weight  in  Figs.  305  and  306,  is  braced  by  the 
diagonals  dc  and  dc".  The  result  is  a  system  of  cross-bracing,  or  counter- 
bracing,  by  crossed  ties,  as  in  Fig.  306.  The  extra  braces  at  the  outer  struts 
ac',  be"  (Fig.  305),  are  not  necessary,  but  they  are  introduced  to  complete 
the  design.  The  maximum  stress  is  imposed  on  each  strut  when  the  weight 
passes  over  its  summit. 

If  the  weight  move  on  the  lower  flange,  the  maximum  stress  on  a  given 
strut  is  imposed  when  the  weight  passes  the  lower  end  of  the  next  strut  on 
the  side  of  the  more  distant  support;  and  the  maximum  stress  on  any  strut, 
by  the  lower  flange,  never  exceeds  half  the  weight.  (Fig.  305.) 

The  tension  in  the  diagonal  d  c  is  resolved  into  compressive  stress  in 
the  upper  bay  c  c  and  the  strut  c  a.  The  compressive  stress  in  the  strut 
c  a  is  resolved  into  tensile  stress  in  the  lower  bay  a  d  and  the  outer 
diagonal  a  c" \  and  lastly,  the  stress  in  the  outer  diagonal  a! c"  is  resolved 
into  compressive  stress  in  the  outer  bay  c"  c  and  in  the  strut  c" a,  of  which 
the  former  is  transmitted  to  the  middle  bay  c  c. 

A  similar  action  takes  place  in  the  other  half  of  the  girder,  and  the 
horizontal  stresses  in  one  half  balance  those  in  the  other. 

The  vertical  stress  in  the  lateral  struts  is  obviously,  by  transmission,  equal 
to  YZ  W.  That  is,  the  stress  in  the  lateral  struts  is  only  half  the  stress  on 
the  central  strut,  which  supports  the  whole  of  the  weight.  The  compressive 
stress  in  the  outermost  struts,  or  half  of  the  weight,  is  received  and  resisted 
by  the  supports  at  a  and  b.  The  outer  bays  of  the  lower  flange,  a  a'  and 
b'  b,  are  not  subjected  to  any  transmitted  stress. 

The  horizontal  stresses  in  the  bays  of  the  upper  and  lower  flanges  are, 
then,  in  the  following  ratios : — 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


No.  of  bay i,         2, 

Compressive  stress  in  upper  flange,  as i,         2, 

Tensile  stress  in  lower  flange,  as o,         i, 


3>         4 

2,  I 

I,  O 


Trigonometrically,  the  weight  is  supposed  to  be  divided  into  two  halves, 
each  of  which  is  represented  by  the  depth  cd,  and  causes  the  diagonal 
stress  on  either  side.  Then, 

Stress  in  strut  cd  \  stress  in  diagonal  dc    :  :  sine  a  :  radius  or  i; 

a  being  the  angle  a  d  c'  formed  by  the  diagonal  with  the  lower  flange. 
Therefore, 

Stress  in  each  diagonal  R  Stress  in  Stmt      T/  W 


or 


Stress  in  each  diagonal  = 


sine  a 
W 


sine  a 


2  sine  a 


(21) 


When  the  distance  apart  of  the  struts  is  equal  to  the  depth  of  the  girder, 
the  angle  a  =  45°,  and  sine  a  =.707;  then, 

W 

Stress  in  every  diagonal  = =  .707  W. 

In  the  upper  flange,  the  stress  caused  by  each  diagonal  being  as  the  half 
weight  dc  to  the  length  of  a  bay  cc" ,  or  as  sine  a  to  cosine  a;  then 


Stress  caused  in  each  upper  bay  = 


cosine  a 


(22) 


sine  a 
When  the  angle  a  =  45°,  cosine  a  -  sine  a;  and 

Stress  caused  in  each  upper  bay  =  y2  W,  and 
Stress  accumulated  in  middle  upper  bay  =  W. 

In  the  lower  flange,  the  middle  lower  bays  are  in  tension  =  ^  W,  due  to 
the  thrust  of  the  struts  at  a  and  V . 

If  the  extreme  bays  of  the  lower  flange  be  removed,  and  the  girder  be 
supported  direct  at  the  ends  of  the  tipper  flange,  as  in  Fig.  307,  the  stresses 


Fig.  307. — Parallel  Strut-Girder. — Lower  end 
bays  removed. 


Fig.  308.— Parallel  Strut-Girder,  loaded 
at  an  intermediate  point. 


in  the  structure  remain  unaltered,  since  there  is  no  horizontal  stress  in  the 
extreme  bays  of  the  parallel  girder,  Fig.  305.  It  was  seen  that  the  func- 
tion of  the  end  struts  was  only  to  support  the  girder. 

STRUT-GIRDER  LOADED  AT  AN  INTERMEDIATE  POINT,  OFF  THE  CENTRE. 

The  strut-girder,  Fig.  308,  four  bays  in  length,  is  loaded  at  cd,  one  bay 
from  the  support  b,  and  three  bays  from  the  support  a.     Let  /=the  total 


FRAMED   WORK. 

number  of  bays,  and  m  and  n  =  the  numbers  of  bays  to  the  left  and  to  the 
right  of  the  weight.  The  loads  on  the  supports  at  a  and  b,  as  well  as  the 
compressive  stress  on  the  struts  to  the  right  and  left,  are  respectively  — 


(23) 


being  in  the  inverse  ratio  of  the  distances  of  the  weight  from  the  supports; 
or  they  are,  in  the  present  example  — 

Stress  in  the  struts  of  the  longer  side  ................   j^  W, 

Do.  do.  shorter  side  ...............  %  W. 

The  stresses  in  the  diagonals  are  in  the  same  proportion,  thus  :  — 

Stress  in  the  diagonals  of  the  longer  side  =  -^-  -.  -  ................  (24) 

/  sine  a 

Do.  do.  shorter  side  =  ^-  -.  -  .................  (25) 

?*  I   sine  a 

If  the  angle  a  =  45°,  then  sine  a  =  .707,  and  the  stresses  are, 

In  the  diagonals  of  the  longer  side  .........   }i  x  -  =  -3535  W, 

.707 

Do.  do.         shorter  side...        .   3^  x  -  =  1.060=;  W. 

.707 

The  unit  or  increment  of  horizontal  stress  in  the  bays  of  the  upper  and 
lower  flanges  is  as  follows  :  — 

Unit  of  stress  in  the  bays  of  the  longer  side,    -r  W  —  :  —  —  ......  (  26  ) 

Do.  do.  shorter  side,  ^W  cosine  a  .....  (  27  ) 

/          sine  a 

When  the  angle  a  =  45°,  cosme  q  =  I?  and  the  unit  of  stress  is, 
sine  a 

In  the  bays  of  the  longer  side  ..............................  %  W, 

Do.         do.         shorter  side  ............................   &:Wfr 

the  accumulated  stress  in  the  several  bays  is,  by  formula  (3),  page  701, 

For  the  longer  side  ............  —  WcosmeaxN  ...............  (28) 

/         sine  a 

For  the  shorter  side  ......  —  W  cosme  q  xN...  ..  (29) 

/         sine  a 

in  which  N  is  the  order-number  of  the  bay,  in  the  upper  flange,  on  either 
side  of  the  weight,  reckoned  from  the  point  of  support.  For  the  lower 
flange,  N  is  the  order-number  less  i,  seeing  that,  as  before  explained,  there 
is  no  transmitted  horizontal  stress  in  the  lower  bays  situated  next  the  points 
of  support. 


712 


STRENGTH   OF  ELEMENTARY  CONSTRUCTIONS. 


The  resultant  stresses  in  the  several  bays  are,  in  the  example,  Fig.  308, 
relatively,  as  follows: — 

No.  of  bay i,         2,         3,         4 

Compressive  stress  in  upper  flange  as i,         2,         3,         3 

Tensile  stress  in  lower  flange  as o,         i,         2,         o 

Here  it  is  apparent  that,  whilst  the  resultant  stresses  in  the  bays  3  and  4 
of  the  upper  flange  balance  each  other,  there  is  no  tensile  stress  in  No.  4 
bay  of  the  lower  flange  to  balance  the  stress  in  No.  3.  But  the  balance  is 
supplied  by  the  difference  of  the  horizontal  components  of  the  diagonal 
stresses  which  meet  below  the  weight  at  d. 

STRUT-GIRDER  UNIFORMLY  LOADED. 

The  general  conditions  of  stress  in  the  strut-girder  uniformly  loaded,  as 
in  Fig.  309,  are  similar  to  those  in  the  Warren-girder,  as  elucidated,  page  704. 


Fig.  309. — Strut-Girder  uniformly  loaded. 

Stress  in  the  Struts. — The  stress  is  calculated  by  an  adaptation  of  formula 
(17),  page  706,  in  which  sine  a  becomes  =  i,  seeing  that  the  angle  of  the 
strut  with  the  flange  is  a  right  angle.  The  formula  becomes, 

Stress  in  a  given  strut  -n"  W,  (  3°  ) 

in  which  n"  —  the  number  of  weights  between  the  strut  and  the  centre  of 
the  girder. 

Stress  in  the  Diagonals. — This  stress  is  found  by  formula  (17) — 


Stress  in  a  given  diagonal  =  -?• 


sme  a 


(so 


For  illustration,  the  diagonals  i  and  8  carry  the  seven  weights  W  to  W7 
suspended  between  them;  and  each  sustains  half  the  number,  or  3^  W, 

and  the  stress  in  each  is  3%  L, Similarly,  the  diagonals  2  and  7  carry 

sine  a 
the  five  weights  W"  to  W6  between  them,  each  sustaining  2^  weights,  or 

2^2  W';  and  the  stress  in  each  is  2*4 •  The  diagonals  3  and  6  carry  the 

sine  a 

weights  W"'  W4  W5  between  them,  each  supporting  i^  W',  and  the  stress  is 
Lastly,  the  diagonals  4  and  5  carry  the  weight  W4  between  them, 


sine  a 
each  supporting 


W,,  and  the  stress  is 


W 


sine  a 


No  diagonal  stress  is 


transmitted  across  the  centre  of  the  girder:  in  this  respect  the  strut-girder 
differs  from  the  Warren-girder. 


FRAMED  WORK.  713 

Stress  in  the  Flanges.  —  The  stress  in  the  flanges  increases  by  diminishing 
increments  at  each  apex  towards  the  centre,  where  it  is  a  maximum.  The 
increments  consist  of  the  horizontal  components  of  the  diagonal  stresses 
developed  at  each  apex.  The  accumulated  horizontal  stress  in  each  bay 
is  expressed  by  the  following  formulas,  which  have  been  constructed  in  a 
manner  similar  to  that  by  which  formulas  (19)  and  (20)  were  constructed:  — 

Let   n  =  the  number  of  bays  in  the  length  of  half  the  girder  ; 

N  =  the  order-number  of  a  given  bay,  in  the  upper  or  the  lower  flange, 
a  -  the  angle  between  the  diagonal  and  the  flange, 
W  =  the  weight  on  one  strut,  for  which  the  unit  of  horizontal  stress  in 

the  flange,  transmitted  through  the  next  diagonal,  is  W  cosme  a> 

sine  a 

For  the  horizontal  stress  in  a  given  bay  :  — 

i  st.  In  the  Upper  Flange:  — 
Stress  in  a  given  bay  =  W'-°,smeaN  («  -  -)  ...............   (32) 

2d.  In  the  Lower  Flange:  — 
Stress  in  a  given  bay  =  W  COsme  a  (N  -  i)  x  (n  -  *Lzl)...  (  33  ) 

When  the  distance  apart  of  the  struts  is  equal  to  the  depth  of  the  girder, 
w,  cosine  <*_W' 
sine  o 

The  gradation  of  stress  in  the  flanges  may  be  given  for  Fig.  309,  con- 
taining 8  bays. 


No.  of  diagonal  and  bays,  .................. 

Increments    of   stress    in    diagonals,  ] 


sine  a 
Increments  of  stress  in  bays  of  upper  j 

flange  (unit  =  W'cosine  °) (^        ^        '#          ^  units' 

sine  a 

Accumulated  stress  in  do.  do. , 3  yz        6  7  y2        8      units. 

Increments  of  stress  in  bays  of  lower  (  T/  /          l7       -f 

a  V      O  ^  7z  2  72  I  72   UnilS. 

flange, ; j  /• 

Accumulated  stress  in  do.  do. , o  3^        6  7^  units. 

STRUT-GIRDER  TRAVERSED  BY  A  LOAD  UNIFORMLY  DISTRIBUTED. 

The  struts  require  to  be  counter-braced,  and  the  stresses  are  calculated 
as  in  the  immediately  preceding  case. 

ROOFS. 

i.  The  weight  of  and  load  upon  a  roof  are  taken  as  uniformly  distri- 
buted over  the  surface  of  the  roof;  and  the  total  weight  on  each  pair 
of  rafters,  couple,  or  truss,  is  equal  to  the  sum  of  the  weight  of  the  truss 
itself,  and  of  so  much  of  the  roof  as  is  carried  between  two  trusses. 


STRENGTH   OF   ELEMENTARY  CONSTRUCTIONS. 


In  the  triangular  roof-truss,  a  be,  Fig.  310,  the  total  weight,  W,  may  be 
considered  as  localized  at  the  supports  a  and  b,  and  the  ridge  c\  a  fourth 


» 


Figs.  310  and  311.  —  Triangular  Roof-Trusses. 

,  and  a  half  at  c.     The  tension  in  a  b  is  that  due  to  the 


each  at  a  and 

weight,  YZ  W,  at  <r,  which  is,  by  equation  ( A ),  page  699, 

which  /  is  the  span,  and  d  the  depth ;  or 

W/ 


W/-=-4</,  in 


(34) 


Or  the  tension  in  the  tie-member  a  b,  under  a  uniformly  distributed  weight, 
is  equal  to  the  product  of  the  weight  by  the  span,  divided  by  8  times  the  rise. 

The  horizontal  thrust  at  the  ridge  c,  is  equal  to  the  tension  in  the  hori- 
zontal tie. 

The  rafters  ca  and  cb  are  subject  to  two  stresses: — ist.  Compressive 
thrust,  as  pillars,  by  the  weight;  the  thrust  is  cumulative,  beginning  as 
nothing  at  the  apex,  and  ending  at  the  maximum  for  the  whole  weight  at  the 

abutments  a  and  b,  where  it  is  equal  to  (^  W  x  _).    2d.  Transverse  stress 

c  d 

from  the  weight,  ^  Wx  uniformly  distributed,  reduced  in  the  ratio  of  the 
slant  height  a  c  to  the  half-span  a  d;  the  moment  of  which  is  equal  to, 

W/ 
4  x  (a  c] 

2.  When  the  horizontal  tie  is  applied  at  any  higher  level,  a'  b' ,  Fig.  311, 
the  tension  in  it  is  inversely  as  the  depth  c  d,  according  to  the  expression 
( 34 ).     In  addition  to  the  stresses  in  the  rafters,  already  noticed  in  the 
previous  case,  there  are  compressive  and  transverse  stresses  excited  by  the 
pull  of  the  tie  a'  b'. 

3.  In  the  A-truss  roof,  Fig.  312,  the  stresses  are  mixed.     Let  the  span  be 
40  feet,  the  rise  10  feet,  and  the  depth  cd  8  feet.     The  rafters  ac  and  be 


Fig.  312.— A-Truss  Roof. 

are  22.5  feet  long,  the  struts  F  are  3.33  feet  long,  and  the  tension  bars  C 
and  D  11.75  ^eet  l°ng'  The  weight  on  the  couple  is  8  tons,  uniformly 
distributed,  of  which  4  tons  is  supported  on  each  rafter,  say  i  ton  at  a  over 
the  abutment,  2  tons  at  F,  and  i  ton  at  the  ridge  c. 


FRAMED   WORK.  715 

The  pressure,  2  tons  on  F,  being  vertical,  is  resolved,  as  indicated  by 
diagram,  into  1.8  tons  stress  on  F,  and  .875  tons  on  A.  The  stress  on  F 
is  resolved  into  3.18  tons  in  C  and  in  D,  tie-rods. 

The  tension  in  E  is  by  formula  (34),  8  tons  *  f>  feet  -  5  tons;  and  it  is 

O  X  O   ICCl 

resolved  into  4^  tons  in  C,  and  .875  tons  in  F.     This  tension  in  F  is 
resolved  into  1.54  tons  in  C  and  in  D. 

The  total  stresses  in  the  three  tension-rods  C,  D,  and  E  are,  then,  as 
follows : — 

Totals. 

In  C,  stress  through  F  by  direct  weight  of  roof,..  3.18  tons. 

stress  by  tensile  force  of  E, 4.75     „ 

stress  from  F  by  do.  do., 1.54     „       9.47  tons. 

In  D,  stress  through  F  by  direct  weight  of  roof,..  3.18     „ 

stress  from  F  by  tensile  force  of  E, 1.54    „      4.72     „ 


In  E,  stress, 5.00     „ 

So  much  by  way  of  analysis.  But  Mr.  Stoney  shows  a  method  of  deducing 
the  stresses,  by  starting  from  the  stress  on  the  abutment  and  working  thence 
towards  the  centre.  Referring  to  Fig.  312,  and  adopting  the  same  data  as 
above,  the  reaction  of  the  left  abutment  is  4  tons,  of  which  i  ton  is  directly 
balanced  by  the  weight  W  concentrated  there,  leaving  3  tons  to  be  resolved 
in  the  directions  of  A  and  C,  into  10.35  tons  and  9.47  tons  respectively. 
The  pressure  of  W",  2  tons,  is  resolved  into  1.8  tons  on  F  and  .875  tons 
on  A;  and  (10.35  -  -$75  = )  9-475  tons  is  tne  thrust  in  B.  At  a,  the  stresses 
in  C  and  F,  which  are  known,  are  resolved  by  the  intermediate  substitution 
of  their  resultant  into  the  stress  4.72  tons  in  D,  and  5  tons  in  E. 

4.  In  Fig.  313,  the  middles  of  the  rafters  are  strutted  by  struts  c'  d  and 
c"  d,  meeting  at  d  in  the  horizontal  tie-bar,  and  tied  to  the  ridge  by  the  ver- 
tical rod  c  d.  The  weight  of  the  roof  is  localized  at  #,  c ',  c,  c",  and  b,  in 
the  proportions  ^  W,  %  W,  ^  W,  ^  W, 
and  y%  W.  In  the  truss  ac'd,  the  weight 
on  c'  is  equally  sustained  by  the  limbs  c'  a 
and  c'  d,  y%  W  being  borne  by  the  abut- 
ment, and  J$  W  being  transmitted  through 
the  tie-rod  cd  to  the  ridge  c.  As  ^  W  is 
also  transmitted  from  c",  in  the  right  hand  Fig  3I3._A.Truss  Roof. 

rafter,  the  total  weight  at  the  ridge  is  (%  + 

^4+^)W=^W.  This  is  just  what  the  ridge  would  have  borne,  without 
the  intervention  of  the  struts;  and  the  function  of  the  struts  is  chiefly  to 
assist  the  rafters  in  resisting  transverse  stress.  The  pull  in  the  vertical 
tie-rod  c  d  is  (^  +  y%  = )  ^  W. 

To  find  the  stresses  in  the  rafters  and  the  horizontal  member  a  b,  Dr. 
Rankine  distinguishes  the  main  truss  a  cb  and  the  secondary  trusses  ac'  d 
and  dc"  b.  The  tension  in  ab  is  the  sum  of  the  tensions  due  to  the  first 
and  second  trusses ;  the  thrust  in  a  c',  likewise,  is  the  sum  of  the  thrusts, 
and  that  in  c'  c  is  the  thrust  of  the  first  truss  only.  Suppose  the  span  /=  20 


STRENGTH   OF  ELEMENTARY  CONSTRUCTIONS. 

feet ;  the  pull  in  a  d  by  the  first  truss  is  by  formula 
by  the  second  truss,  ^ *  '*    =  */6  W.      The 

sum  of  these  pulls  is  (j^  +  x/6  = )  yz  W,  the  resultant  tension  in  a  b  or  a  d. 
The  thrust  in  a  c  by  the  first  truss,  is  to  the  relative  tension  in  a  d,  as  a  c 

to  a  d.     The  length  of  a  c  =  \/(a  d)2  +  (cd)*  =12.5  feet,  and  the  thrust  is, 

y§  W  x  —5  =.4! 7  W.     In  the  second  truss,  the  thrust  in  a  c'  is  the  same 
10 


fraction  of  the  tension  in  ad,  due  to  the  truss;  or  it  is  T/e  Wx 


12.5  = 
10 

.242  W.    The  sum  of  the  first  and  second  thrusts,  or  .658  W,  is  the  resultant 
thrust  in  a  c  '  . 

The  value  of  the  thrust  in  a  c,  by  the  first  truss,  may  be  found  in  terms 
of  the  relative  tensions  in  a  d  and  c  d;  for  it  is  equal  to 

^/tension  in  (0£)2  +  tension  in  (cd)*=  \J  (YzY  +  (Y%Y  =  -4J7;  as  nas 
already  been  found. 

5.  In  Fig.  314,  each  rafter  is  divided  into  three  equal  parts,  which  are 
supported  by  two  struts,  c'  d  and  c'"  d'  for  the  left-hand  rafter,  and  c4  d" 
and  c"  dim  the  right-hand  rafter;  suspended  by  vertical  rods  c'  d\  c  d,  and 
c"  d"  ';  united  by  the  main  tie-rod  a  b.  The  total  weight  W,  uniformly  distri- 
buted, is  localized  at  a  c'"  c'  c  c"  c  b 


in  the  proportions, 


c  c  c  c 

'/6  W,    '/6  W,    «/6  W,    «/e 


«/6  W,    '/I2  W. 


Fig.  314.— Trussed  Roof. 


Three  trusses  are  recognized  here:  the  first  truss  a  c  b,  the  second  a  c  d,  and 
the  third  a  c'"  d'.  Half  the  stress  at  c'"9  the  summit  of  the  isosceles  or  third 
truss,  is  transmitted  by  the  vertical  rod  c'  d'  to  c  ',  where  the  stress  is  increased 
to  (  J/6  +  Yi2  =  )  }i  W.  This  load  is  transmitted  to  a  and  d'  at  the  base 
of  the  truss,  in  the  inverse  ratio  of  the  segments  a  d'  and  d'  d;  that  is,  two 
thirds,  or  (^  x  2^  =  )  i/6  W,  is  transmitted  through  the  strut  cd  and  rod  cd, 
to  c.  An  equal  quantity,  x/6  W,  is  transmitted  from  the  right-hand  rafter,  and 
the  sum  x/3  W  added  to  T/6  W,  makes  ^  W,  the  resultant  load  at  the  ridge. 
Suppose  the  span  /=  60  feet  and  the  rise  d  =  15  feet,  the  pull  in  a  b  or  a  d 

due  to  the  first  truss,  is  by  formula  (A),/^         =  ^  W;  by  the  second  truss, 

4" 
the  pull  is  8/9  ths  of  what  it  would  be  if  the  truss  were  isosceles,  or  it  is 

*wx  %  =  w/  the  third  'A  w  x  y3  / 


24  </ 
=  '/6  W.     The  sum  of  the  three  pulls,  or 


4  x 


d 


*/6  +  '/e  =)  2%4  w> 


FRAMED  WORK.  717 

is  the  resultant  stress  in  a  d'  and  d"  b;  and  the  sum  of  the  first  and  second 
pulls  is  the  resultant  stress  in  d'  d",  or  fe  W. 

The  pull  in  the  main  tie-rod  a  fr,  due  to  the  second  strut,  was  said  to  be 
8/9  ths  of  what  it  would  have  been  for  an  isosceles  truss.  In  general 
terms,  the  fraction  of  what  it  would  be  for  an  isosceles  truss  of  the  same 
height  and  length  is  the  ratio  of  the  product  of  the  segments  into  which  the 
tie-rod  or  base  of  the  truss  is  divided  by  the  vertical  rod  from  its  apex, 
to  the  square  of  half  the  base.  In  this  instance  the  base  is  divided  into  */$ 
and  I/3  ,  and  2/3  x  I/3  -  2/9  ;  also  ^  x  y2  =  %  or  2/8  j  and  the  ratio  of 
2/9  to  2/s  is  8  to  9,  or  8/9  ths.1 

The  thrusts  in  the  rafters  may  be  found  by  the  method  already  applied, 
in  the  previous  case;  and  the  same  general  process  is  applicable  to  roofs 
of  more  extensive  construction.  Professor  Rankine  gives  general  equations, 
for  the  stresses  in  roofs  of  the  strut-and-rod  class,  Fig.  314;  and  Mr.  Stoney 
shows  how  the  stresses  may  be  found  in  employing  the  parallelogram  of 
forces,  from  the  pressure  on  an  abutment.2 

By  the  application  of  the  principle  of  the  parallelogram  of  forces,  the 
stresses  in  crescent  and  other  forms  of  girders  and  roofs  may  be  determined. 

1  See  page  508,  at  top. 

2  Civil  Engineering,  page  472.     The  Theory  of  Strains,  page  159. 


WORK,   OR  LABOUR. 


UNITS   OF   WORK   OR   LABOUR. 

The  fundamental  units  of  work — the  foot-pound  and  the  kilogrammetre — 
have  been  denned  at  page  312;  and  their  relations  with  those  of  horse- 
power have  been  stated  at  page  158. 

Horse-power. — Horse-power  is  a  measure  of  the  rate  at  which  work  is 
performed.  One  horse-power  is  the  expression  of  33,000  foot-pounds  of 
work  done  per  minute,  or  550  foot-pounds  of  work  done  per  second.  It 
is  nearly  identical  with  the  French  horse-power  (cheval-vapeur,  or  cheval\ 
which  is  equal  to  75  kilogrammetres  of  work  done  per  second.  As  a  kilo- 
grammetre is  equal  to  7.233  foot-pounds,  the  "  c/iei>a/"  is  equal  to  (75  x 
7.233  =  )  542.5  foot-pounds  of  work  per  second,  which  is  1,37  per  cent, 
less  than  the  English  measure  of  a  horse-power. 

Mechanical  equivalent  of  heat. — The  values  of  the  mechanical  equivalent 
of  heat  in  English  and  in  French  measures  are  denned  at  page  332,  and 
their  relations  are  stated  at  page  159.  An  English  unit  of  heat  is  the 
quantity  which  is  required  to  raise  the  temperature  of  water  at  or  near 
39.1°  F.,  the  temperature  of  its  maximum  density,  through  i°  F. ;  and  its 
mechanical  value  or  equivalent  is  equal  to  772  foot-pounds.  One  horse- 
power is  therefore  equivalent  to  (33,000^772  =  )  42^  heat-units  per 
minute. 

A  French  unit  of  heat  is  equal  to  that  which  is  required  to  raise  the 
temperature  of  i  kilogramme  of  water  through  i°C. ;  and  its  mechanical 
equivalent  is  424  kilogrammetres  =  3063. 5  foot-pounds. 

LABOUR  OF  MEN. 

Mr.  Smeaton  concluded  that  the  power  of  an  ordinary  labourer  at  ordi- 
nary work  was  equivalent  generally  to  work  done  at  the  rate  of  3762  foot- 
pounds per  minute.  But,  according  to  a  particular  estimate  made  by  him 
for  pumping  up  water  4  feet  high,  by  good  English  labourers,  their  power 
was  equivalent  to  3904  foot-pounds  of  work  per  minute;  and  this  he 
estimated  as  twice  that  of  ordinary  persons  "  promiscuously  picked  up." 

Mr.  John  Walker  found  that  the  force  exerted  by  an  ordinary  labourer  in 
raising  weights  for  driving  piles,  average  daily  work,  was  12  Ibs.  In 
working  daily  at  a  winch  or  a  crane-handle,  the  average  force  was  14  Ibs. 
moving  at  the  rate  of  220  feet  per  minute,  equivalent  to  (14  x  220  = )  3080 
foot-pounds  per  minute. 

Mr.  Glynn  says  that  a  man  may  exert  a  force  of  25  Ibs.  at  the  handle  of 
a  crane  for  short  periods;  but  that,  for  continuous  work,  a  force  of  15  Ibs.  is 
all  that  should  be  assumed,  moving  through  220  feet  per  minute.  The 
power  of  a  man  would  thence  be  (15  x  220  =  )  3300  foot-pounds  per  minute. 


LABOUR   OF   MEN. 


719 


Mr.  G.  B.  Bruce  states  that,  in  average  work  at  a  pile-driver,  a  labourer 
exerts  a  force  of  16  Ibs.,  plus  the  resistance  of  the  gearing,  at  a  velocity  of 
270  feet  per  minute,  for  10  hours  a  day,  making  one  blow  every  four  minutes. 
The  power  is  (16  x  270  =  )  4320  foot-pounds  per  minute. 

Mr.  Joshua  Field,  in  1826,  tested  the  performances  of  men  at  a  crane 
of  rough  construction,  in  ordinary  use.  The  barrel  was  n^  inches  in 
diameter  to  the  centre  of  the  chain;  and  the  driving -gearing  consisted  of 
two  pinions  and  two  wheels,  geared  successively,  with  an  1 8-inch  handle. 
The  ratio  of  the  power  to  the  weight  was  i  to  105.  The  loads  were  thus 
so  proportioned  as  to  be  reduced  successively  to  from  10  to  35  Ibs.  at  the 
handle ;  frictional  resistance  being  additional.  The  load  was  raised  through 
a  height  of  16^  feet  in  each  experiment.  The  results  were  as  follows: — 

Table  No.  251. — POWER  OF  MEN  AT  A  CRANE. 


Statical 

Equiva- 

No. 
of 

Resist- 

T   s\r*A 

Time 

lent 
Power  in 

Ex- 
peri- 

ance of 
the  Load 
at  the 

-LiOao. 
Raised. 

in 
Raising. 

Foot- 
pounds 

REMARKS. 

ment. 

Handle. 

per 

Minute. 

Ibs. 

Ibs. 

minutes. 

ft.  -Ibs. 

I 

10 

1050 

i-5 

11,550 

Easily  done  by  a  stout  Englishman. 

2 

15 

1575 

2.25 

11,505 

Tolerably  easily  by  the  same  man. 

3 

20 

2100 

2.0 

17,325 

Not  easily  by  a  sturdy  Irishman. 

4 

25 

2625 

2-5 

17,329 

With  difficulty  by  a  stout  Englishman. 

5 

30 

3150 

2.5 

20,790 

Do.           by  a  iTondon  man. 

6 

35 

3675 

2.2 

27,562 

With  the  utmost  difficulty  by  a  tall  Irishman. 

7 

55 

55 

2-5 

24,255 

Do.            do.             by  a  London  man. 

8 

55 

)) 

2.83 

21,427 

With  extreme  labour  by  a  tall  Irishman. 

9 

10 

55 
,5 

5, 
J> 

3-0 
4.05 

20,212 
15,134 

Withvery  great  exertion  by  a  sturdy  I  rishman. 
With  the  utmost  exertion  by  a  Welshman. 

ii 

„ 

J5 

— 

Given  up  at  this  time  by  an  Irishman. 

Mr.  Field  states  that  No.  4  gave  a  near  approximation  to  the  maximum 
power  of  a  man  for  2  ^  minutes.  In  all  the  succeeding  trials,  the  men  were 
so  much  exhausted  as  to  be  unable  to  let  down  the  load. 

It  would  appear  from  this  table  that  the  maximum  net  pressure  at  the 
handle  for  constant  working  would  not  exceed  15  Ibs.,  exclusive  of  frictional 
resistance.  The  loads  were  only  from  y2  to  i  ^  tons, — much  below  the 
capacity  of  the  crane;  and  the  frictional  resistance  was  disproportionally 
great  for  the  work  of  one  man.  The  author  has  found  that  when  cranes 
were  worked  up  to  their  capacity,  the  men  could  without  difficulty  exert  a  net 
pressure  of  30  Ibs.  at  the  handle,  exclusive  of  frictional  resistance,  for  a 
short  time.  In  one  instance,  he  observed  that  a  very  strong  man  raised 
23  cwt.  by  a  30  cwt.  crane,  when  he  exerted  a  net  force  of  100  Ibs.  at  the 
handle.  An  ordinary  man  at  the  same  crane,  raised  14  cwt.  with  difficulty, 
with  a  net  force  of  56  Ibs.  at  the  handle;  and  the  same  man  raised  without 
difficulty  10  cwt.,  with  a  force  of  40  Ibs.  at  the  handle. 

A  man  can  exert  on  the  handle  of  a  screw-jack,  of  say  1 1  inches  radius, 
a  net  force  of  20  Ibs.,  without  difficulty. 

From  the  foregoing  data,  it  appears  that  the  average  net  daily  work  of 
an  ordinary  labourer  at  a  pump,  a  winch,  or  a  crane,  may  be  taken  at 


720  WORK,   OR   LABOUR. 

3300  foot-pounds  per  minute;  and,  allowing  one-third  more  for  the  frictional 
resistance  of  the  machine  or  apparatus,  the  total  work  done  would  be  at 
the  rate  of  4400  foot-pounds  per  minute. 

It  may  be  added  that,  taken  generally,  well-fed  English  labourers  can 
turn  a  crank  by  hand,  at  the  rate  of  from  25  to  30  turns  per  minute,  for  a 
continuance,  exerting  a  pressure  of  20  Ibs.  at  the  handle;  or  they  can  apply 
a  pressure  of  28  or  30  Ibs.  for  a  short  time,  or  from  50  to  56  Ibs.  at  an 
emergency. 

M.  Cornet's  estimate  of  the  work  of  a  labourer  in  France,  turning  a 
crank,  amounts  to  6  kilogrammetres  per  second,  equivalent  to  2604  foot- 
pounds per  minute,  for  8  hours  a  day  above  ground,  or  6  hours  a  day  in  a 
mine.  This,  it  is  presumed,  is  the  net  work. 

LABOUR  OF  HORSES. 

According  to  Messrs.  Boulton  &  Watt's  estimate  of  the  power  of  a  dray- 
horse,  it  could  do  work  equivalent  to  33,000  foot-pounds  per  minute,  for 
8  hours  a  day. 

Tredgold  estimated  the  work  of  a  horse  at  27,000  foot-pounds  per 
minute,  for  8  hours  a  day. 

Simms  tested  the  labour  of  horses  in  raising  water : — 

23,412  foot-pounds  per  minute,  for  8  hours  a  day. 
24,360         „  „  6 

27,056         „  „  4%  „       „ 

32,943   .     »  »  3       »       „ 

He  preferred  the  performances  for  6  hours  and  3  hours  a  day,  as  they  were 
unobjectionable  to  the  health  and  durance  of  the  horses. 

Rennie  found  that  a  horse  weighing  1 1  cwts.  could  draw  a  canal  boat  at 
a  speed  of  2^  miles  per  hour,  with  a  pull  of  108  Ibs.,  over  a  distance  of 
20  miles  per  day.  This  performance  is  equivalent  to  a  work  of  23,760  foot- 
pounds per  minute.  He  estimated  that  the  average  work  of  horses,  strong 
and  weak,  is  at  the  rate  of  22,000  foot-pounds  per  minute. 

Mr.  Beardmore  found  that  a  horse  eight  years  old,  weighing  10^  cwts., 
performed  39,320  foot-pounds  of  work  per  minute,  for  8  hours  a  day. 

It  is  inferred  from  the  foregoing  data,  that  the  maximum  work  done  by  an 
average  horse,  per  day  of  8  hours,  is  at  the  rate  of  25,000  foot-pounds  per 
minute.  At  the  same  time,  it  appears  from  the  results  of  trials  at  Bedford, 
to  be  noticed  subsequently,  that  the  average  work  of  a  horse  is  20,000  foot- 
pounds per  minute.  See  page  963. 

Good  horses  can  draw  a  load  of  i  ton  at  the  rate  of  2  ^  miles  per  hour, 
during  from  10  to  12  hours. 

Mr.  A.  Wilson  found  that,  in  India,  a  pair  of  well-fed  bullocks  raised 
82  bags  of  water  22  feet  high  in  i  hour,  for  a  morning's  work  of  4^  hours. 
Each  bag  contained  4^  cubic  feet  of  water,  and  the  work  was  equivalent 
to  8000  foot-pounds  per  minute. 

WORK  OF  ANIMALS  IN  CARRYING  LOADS.1 

Men— Carrying  by  Hand. — Labourers  wheeling  millstone  in  barrows,  on 
the  quays  of  Paris,  a  distance  of  22  to  27  yards,  making  25  to  30  trips  per 

1  Data  derived  from  Les  Moyens  de  Transport,  by  M.  Alfred  Evrard,  1872,  vol.  i. 


LABOUR   OF  ANIMALS.  72 1 

hour;  the  daily  work  performed  is  equivalent  to  the  carriage  of  from  330  to 
400  Ibs.  i  mile. 

The  following  are  other  cases  of  daily  labour,  showing  the  useful  weight 
carried  i  mile: — 

In  Belgium,  working  in  couples,  one  man  carries  560  Ibs.  i  mile. 

At  Port  Royal,  loading  pig-iron,  one  man  carries  160  Ibs.  i  mile. 

At  Paris,  loading  sugar-loafs,  86  Ibs. 

Men — Carrying  on  the  Back. — Carrying  tiles  or  bricks,  net  load  106  Ibs.; 
day's  work  600  Ibs.  carried  i  mile. 

Carrying  coal  in  mines,  net  load  90  to  100  Ibs.;  day's  work  344  Ibs. 
i  mile.  Another  case,  net  load  100  to  130  Ibs.;  day's  work  340  Ibs. 
i  mile. 

Loading  coke  into  waggons,  net  load  100  Ibs. ;  day's  work  270  Ibs. 
i  mile. 

Discharging  coke  on  the  ground,  net  load  100  Ibs.;  day's  work  330  Ibs. 
i  mile. 

Discharging  coal  on  the  ground,  Port  Royal;  net  load  106  Ibs.;  day's 
work  370  Ibs.  i  mile. 

Discharging  coal  on  the  ground,  Paris;  net  load  no  Ibs.;  day's  work 
560  to  960  Ibs.  T  mile. 

Charging  small  coal  into  boats,  Rive-de-Gier;  190  Ibs.  net  load;  day's 
work  1230  Ibs.  i  mile. 

On  the  back  of  a  Horse. — The  load  carried  by  a  horse  on  its  back  varies 
generally  from  220  to  390  Ibs.,  about  27^  per  cent,  of  the  weight  of  the 
animal. 

Carrying  a  man  of  176  Ibs.  weight,  at  about  3^  miles  per  hour;  day's 
work  4400  Ibs.  i  mile. 

Carrying  a  load  of  260  Ibs.  for  10  hours  at  2^  miles  per  hour,  6540  Ibs. 
i  mile. 

Trotting  with  a  man  of  176  Ibs.  weight,  at  5  miles  per  hour,  for  7  hours; 
day's  work,  6100  Ibs.  i  mile. 

On  the  back  of  a  Mule. — At  Buenos  Ayres;  net  load,  170  to  220  Ibs.; 
day's  work  6400  Ibs.  i  mile. 

In  Spain,  net  load  400  Ibs.  at  2.9  miles  per  hour;  day's  work  5300  Ibs. 
i  mile. 

In  France,  net  load  330  Ibs.  at  2  miles  per  hour;  day's  work  5000  Ibs. 
i  mile. 

On  the  back  of  an  Ass. — Load  176  Ibs.,  carried  19  miles;  day's  work 
3300  Ibs.  i  mile. 

The  asses  in  Syria  can  carry  from  450  to  550  Ibs.  of  grain.  - 

On  the  back  of  a  Camel. — Load  550  Ibs.  carried  30  miles  per  day  for 
4  days,  resting  on  the  5th  day.  For  4  days,  day's  work  16,500  Ibs.  i  mile. 
For  5  days,  13,000  Ibs.  i  mile. 

The  ordinary  load  for  a  dromedary  is  770  Ibs. 

On  the  back  of  a  Lama. — Load  no  Ibs.;  day's  work  2000  to  3000  Ibs. 
i  mile. 


46 


FRICTION    OF    SOLID    BODIES. 


The  friction  between  surfaces  pressed  together,  whether  flat  or  round,  of 
which  one  is  moved  on  the  other,  is  said  to  be  in  the  direct  ratio  of  the 
pressure,  and  to  be  independent  of  the  velocity,  and  of  the  area  of  the 
surfaces  pressed  together,  up  to  what  may  be  called  the  elastic  limit,  or  the 

Table  No.  252. — FRICTION  OF  JOURNALS  IN  THEIR  BEARINGS. 

Diameters  from  2  to  4  inches.     Speeds  varied  as  I  to  4.     Pressures  up  to  2  tons  nearly. 
(Reduced  from  M.  Morin's  data. ) 


Description  of  Surfaces  in  Contact. 

LUBRICANT. 

Coefficient  of  Friction. 

Ordinary 
Lubrication. 

Continuous 
Lubrication. 

JOURNALS.                  BEARINGS. 

Cast  iron  on  cast  iron  

Cast  iron  on  gun  metal  

Cast  iron  on  lignum  vitae 

Wrought  iron  on  cast  iron 
Wrought  iron  on  gun  metal 

Wrought  iron  on  lignum  ) 
vitae              ( 

(  Lard,  olive  oil,  or  tallow 
The   same  lubricants,  ) 
and  wetted                  \ 

pressure=i 

.07  to  .08 

.08 

.054 
.14 

.14 

.07  to  .08 

.16 
.16 

.10 

.14 

.07  to  .08 
.07  to  .08 

.09 
.19 
.25 

.11 

.19 

.10 

.09 

.12 
•IS 

pressure^  i. 

.03  to  .054 

'  Asphalte. 

Surfaces  unctuous 

(^  Unctuous  and  wetted  .  .  . 
(  Lard,  olive  oil,  or  tallow 
i  Surfaces  unctuous 

.03  to  .054 

\  Unctuous  and  wetted  .  .  . 
(  Slightly  unctuous 

.09 

.03  to  .054 
.03  to  .054 

f  Wood  slightly  unctuous 
|  Oil,  or  lard  

•{  Unctuous 

Unctuous,  with  mixture  ) 
[     of  lard  and  plumbago  ) 
Olive  oil,  tallow,  or  lard... 
(  Olive  oil,  tallow,  or  lard 
I  Grease 

y  Unctuous  and  wetted.... 
(  Slightly  unctuous 

— 

Oil,  or  lard  

Unctuous 

— 

Gun  metal  on  gun  metal... 

Lignum  vitae  on  cast  iron 
Lignum  vitae  on  lignum  ) 

fOil  

\  Lard  

(  Lard  

.07 

(  Unctuous  

Lard  

vitae  \ 

FRICTION:  M.  MORIN'S  EXPERIMENTS. 


723 


limit  beyond  which  abrasion  takes  place.  Friction  is  of  two  kinds : — sliding 
or  rubbing,  and  rolling;  and  the  frictional  resistance  to  the  commencement 
of  motion,  after  two  bodies  have  rested  sometime  in  contact,  is  greater  than 
the  friction  between  bodies  of  which  one  is  already  in  motion  upon  the 
other. 

M.  Morin's  experiments  afford  the  principal  available  data  for  use. 
Though  the  constancy  of  friction  holds  good  for  velocities  not  exceeding 
15  or  1 6  feet  per  second;  yet,  for  greater  velocities,  the  resistance  of  friction 
appears,  from  the  experiments  of  M.  Poiree,  in  1851,  to  be  diminished  in 
some  proportion  as  the  velocity  is  increased. 

Table  No.  253. — FRICTION: — RESULTS  OF  EXPERIMENTS. 

(Reduced  from  M.  Morin's  data.) 


Description  of  Surfaces  in  Contact. 

Disposition  of 

Fibres. 

State  of  the  Surfaces. 

Coefficient  of 
Friction. 

Oak  on  oak     

parallel 

dry 

pressure=i. 
48 

Do 

soaped 

.16 

Do 

perpendicular 

dry 

.14- 

Do         

on  end,  on  side 

.19 

Different  woods  on  oak  

parallel 

.36  to  .40 

\Vroucrht  iron  on  oak 

.62 

Do             do         ... 

wetted 

.26 

Do.            do  

soaped 

.21 

Cast  iron  on  oak 

dry 

.40 

Do             do 

y 

wetted 

.22 

Do             do  

soaped 

•  IQ 

Brass  on  oak 

drv    • 

.62 

\Vrought  iron  on  elm 

.2; 

Cast  iron  on  elm  

.20 

Leather  on  oak 

.30  to  .3C 

Do         do       

wetted 

.20 

Leather  belt  on  oak  (flat)  

dry 

.27 

Do.          on  oak  pulley  

Leather  on  cast  iron  &  on  bronze 
Do.           do.            do. 
Do.           do.            do. 
Do.           do.            do. 
Wrought  iron  on  cast  iron  

perpendicular 

» 

wetted 
unctuous  and  wet 
oiled 
slightly  unctuous 

•47 

(after  resting 
in  contact.) 
.56 
.36 
•23 
•15 
.18 

Do              on  bronze 

.18 

Cast  iron  on  cast  iron  

.15 

Do.      on  bronze  



-15 

Bronze  on  bronze  

drv 

.20 

Do  on  cast  iron 

22 

Do  on  wrought  iron 

slightly  unctuous 

.16 

Oak,  elm,  yoke  elm,  cast  iron,  ) 
wrought     iron,    steel,    and  ( 
bronze,     sliding     on     each  ( 
other,  or  on  themselves  j 

— 

'lubricated   in^j 
the    ordinary  ! 
way  ,with  lard,  f 
<{  tallow,  oil,  &c.J 
continuously  ) 
lubricated     } 
[_  slightly  unct'ous 

.07  to  .08 

.05 
•15 

724 


FRICTION   OF   SOLID   BODIES. 


It  appears  from  the  table  No.  252  that  the  frictional  resistance  of  metal 
journals  revolving  in  metal  bearings  is  uniform  for  all  metals,  with  one 
exception;  and  that  the  resistance,  with  continuous  lubrication,  is  only 
56  per  cent,  of  the  resistance  with  ordinary  lubrication: — 


Lubrication. 


Coefficient. 


JOURNAL.  BEARING. 

Cast  iron  in  cast  iron N     ordinanr     /  -°7  to  .08,  or  */14  to  '/I2. 

Cast  iron  in  gun  metal —  /  I         mean  .075,  or  I/i3.3 

Wrought  iron  in  cast  iron  V   continuous   f  -03  to  .054,  or  r/33  to 
Wrought  iron  in  gun  metal )  \          mean  .042,  or  I/a4 

Gun  metal  in  gun  metal;  ordinary,  .10,  or  I/I0ih. 


2.5 


Additional  data  derived  from  the  friction  of  mill-shafting,  are  given  under 
SHAFTING. 

FRICTION  ON  RAILS. 

M.  Poire'e's  experiments  on  the  Paris  and  Lyons  railway  were  made  with 
a  waggon,  presumably  having  four  wheels,  of  which  the  brake  was  screwed 
up,  so  that  the  wheels  were  skidded.  The  resistance  to  traction,  or  the 
friction  on  the  rails,  at  various  velocities,  was  as  follows : — 

Table  No.  254. — SLEDGING  FRICTION  OF  A  WAGGON  ON  RAILS. 

M.  Poiree's  Coefficients. 


Empty  Waggon,  3.40  tons. 

STATE  OF  THE  RAILS. 

Dry. 

Very  Dry. 

Damp. 

Dry  and 
Rusty. 

Dry.    Springs 
gagged. 

Velocity  of  Waggon. 

feet  per  second. 
13  tO  20 
20  tO  26 

26  to  33 
33  to  46 
46  to  60 
60  to  72 

miles  per  hour. 

9  to  14 
14  to  18 

l8  tO  22 
22  tO  30 

30  to  40 
40  to  50 

weight=i. 
.208 

.179 
.167 

.144 

weight=i. 
.246 

.222 
.202 
.I87 

weight=i. 
.110 
.083 

weight=i. 
.201 
.182 

•175 
.162 

•~36 

weight=i. 
.200 

.172 
.154 
.132 

Comparative  Coefficie 

Speed,  30  feet  per  second  ; 
or  20  miles  per  hour. 

nts  of  Frictioi 

•175 
.169 

i  for  Diffe 

rent  Weigh 

ts  ;  Rails  I 

)ry. 

Empty  waggon,  3.40  tons 
Loaded  waggon,  6.45     „ 

At  speeds  under  20  miles  per  hour,  it  appears  from  the  table  that,  when 
the  rails  are  dry,  the  coefficient  of  friction,  or  the  adhesion,  is  one-fifth  of 
the  weight,  and  that  on  very  dry  rails  it  is  one-fourth.  As  the  speed  is 
increased,  the  adhesion  is  reduced. 

These  data  are  corroborative  of  the  results  of  the  author's  experiments 
on  the  ultimate  tractive  force  of  locomotives  on  dry  rails,  from  which  he 
obtained  a  coefficient  of  friction  equal  to  one-fifth  of  the  weight,  at  speeds 
of  about  10  miles  per  hour. 


WORK   ABSORBED   BY   FRICTION.  725 

They  are  corroborated  by  the  experience  of  train-brakes  on  the  District 
Railway,  London.  The  gripping  force  of  the  brake  is  greatest  just  before 
the  train  is  brought  to  a  state  of  rest. 

It  is  seen  from  the  table  that,  on  damp  rails,  the  coefficient  of  friction,  at 
20  miles  per  hour,  was  reduced  to  one-ninth. 

In  the  second  part  of  the  table,  it  is  seen  that  the  coefficient  of  friction 
increases  in  a  ratio  rather  less  than  that  of  the  weight. 

WORK  ABSORBED  BY  FRICTION. 

The  product  of  the  total  pressure  between  the  rubbing  surfaces  by  the 
coefficient  of  friction,  is  the  total  frictional  resistance;  and  the  product  of 
this  resistance  by  the  space  through  which  it  acts,  is  the  work  done,  or 
absorbed,  by  friction.  Let — 

W  =  the  load  or  pressure,  in  pounds, 

/=the  coefficient  of  friction  between  the  two  surfaces, 
s  =  the  space  passed  through  by  one  surface  on  the  other,  in  feet, 
t  —  the  time  in  minutes, 

v  =  the  velocity  at  the  surface,  in  feet  per  minute, 
S  =  the  speed  of  revolution,  or  number  of  turns  per  minute, 

U  =  the  work  absorbed,  in  foot-pounds, 

H  =  the  horse-power  absorbed, 

d=\he  diameter  of  the  axle-journal,  or  of  the  pivot,  in  inches, 
r  =  ihe  radius  of  the  axle-journal,  or  of  the  pivot,  in  inches, 
a  =  half  the  angle  at  the  apex  of  a  conical  journal  or  a  conical  pivot, 
/=the  axial  length  of  the  rubbing  surface  of  a  conical  pivot,  in  inches. 

The  space  described  by  a  cylindrical  journal  for  one  turn,  is  equal  to  3. 14  d, 
in  inches,  or  to  .26  d,  in  feet;  and  by  the  flat  end  of  a  cylindrical  pivot, 
the  space  described  is  equal  to  two-thirds  of  this  quantity,  in  the  ratio  of 
the  mean  diameter  to  the  extreme  diameter,  or  .175  */,  in  feet.  For  a 
conical  pivot,  the  mean  diameter  is,  as  for  a  flat  pivot,  two-thirds  of  the 
'  extreme  diameter  of  the  rubbing  surface  of  the  pivot,  and  the  space 
described  for  one  turn  is  expressed  by  .175  d,  in  feet.  But  the  pressure  on 
the  surface  of  the  conical  pivot,  compared  with  the  pressure  on  a  flat  pivot, 
is  greater  in  the  ratio  of  the  slant  length  of  the  pivot  to  the  extreme  radius 

of  the  rubbing  surface,  or  as  radius  to  sine  a;  or  as  A/  r2  +  /2  to  r. 

The  pressure  on  the  surface  of  a  conical  journal,  is  to  that  on  a  cylin- 
drical journal  of  the  same  length  and  extreme  diameter,  as  radius  to 
cosine  a. 

Work  Absorbed  by  Friction. 

1.  On  a  flat  surface U=/Wx  s (  i  ) 

2.  On  a  cylindrical  journal,  for  one  turn U  =/"W  x  .26  d.....  (  2  ) 

3.  On  the  square  end  of  a  cylindrical  pivot,  I  TT  =  / W  x  1 7  <:  </      (O 

for  one  turn j 

4.  On  a  conical  journal,  for  one  turn \  U=^- '- (  4  ) 

(  cos  a 

5.  On  a  conical  pivot,  for  one  turn <  U="—   .x  '*^     ...  (  5  ) 

The  second  formula,  (  2  ),  is  applicable  to  the  cases  of  friction-couplings 
or  friction-brakes. 


726  FRICTION   OF   SOLID   BODIES. 

The  horse-power  absorbed  by  continuous  frictional  action  on  a  flat  sur- 
face, or  surface  of  other  form,  is  equal  to  the  product  of  the  resistance  by 
the  velocity,  divided  by  33,000.  On  a  journal  or  a  pivot,  it  is  equal  to  the 
work  absorbed  in  one  turn,  multiplied  by  the  speed,  and  divided  by  33,000. 

Horse-Power  Absorbed  by  Friction. 
i.  On  a  flat  surface <  H  =  TTT^T (  6  ) 

«J«J5 


2.  On  a  cylindrical  journal  <  H  = —     QQQ —       "127  ooo ^^ 

(8) 


/WSx.i75  </_/W 


r 

3.  On  a  cylindrical  pivot,  j  H  =        33j000 

/WSx.2_ 

4.  On  a  conical  journal....  j  H  =  33)0oo  x  cos  a  ~  I27)000xCOSa 

/WSx.i75^_      /WS^ 

5.  On  a  conical  pivot |  H  =  x  sin  a  ~  l89)OOOx  sin  a (I0) 


MILL-GEARING. 


TOOTHED   GEAR. 

In   a   pair   of   toothed   wheels,  or  pinions,   geared    together,  the   re- 
lative  angular   velocities   or   speeds,    or   number   of    turns    per    minute, 
are   inversely  as   their   diameters,  or  as   their   radii;    and  in  a  train  of 
wheels  and  pinions,  like  that  in  Fig.  315, 
suppose   the  axle  A  makes  27   turns   per 
minute,  and  that  the  diameters  of  the  wheel 
and    pinion   through   which   it   drives    the 
axle  C,  are  as  7  and  3,  the  axle  C  makes 

(27  x  Z  = )  63  turns  per  minute.    Again,  the 

axle  C  drives  the  axle  D  by  a  wheel  and  a 
pinion  having  diameters  also  as  7  to  3,  and 

the  speed  of  D  is  (63  x  I  = )  147  turns  per  Fi^  sis-Toothed  Gear. 

*J 

minute.  Finally,  the  axle  B  is  driven  by  D  by  gearing  as  7  to  3,  and 
D  makes  (147  x  2  =  )  343  turns  per  minute.  The  successive  accelerations 
of  speed  are,  then,  as  follows : — 

Axle  A  driving  axle  C as  3  to  7 

«     C  „  D „  3  to  7 


D 


B 


3  to  7 


Total  acceleration . .  . ..as 


27  to  343 

Here,  the  total  ratio  of  acceleration  is  found  by  multiplying  together  the 
first  terms  of  the  ratios,  and  the  last  terms  of  the  ratios;  equivalent  to  the 
ratio  of  i  to  12.7,  since  343-^-  27  =  12.7. 

It  is  seen  that,  in  this  example,  the  acceleration  of  speed  takes  place  by 
equal  additions,  in  the  ratio  of  3  to  7.  To  find  what  this  common  ratio 
must  be,  when  the  initial  and  final  speeds  alone  are  given,  take  the  cube 
root  of  the  compound  ratio  of  total  acceleration,  that  is, 


=    ;or 

27      3'      ' 

giving  the  common  ratio  3  to  7,  as  already  employed,  or  i  to  2*4,  which  is 
the  same. 

The  third  root  is  taken,  because  there  are  three  accelerations  of  speed. 
If  there  were  only  two  accelerations,  the  square  root  would  be  taken;  if 


728 


MILL-GEARING. 


four  accelerations,  the  fourth  root.  In  general,  to  find  the  common  ratio, 
take  that  root  of  the  total  ratio,  the  index  of  which  is  equal  to  the  number 
of  accelerations.  The  same  rule  applies  in  cases  of  reduction  of  speed. 

When  the  speeds  are  increased  by  ratios  which  are  different  from  each 
other,  the  ratios  are  nevertheless  to  be  multiplied  together,  as  already 
exemplified,  to  find  the  resultant  ratio. 

In  making  this  multiplication,  in  the  above  example,  the  ratio  3  to  7  might 
have  been  replaced  by  the  equivalent  ratio  i  to  aj$.  This  ratio  multiplied 
twice  by  itself,  produces  the  total  ratio  i  to  12.7;  and  27  x  12.7  =  343,  the 
final  speed  in  turns  per  minute. 


Fig.  316.— Toothed  Gear. 

The  numbers  of  teeth  in  the  several  wheels  and  pinions  of  each  pair,  are 
necessarily  proportional  to  their  diameters;  and  they  may  be  employed 
instead  of  the  diameters,  to  express  the  ratios ;  and  be  multiplied  together 
to  find  the  resultant  ratio. 

PITCH  OF  THE  TEETH  OF  WHEELS. 

The  pitch  of  the  teeth  of  wheels  is  the  distances  apart  from  centre  to 
centre  of  the  teeth,  measured  on  the  pitch-circle,  ^^pitch-circle,  ox  pitch- 


PITCH   OF   THE   TEETH   OF   WHEELS. 


729 


line,  is  the  circle  passing  through  the  body  of  the  teeth,  which  expresses 
the  virtual  or  normal  circumference  of  the  wheel;  it  would  be  the  actual 
circumference  if  the  teeth  were  indefinitely  small,  when  the  wheel  and 
pinion  might  work  together  by  frictional  contact.  In  the  annexed  Fig.  316, 
showing  the  halves  of  a  wheel  and  a  pinion  in  gear,  A  B  is  the  line  of 
centres,  and  C  C  and  C  C  are  the  pitch  circles  touching  at  c\  the  divisions 
A  c  and  B  c,  of  the  line  of  centres,  being  the  pitch-radii  of  the  wheels.  The 
arc  of  the  pitch-circle,  between  p  and  /,  is  the  pitch  of  the  teeth,  and  it 
comprises  a  tooth  and  a  space. 

The  numbers  of  teeth  in  the  wheel  and  the  pinion  are  in  the  same  ratio 
as  the  diameters  of  their  pitch-circles. '  Let  N  —  the  diameter  of  any  wheel 
at  the  pitch-line,  P  =  the  pitch,  n  =  the  number  of  teeth ;  then, 


D 


3.1416 


(i),  (»),  (3)- 


In  ordinary  practice,  the  pitches  most  commonly  used  range  from  i  inch  to 
4  inches,  advancing  by  eighths  to  2  inches,  and  thence  by  fourths  of  an 
inch.  Below  i  inch,  the  pitches  decrease  by  eighths  down  to  %  inch. 
Sir  William  Fairbairn  employed  the  following  pitches  :  — 

Spur  Fly-  Wheels.  —5,  4^,  4,  3^,  3^,  3,  2%,  2,  i^  inches. 

Spur  and  Bevil  Wheels  for  Millwork.  —  5,  4^,  4,  3^,  3^,  3,  2^,  2^, 
2^,  2/8,  2,  1  24,  i^,  i3/8,  i%,  i/8j  i,  /8  inches. 

For  toolwork  the  pitches  usually  range  from  i  inch  to  %  inch. 

To  save  calculation  in  the  application  of  the  formulas  (i)  &  (3),  the  values 


t  and 


3.1416 


for  particular  pitches,  are  given  in  the  table  No.  255. 


Table  No.  255. — TOOTHED  WHEELS — MULTIPLIERS  FOR  THE  NUMBER 
OF  TEETH  AND  THE  DIAMETER. 

(Rules  i  and  3  ). 


Pitch. 

Multiplier  for 
the  Number 
of  Teeth. 

Multiplier 
for  the 
Diameter. 

Pitch. 

Multiplier  for 
the  Number 
of  Teeth. 

Multiplier 
for  the 
Diameter. 

inches. 

3-1416 
pitch. 

pitch 

inches. 

3.1416 
pitch. 

pitch 

3.1416. 

3.1416. 

6 

.5236 

1.9095 

it* 

1.9264 

.5141 

5 

.6283 

I.59I5 

i% 

2.0944 

•4774 

±y* 

.6981 

1.4720 

i*/s 

2.2848 

•4377 

4 

.7854 

1.2732 

iX 

2.5132 

•3979 

3^ 

.8976 

I.II4I 

i# 

2.7926 

.3568 

3X 

.9666 

1-0345 

i 

3.I4I6 

•3183 

3 

1.0472 

•9547 

7/s 

3-5904 

.2785 

2^ 

I-I333 

.8754 

X 

4.1888 

.2387 

2^ 

1.2566 

.7958 

X 

5.0266 

.1989 

2# 

I-3963 

-7135 

y* 

6.2832 

.1592 

2 

1.5708 

.6366 

X 

8.3776 

.1194 

l» 

1.6755 

-5937 

X 

12.5664 

.0796 

I# 

1.7952 

•5570 

730 


MILL-GEARING. 


Table  No.  256. — DIAMETER  OF  TOOTHED  WHEELS. 

(Given  the  pitch  and  the  number  of  teeth. ) 


Num- 
ber of 
Teeth. 

PITCH  IN  INCHES. 

I 

I* 

i# 

& 

2 

2X 

2^ 

*X 

3 

3X 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

I 

.318 

.398 

•477 

•557 

.637 

.714 

.796 

.875 

•955 

1.04 

2 

•  636 

.796 

•955 

1.14 

1.27 

1-43 

i-59 

i-75 

l.oi 

2.07 

3 

•955 

I.I93 

i.43 

1-67 

I.9I 

2.14 

2.39 

2.63 

2.86 

3.10 

4 

1.27 

I.59I 

1.91 

2.23 

2-55 

2.85 

3.18 

3-5o 

3-82 

4.14 

5 

1.989 

2.39 

2.79 

3-18 

3-57 

3-98 

4-38 

4-77 

5-17 

6 

1.91 

2.387 

2.86 

3-34 

3.82 

4.28 

4.78 

5-25 

5-73 

6.21 

7 

2.23 

2.785 

3-34 

3-90 

4-45 

4-99 

5.57 

6.13 

6.68 

7.24 

8 

2-55 

3.182 

3-82 

4.46 

5-°9 

5-7i 

6.37 

7.00 

7.64 

8.28 

9 

2.86 

3.580 

4-30 

S.oi 

5-73 

6.42 

7.16 

7.88 

8-59 

9.31 

10 

3.18 

3-98 

4-77 

5-57 

6.37 

7.14 

7.96 

8.75 

9-55 

10-35 

20 

6.36 

7.96 

9-55 

11.14 

12.73 

14.27 

15-92 

17.51 

19.10 

20.69 

30 

9-55 

u-93 

14-32 

16.71 

19.10 

21.41 

23-87 

26.26 

28.64 

31.04 

40 

12.73 

15.91 

19.10 

22.28 

25.46 

28.54 

31-83 

35-02 

38.19 

41.38 

50 

19.89 

23.87 

27-85 

31-83 

35-67 

39-79 

43-77 

47-74 

51-73 

60 

19.09 

23-87 

28.64 

33-42 

38-20 

42.81 

47-75 

52-52 

57.29 

62.O7 

70 

22.27 

27.85 

33-42 

38.99 

44.56 

49-94 

55-71 

61.28 

66.84 

72.42 

80 

25.46 

31.82 

38.19 

44-56 

50.93 

57.08 

63.66 

70.03 

76.38 

82.76 

90 

28.64 

35-8o 

42.97 

50.13 

57.29 

64.21 

71.62 

78.78 

85-93 

93-  " 

IOO 

31-83 

39-78 

47-74 

55-70 

63.66 

71.35 

79-58 

87.54 

95.48 

103-45 

110 

43.76 

52-51 

61.27 

70.03 

78.48 

87-54 

96.29 

105.03 

113.80 

120 

38.18 

47-74 

57.28 

66.84 

76.39 

85.62 

95-50 

105.05 

114.58 

124.14 

130 

41.36 

62.06 

72.41 

82.76 

92.75 

103.45 

113.80 

124.12 

134-50 

I4O 

44-54 

55-70 

66.84 

77.98 

89.12 

99.89 

111.41 

122.56 

I33-67 

144.83- 

IS0 

47-73 

59-67 

71.61 

83.55 

95-49 

107.03 

H9-37 

W-31 

143.22 

I55-I8 

1  60 

50.91 

63-65 

76.38 

89.12 

101.86 

114.16 

127-33 

140.06 

152.77 

165.52 

170 

54.10 

67-63 

81.16 

94.69 

108.22 

121.29 

I35-29 

148.82 

162.32 

175.87 

180 

57.28 

71.60 

85-93 

100.26 

H4.59 

128.43 

H3-24 

157-57 

171.86 

186.21 

190 

60.46 

75-58 

90.71 

105.83 

120.95 

J35-57 

151.20 

166.33 

181.41 

196.56 

200 

63-64 

79.56 

95.48 

111.40 

127.32 

142.70 

159.16 

175.08 

190.96 

206.90 

2IO 

66.81 

83-55 

100.26 

116.97 

133-68 

149.82 

167.13 

163.84 

200.52 

217.26 

2  2O 

70.00 

87-52 

105.02 

122.54 

140.06 

156.96 

175.08 

192.58 

210.06 

227.60 

230 

73.19 

91.49 

109.80 

128.11 

146.42 

164.11 

183.03 

201.34 

219.60 

237.94 

240 

76.36 

95-48 

114-56 

133.68 

152-78 

171.24 

191.00 

2IO.IO 

229.16 

248.28 

250 

79-55 

99-45 

119-35 

139.25 

J59-I5 

178.37 

198.95 

218.85 

238.70 

258.63 

260 

82.72 

103.44 

124.12 

144.82 

165-52 

185.50 

206.90 

227.60 

248.24 

269.00 

270 

85-92 

107.40 

128.91 

150.39 

I73-87 

192.63 

214.86 

236.34 

257-79 

279-33 

280 

89.08 

111.40 

133-68 

I55-96 

178.24 

199.78 

222.82 

245.12 

267.34 

289.66 

290 
300 

92.28 
95-49 

II5-36 
H9-34 

138.45 
143.22 

161.53 
167.10 

184.61 
190.98 

206.91 
214.05 

230.78  253.86 
238.74!  262.62 

276.89 
286.44 

300.01 

APPENDIX  TO  TABLE.— To  find  the  diameters  for  pitches  under  i  inch,  namely, 

X    I    5/i6    |     #    |     7/i6    |     ft    |    9/i6    |     K    I    «/i6    I     %    I    '3/i6    |     %  inch, 
refer  to  the  columns  in  the  table,  respectively,  for 

i    I    i#   I  i%   I    i%   !     *     I    *X   I  »K  I    2%    I   iKI    3#    I   i%  inches; 
and  divide  the  quantities  in  these  columns  by  the  corresponding  divisors,  following : — 

4l4l4l4|a|4|2|4|a|4|a 
To  find  the  diameter  for  pitches  above  3^  inches,  namely, 

3l/2    I      4      I    4^    I      5      I    5%    I    6  inches, 
refer  to  the  columns  in  the  table,  respectively,  for 

iK    I      2      |     2#    |    2^    I    2%    |    3  inches; 
and  multiply  the  quantities  in  these  columns  by  2. 

To  find  the  diameter  for  any  number  of  teeth  between  10  and  300,  not  specified  in  the  table.  Find  the  dia- 
meters for  the  tens  and  hundreds  in  the  number,  and  for  the  units;  the  sum  of  these  diameters  is  the  required 
diameter.  For  a  wheel  of  i-inch  pitch,  with  135  teeth,  for  example: — the  diameter  for  130  teeth  is  41.36 
inches,  and  for  5  teeth  it  is  1.59  inches;  and  41.36  +  1.59=42.95  inches,  the  required  diameter  for  135  teeth. 


FORM   OF   THE   TEETH   OF   WHEELS. 


731 


RULE  i.  To  find  the  Number  of  Teeth  in  a  wheel  of  a  given  diameter  and 
pitch. — Multiply  the  diameter  in  inches,  by  the  multiplier  in  the  second 
column  of  the  table,  opposite  the  given  pitch.  The  product  is  the  number 
of  teeth. 

RULE  2.  To  find  the  Diameter  of  a  Wheel  having  a  given  number  and  pitch 
of  teeth. — Multiply  the  number  of  teeth  by  the  multiplier  opposite  the  given 
pitch,  in  the  third  column  of  the  table.  The  product  is  the  diameter  in 
inches. 

Note  to  Rule  i. — When  the  answer  contains  a  fraction,  the  diameter  re- 
quires to  be  slightly  altered,  so  as  to  bring  out  a  whole  number  of  teeth. 
For  this  purpose,  take  the  nearest  whole  number  of  teeth  to  the  answer 
given  by  the  rule,  and  find  the  modified  diameter  for  that  number,  by 
Rule  2. 

FORM  OF  THE  TEETH  OF  WHEELS. 

That  the  teeth  of  wheels  and  pinions  may  work  together  smoothly, 
steadily,  without  rubbing  friction,  and  with  uniform  motion,  they  should 
be  shaped  to  epicycloidal  forms  on  the  faces,  and  hypocycloidal  forms  on 
the  flanks, — being  the  portions  of  the  sides  of  the  teeth  respectively  above 
and  below  the  pitch-line.  These  forms  are  explained  and  exemplified  at 
pages  19  and  20. 

With  respect  to  the  formation  of  the  flanks  of  the  teeth  of  the  wheel, 
when  the  generating  circle  has  half  the  diameter  of  the  pitch  circle,  the 
hypocycloid  described  by  it  is  a  straight  diametrical  line,  and  therefore  the 
flanks  are  straight  and  radial  to  the  centre,  as  in  the  Fig.  318,  next  page. 


Fig.  317.— Formation  of  Teeth  of  Wheels. 

Similarly,  with  a  generating  circle  having  half  the  diameter  of  the  pitch- 
circle  of  the  pinion,  the  flanks  of  the  teeth  of  the  pinion  are  also  straight 
and  radial.  If  now  the  same  generating  circles  be  employed  to  form  the 
faces  of  the  teeth  of  the  other  wheel  or  pinion  respectively,  the  wheel  and 
pinion  will  work  truly  together.  For  example,  let  the  template  B,  Fig.  317, 
cut  to  the  pitch-circle  of  the  wheel,  be  screwed  to  a  hardwood  board  C.  Set 
off  the  thickness  a  c  of  a  tooth,  on  the  pitch-circle  d  d,  and  apply  the  segment 
D,  of  which  the  radius  is  equal  to  half  the  radius  of  the  pinion.  With  a 
tracing  point  /,  inserted  obliquely  at  the  edge  of  D,  roll  the  segment  D 
on  the  template  B,  so  as  to  describe  the  epicycloidal  curve  a  t;  and,  in  the 
same  way,  describe  the  counterpart  ct.  The  two  faces  of  the  tooth  are 


732 


MILL-GEARING. 


thus  formed.  To  draw  the  flanks  of  the  tooth,  trace  the  arc  dd  on  the 
board  C  C,  and  remove  B;  screw  the  board  to  a  slip  of  deal  h,  Fig.  318, 
and  find  the  centre  of  the  pitch-circle,  on  the  slip  h\  draw  the  radii  a  b 
and  cb,  to  form  the  flanks,  which  are  slightly  rounded  at  the  base  b,  to  join 
the  arc  //,  drawn  through  the  roots  of  the  teeth.  The  end  of  the  tooth  is 
defined  by  the  arc  gg. 


L — •*> — *—._/•>— J 


Fig.  318.— Formation  of  Teeth  of  Wheels. 

Performing  the  same  operation,  conversely,  for  the  teeth  of  the  pinion, 
the  wheel  and  the  pinion  so  constructed  work  truly  together.  In  the 
wheel  and  pinion,  Fig.  316,  are  shown  the  generating  circles,  having 
diameters  equal  to  the  radii  of  the  respective  wheels.1 

Whilst  a  wheel  and  pinion,  or  two  wheels,  having  their  teeth  so  con- 
structed, work  truly  together,  they  do  not  work  truly  with  wheels  or  pinions 
of  other  diameters.  For  bevil-wheels,  this  peculiarity  is  of  no  moment, 
since  they  can  only  work  by  pairs ;  but,  for  spur-wheels,  it  is  convenient  that 
any  two  wheels  of  the  same  pitch  should  be  capable  of  working  truly  together. 
This  property  of  interchangeableness  is  secured  by  the  employment  of  the 
same  generating  circle  for  both  flanks  and  faces,  and  for  all  diameters,  as 
well  as  for  racks.  The  minimum  number  of  teeth  may  be  taken  as  12,  and 
the  diameter  of  the  common  generating  circle  may  be  taken  as  equal  to  the 
radius  of  the  pinion  of  12  teeth.  Teeth  of  wheels  so  formed  have  the 
flanks  rather  excessively  tapered;  but,  by  reducing  the  taper,  or  thinning 
the  tooth,  for  a  distance  of  half  the  height  of  the  flank  measured  from  the 
root  of  the  tooth,  the  working  durability  of  the  tooth  is  much  increased, 
and  steadiness  of  action  is  promoted  even  when  the  tooth  has  become 
considerably  worn.  If  the  full  form  of  the  flank  of  the  tooth  be  retained, 
a  shoulder  is  gradually  worn  into  it,  by  which  a  tendency  is  induced  to 
force  the  wheel  and  pinion  out  of  gear;  but  in  cutting  away  the  flanks  near 
the  base,  whilst  the  working  portion  of  the  flank-surface  remains,  shoulders 
can  only  be  formed  after  very  excessive  wear.2 

Involute  Teeth. — The  teeth  of  two  wheels  will  work  truly  together,  when 
their  acting  surfaces  are  involutes.  The  involute  curve  may  be  described 

1  There  is  an  error  in  the  form  of  the  flanks  of  the  teeth  as  shown  in  Fig.  316;  they 
should  have  been  straight  and  radial. 

2  Mr.  Robert  Wilson,  of  Patricroft,  employs  this  method  of  forming  the  teeth  of  spur- 
wheels.     The  author  is  indebted  to  him  for  the  above  particulars  about  it. 


FORM   OF   THE   TEETH   OF   WHEELS. 


733 


mechanically: — Let  A,  Fig.  319,  be  the  centre  of  a  wheel,  and  mna  a 
thread  lapped  round  its  circumference.  The  curve  a  b  h,  described  by  a 
pin  at  the  end  of  the  thread  when  unwound  from  the  circle,  is  an  involute. 


Fig.  319. — Formation  of  Teeth  of  Wheels. 

The  curve  is  also  formed  by  causing  a  straight  ruler  R  to  roll  on  the  circle, 
Fig.  319,  with  a  pin  at  the  end,  tracing  the  involute  qp. 

That  two  wheels  with  involute  teeth  should  work  truly,  the  circles  from 


Fig.  320. 


Formation  of  Teeth  of  Wheels. 


which  the  involute  forms  for  each  wheel  are  generated,  must  be  concentric 
with  the  wheels,  with  diameters  in  the  same  ratio  as  those  of  the  wheels. 
Let  AT,  BT,  Fig.  320,  be  the  pitch-radii  of  two  wheels  to  work  together; 


734 


MILL-GEARING. 


through  T  draw  any  straight  line  D  E,  and  with  the  perpendiculars  A  D 
and  B  E,  describe  the  circles  D  H  and  E  F.  The  involutes  K  T  H  and 
G  T  F  give  the  forms  of  the  teeth. 

To  describe  the  teeth  of  a  pair  of  wheels,  of  which  A  c  and  B  c,  Fig.  321, 
are  the  pitch  radii,  draw  c  d  and  c  d  perpendicular  to  the  radials  B  d  and 
A  d;  these  radials  are  the  radii  of  the  involute  circles  from  which  the  acting 
faces  of  the  teeth  are  formed. 

Involute  teeth  have  the  disadvantage  of  being,  when  in  contact,  too  much 
inclined  to  the  radial  line,  by  which  an  undue  pressure  is  excited  on  the 
bearings.  But  they  have  the  advantage  of  working  truly,  even  at  varying 
distances  apart  of  the  centres,  and  any  two  wheels  of  a  pitch  will  work 
together  in  sets,  however  different  the  diameters. 

PROPORTIONS  OF  THE  TEETH  OF  WHEELS. 

Referring  to  the  annexed  Fig.  322,  the  leading  dimensions  are  indicated 
by  literal  references,  and  are  thus  distinguished  in  the  table  No.  257,  in 


Fig.  322.— Proportions  of  Teeth  of  Wheels. 


whj^Tftie  first  and  second  scales  of  proportions  are,  both  of  them,  used  by 
engineers  in  good  practice;1  and  the  third  is  the  scale  of  Sir  Wm.  Fairbairn.2 

TABLE  No.  257. — TEETH  OF  WHEELS: — PROPORTIONAL  DIMENSIONS. 

(Fig.  322.) 


ELEMENTS. 

ist  Scale. 

2d  Scale. 

3d  Scale. 

A£+C£  Pitch  of  teeth,  

I 

15      or  I 

1.  00 

C&            Thickness  of  teeth,  

S/TT  or  .4.S 

7      or  .47 

.41 

Ad            Width  of  space  

61     or   1  s 

8     or  $1 

re 

Ad     C^  Play  

i      or  07 

.IO 

a  c             Length  above  pitch-line,  

s/IO  or  .30 

5X°r-37 

•  W 

C  c            Length  below  pitch-line 

4/10  or  40 

6  */£  or  4.3 

4.O 

A  a  -\-fic   Working  length  of  tooth,  

6/IO  or  60 

ii      or  73 

7O 

Ca            Whole  length  of  tooth,  

7/10  or  .70 

12     or  .80 

.7C 

C  c—  A  a  Clearance  at  root,  

I/IO  or  .10 

i      or  .07 

.05 

C  b            Thickness  of  rim,  

7     or  .4.7 

Note  to  table. — The  proportion  of  clearance  at  the  root  of  the  tooth  is 
usually  varied  from,  say,  x/I0th  of  the  pitch  for  the  smaller  wheels,  to  x/aoth 
for  the  larger  wheels. 

1  Engineer  and  Machinist's  Assistant,  page  89.        2  Mills  and  Millwork,  Part  2,  page  33. 


PROPORTIONS  AND  STRENGTH  OF  THE  TEETH  OF  WHEELS.     735 


STRENGTH  OF  THE  TEETH  OF  WHEELS. 

The  tooth  of  a  wheel  in  action,  is  a  beam  fixed  at  one  end  and  loaded  at 
the  other;  and,  as  the  available  strength  of  a  wheel-tooth  is  that  of  its 
weakest  exposure,  the  strength 
should  be  calculated  for  the  con- 
tingency that  the  whole  of  the  force 
may  be  applied  to  the  tooth  at  one 
corner,  D,  Fig.  323.  The  tooth, 
it  is  conceivable,  may  be  broken 
across  at  any  of  the  lines  B  b,  B  c, 
or  B  d;  but,  under  uniform  con- 
ditions, the  actual  line  of  fracture 
would  be  B  ^,  which  is  the  base  of 
a  right-angle  triangle  having  the 
equal  sides  D  B  and  D  b,  and  of 
which  the  height  D  e  is  equal  to 
half  the  base  B  b.  Now,  the  height  of  the  tooth  D  B,  is  the  slope  of  the 
right-angled  triangle  D  B  e,  and  is  1.414  times  D  e\  and  the  values  of  the 
elements  for  the  application  of  the  formula,  according  to  the  3d  scale  of 
proportions  in  table  No.  257,  are:  — 

P  =  the  pitch  =  i. 

/=the  length  of  the  "beam"  =  D  e  =  .75  P  4-  1.414  =  .53  P. 
b  -  the  breadth  of  the  beam  =  B£  =  .53Px2  =  1.06  P. 
d=  the  depth  of  the  beam,  or  the  thickness  of  the  tooth  =  .45  P. 
s  =  the  tensile  strength  of  the  material  in  tons  per  square  inch. 
W  =  the  breaking  weight  at  the  corner  of  the  tooth,  in  pounds. 

Adapting   the    general   formula  (4),   page    507,   the  coefficient  .289^ 
becomes  (.289  x  2240  s  =  )  647,  and,  in  terms  of  the  above  symbols,  — 


Fig.  323.— Strength  of  Teeth  of  Wheels. 


w_  (647  ........................... 

To  express  the  breaking  strength  of  the  tooth  in  terms  of  the  pitch,  sub- 
stitute the  equivalent  values  of  b,  d,  and  /,  in  formula  (  4  ):  — 

-          W  =  647^i.o6^x(.4SP)-  =  647fx.40S^.or 

W  =  (262  s)  P2  ................................................  (5) 

VALUES  OF  THE  NUMERICAL  COEFFICIENT  IN  FORMULA  (  5  ). 

Ultimate  Tensile  Strength  Coefficient 

per  Square  Inch.  (262  s). 

tons. 

7.  Cast  iron,  ........................  262  x   7=  1834,  say  1800 

8.  Do.,      ........................  262  x   8  =  2096,    „    2100 

9.  Do.,      ........................  262  x   9  =  2358,    „    2400 

10.  Do.,      ........................  262x10  =  2620,  „  2600 

11.  Do.,      ........................  262x11  =  2882,  „  2900 

12.  Do.,      ........................  262x12  =  3144,  „  3100 

12.  Gun-metal,  .......................  262x12  =  3144,  „  3100 

20.  Wrought  iron,  ..................  262x20  =  5240,  „  5200 

30.  Steel,  ..............................  262x30  =  7860,  „  8000 


736  MILL-GEARING. 

For  wheels  of  ordinary  cast  iron,  when  the  tensile  strength  is  not  given, 
assume  7  tons  per  square  inch  as  the  tensile  strength,  and  adopt  the  co- 
efficient 1800.  Then, 

The  Ultimate  Transverse  Strength  of  the  Teeth  of  ordinary  Cast-Iron 
Wheels,  in  terms  of  the  pitch,  is, — 

W-i8oo  P2 (6) 


Inversely,  P=         -- (7) 


To  express  the  breaking  strength  of  the  tooth  in  terms  of  the  thickness 

of  the  tooth,  d,  which  is  equal  to  .45  P.     The  pitch  P  =  —  ,  and  by  sub- 

•45 

stitution  in  formula  (  5  ),  W=  262  sx  (  —  )2;  or, 

^2  .........................   (8) 


VALUES  OF  THE  NUMERICAL  COEFFICIENT  IN  FORMULA  (  8  ). 

Ultimate  Tensile  Strength  Coefficient 

per  Square  Inch.  (1294  s). 

tons. 

7.  Cast  iron,  ..................  i2Q4x    7=    9058,  say     9000 

8.  Do.,      ..................  I294X    8=10,352,    „    10,000 

9-          Do.,      ..................  I294X    9=11,646,    „    12,000 

10.  Do.,      ..................  1294x10=12,940,  „  13,000 

11.  Do.,      ..................  1294x11  =  14,234,  „  14,000 

12.  Do.,      ..................  1294x12  =  15,528,  „  16,000 

12.  Gun-metal,  .................  1300x12=15,528,  „  16,000 

20.  Wrought  iron,  ............  1294x20  =  25,880,  „  26,000 

30.  Steel,  ..............  ..........  1294x30=38,820,  „  39,000 

Again  assuming  a  tensile  strength  of  7  tons  per  square  inch  for  ordinary 
castings,  — 

The  Ultimate  Transverse  Strength  of  the  Teeth  of  ordinary  Cast-Iron 
Wheels  in  terms  of  the  thickness,  is, 


/~~W 

Inversely,  =  d  A/-  -  ...........................   (  10  ) 

v  9000 

The  excess  of  transverse  strength  of  cast  iron,  in  thicknesses  of  from 
i  to  3  inches,  above  that  which  is  calculated  from  the  tensile  strength,  as 
detailed  at  page  555,  affords  a  margin  for  weak  forms  of  teeth,  and  for  the 
loss  of  strength  by  trimming  or  by  wear,  and  especially  by  the  removal  of  the 
skin.  The  excess,  it  is  true,  diminishes  as  the  thickness  increases  ;  but,  on 
the  contrary,  the  diminution  of  strength  is  less  by  the  wear  of  the  thicker 
teeth  than  by  that  of  the  thinner  teeth.  Thus,  there  is  a  natural  adjustment 
of  the  supply  of  strength  in  excess  to  the  requirement. 

WORKING  STRENGTH  OF  WHEEL-TEETH. 

It  is  usual  to  act  on  a  factor  of  safety  of  10,  for  wheel-teeth:  Formulas 
(  6  ),  (  7  ),  (  9  ),  and  (  10  ),  thus  adapted,  become,  — 


HORSE-POWER   OF  TOOTHED  WHEELS.  737 

For  the  Working  Strength  of  Wheel-teeth  of  ordinary  Cast  Iron  :  — 
In  terms  of  the  pitch,  .........  W  =  180  1^.  ......................   (  n  ) 

Do"  "P=V^-  -(I2> 

In  terms  of  the  thickness,.  .  .  .  W  =  900  d2  ........................  (  13  ) 

~ 


For  wheels  of  cast  iron  of  greater  strength,  or  of  gun-metal,  wrought  iron, 
or  steel,  the  coefficients  for  a  factor  of  safety  of  10,  are  one-tenth  of  those 
which  are  given  at  pages  735  and  736,  and  they  may  be  substituted  in  the 
above  formulas,  thus,  — 

FOR  WORKING  STRENGTH.  In  formulas  (n)  and  (12).     In  formulas  (13)  and  (14). 

Coefficient  for  gun-metal,  ..............  310     ............     1600 

Do.  wrought  iron,  ..........   520     ............     2600 

Do.  steel,  .....................  800     ............     3900 

Sir  William  Fairbairn  states  that,  for  wooden  teeth,  a  thickness  i^  times 
that  of  cast-iron  teeth  is  sufficient;  3/s  ths  of  the  pitch  goes  to  the  thick- 
ness of  the  wooden  cogs,  and  2/5  ths  to  that  of  the  iron  teeth  of  the  wheel 
geared  with  the  wooden  teeth.  There  is  no  clearance,  as  the  teeth  are 
accurately  trimmed. 

BREADTH  OF  THE  TEETH  OF  WHEELS. 

When  the  breadth  of  a  tooth  is  just  twice  its  whole  length,  its  ultimate 
resistance  to  transverse  stress  is  approximately  equal  to  its  resistance  to 
diagonal  stress  applied  at  one  corner.  A  greater  breadth  than  twice  the 
length,  therefore,  is  not  reckoned  to  add  to  the  transverse  resistance  of  the 
tooth;  but  it  is  necessary  for  durability. 

Breadth  in  relation  to  Working  Stress.  —  Tredgold  fixed  the  maximum 
average  working  stress  at  the  pitch-line  at  400  pounds  per  inch  of  breadth 
of  teeth.  Sir  William  Fairbairn  adopted  this  datum. 

HORSE-POWER  TRANSMITTED  BY  TOOTHED  WHEELS. 

A  horse-power  is  work  done  at  the  rate  of  33,000  pounds  through  i  foot, 
per  minute;  or  550  foot-pounds  per  second.  Let,— 

H  =  the  horse-power  transmitted. 

W  =  the  stress  in  pounds,  at  the  pitch-line. 

v  =  the  velocity  at  the  pitch-line,  in  feet  per  second. 

S  =  the  speed  in  turns  per  minute. 

D  =  the  diameter  in  feet. 


That  is  to  say;  —  ist.  The  horse-power  transmitted  is  equal  to  the  product 
of  the  stress  by  the  velocity  at  the  pitch-line,  divided  by  550.  2d.  The 
stress  at  the  pitch-line  is  equal  to  550  times  the  horse-power  divided  by  the 
velocity  at  the  pitch-line.  3d.  The  velocity  at  the  pitch-line  is  equal  to  550 
times  the  horse-power,  divided  by  the  stress. 


738  MILL-GEARING. 

The  speed  of  the  wheel,  in  turns  per  minute,  is  equal  to     v  x    °   ;  or 


That  is  to  say;  —  ist.   The  speed  of  the  wheel  is  equal  to  19.1  times  the 
velocity  at  the  pitch-line,  divided  by  the  diameter.     2d.   The  velocity  at  the 
pitch-line  is  equal  to  the  product  of  the  diameter  by  the  speed,  divided  by 
19.1. 
To  find  the  horse-power  in  terms  of  the  stress,  the  diameter  and  the  speed  ; 

W  v 
or  W,  D,  and  S.     By  formula  (15),  H  =  -  ;  and,  substituting  the  value  of 


10,500 


(20) 


That  is  to  say,  the  horse-power  transmitted  is  equal  to  the  product  of  the 
stress  at  the  pitch-line,  by  the  diameter  and  by  the  speed;  divided  by 
10,500. 

For  the  horse-power  in  terms  of  the  pitch.    Substitute  in  formula  (20),  the 
value  of  W  in  terms  of  the  pitch,  that  is,  by  formula  (n),  for  cast  iron, 


10,500 
(for  cast  iron)       H  =  — - — --   (  22  ) 

That  is  to  say,  the  horse-power  that  may  be  transmitted  by  a  cast-iron  wheel 
is  equal  to  the  product  of  the  square  of  the  pitch  by  the  diameter  and  by 
the  speed;  divided  by  58.3. 

The  horse-power  per  foot  of  diameter  and  per  turn  per  minute,  is 

P2 
(for  cast  iron)       H  = ;   (  23  ) 

being  equal  to  the  square  of  the  pitch  divided  by  58.3. 

The  formulas  (  22  )  and  (  23  )  are  available  for  the  calculation  of  the 
horse-power  of  wheels  made  of  other  metals,  by  using  the  proper  constants, 
as  follows: — 

FOR  HORSE-POWER.  In  Formulas  (22)  and  (23). 

Coefficient  for  gun-metal 10,500  ^310  =  33.9 

Do.          wrought-iron 10,500-^-520  =  20.2 

Do.         steel 10,500^-800  =  13.1 

Table  No.  258  gives  particulars  of  dimensions,  stress,  and  horse-power 
of  toothed  spur-wheels  of  ordinary  cast  iron,  for  various  pitches.  The 
thickness,  column  2,  is  given  as  45  per  cent  of  the  pitch;  the  working 


WEIGHT   OF   TOOTHED   WHEELS. 


739 


stress  at  the  pitch-line,  column  3,  is  calculated  from  the  pitch  by  formula 
(n);  the  horse-power  transmitted  at  i  foot  per  second,  column  6,  is 
calculated  from  the  stress  by  formula  (15),  and  the  horse-power  per  foot 
of  diameter  and  per  turn  per  minute,  in  the  last  column,  is  calculated  by 
formula  (23).  This  table  affords  data  for  performing  all  the  usual  calcula- 
tions for  the  horse-power  of  toothed  wheels.1 

Table  No.  258. — STRENGTH  AND  HORSE-POWER  OF  SPUR-WHEELS, 

Made  of  ordinary  cast  iron. 


BREADTH  OF  TEETH. 

HORSE-POWER 

TRANSMITTED. 

Pitch 

of 
Teeth. 

Thickness 
of 
Teeth. 

Working 
Stress  at 
Pitch-line. 

Least 

Usual 

At  i  Foot 
per  Second 

Per  Foot  of 
Diameter 

Breadth. 

Breadth. 

at  the 

and  per  Turn 

Pitch-line. 

per  Minute. 

inches. 

inches. 

inches. 

pounds. 



inches. 

H.  P. 

H.  P. 

thickness  X  2. 

I 

•45 

1  80 

.90 

2X 

•327 

.0172 

& 

1 

28l 
405 

1.  12 
1-34 

2.l/2  and  3 
3X  and  3^ 

.511 
.736 

.0268 
.0386 

l$£ 

•79 

551 

1.58 

4X 

1.  000 

.0525 

2 

.90 

720 

1.  80 

6 

1.310 

.0686 

for  400  Ibs. 

per  inch. 

2 

.90 

720 

1.  80 

6 

1.310 

.0686 

2X 

.01 

911 

2.28 

6 

1.656 

.0868 

2>£ 

.12 

II25 

2.8  1 

6 

2.045 

.1007 

2/4^ 

•24 

1361 

3-40 

7 

2.474 

.1297 

3 

•35 

l620 

4.05 

9 

2.945 

.1544 

3X 

.46 

1901 

4-75 

9 

3.456 

3/^s 

•57 

2205 

5.51 

10 

4.010 

.2100 

4 

.80 

2880 

7.20 

14 

5.236 

.2744 

4X 

2.02 

3645 

9.11 

6.627 

•3472 

5 

2.25 

4500 

11.25 

15  to  16 

8.182 

4028 

6 

2-73 

6480 

16.20 

16  to  18 

11.782 

.6176 

Note. — For  mitre  and  bevil  wheels,  the  mean  diameter,  breadth,  and  thickness  of  teeth 
are  to  be  used  in  calculation. 

WEIGHT  OF  TOOTHED  WHEELS. 

Spur-wheels. — The  weight  of  spur-wheels  of  usual  proportions,  per  inch 
of  breadth,  increases  in  a  ratio  greater  than  the  diameter,  but  less  than  the 
square  of  the  diameter.  As  ascertained  by  plotting  the  particulars  of  a 
great  number  of  wheels,  the  relation  of  the  diameter  and  the  weight  per 
inch  wide,  is  represented  by  the  general  expression  a  d+  b  d2,  in  which  d  is 
the  diameter,  and  a  and  b  are  constants  for  each  pitch. 

Within  the  ordinary  practical  limits  of  breadth  for  each  pitch,  the  weight 
varies  as  the  breadth  of  the  wheel,  and  may  be  taken  at  a  constant  per 
inch  of  breadth. 


1  The  author  is  aware  that  Tredgold,  Fairbairn,  and  others,  give  higher  values  for  the 
strength  and  power  of  the  teeth  of  wheels  than  he  gives  in  the  text. 


740 


MILL-GEARING. 


Table  No.  259. — WEIGHT  OF  CAST-IRON  SPUR-WHEELS, 

Per  Inch  of  Breadth. 


T}-4._T_ 

DIAMETERS  IN  FEET. 

.rltCn. 

•50 

•75 

i 

,.5  |    , 

2-5 

3 

4 

Usual  Breadth. 

inches. 

cwt. 

cwt. 

cwt. 

cwt. 

cwt. 

cwt. 

cwt. 

cwt. 

inches. 

6 

— 

— 

— 

— 

1.27 

1.66 

2.07 

2.97 

i6to  18 

5 

— 

— 

— 

— 

1.  08 

1.40 

•75 

2.52 

15  to  16 

4^ 

— 

— 

— 

.707 

.984 

1.28 

.60 

2.30 

H 

4 

— 

— 

— 

.638 

.888 

1.16 

•44 

2.07 

14 

3^ 

— 

— 

— 

.569 

•792 

1.03 

.29 

1.85 

10 

3X 

— 

— 

— 

•535 

•744 

.968 

.21 

•74 

9 

3 

— 

— 

.319 

.500 

.696 

.905 

•13 

.62 

9 

1% 

— 

.218 

.297 

.466 

.648 

.844 

.05 

•5i 

7 

2^ 

— 

.202 

.275 

•431 

.600 

.781 

.98 

.40 

6 

2X 

— 

.185 

•253 

•397 

.552 

.719 

.90 

.29 

6 

2 

.110 

.I69 

.231 

.362 

.504 

.656 

.82 

.18 

6 

i# 

.100 

•153 

.209 

.328 

.456 

•594 

•74 

.06 

4X 

i# 

.089 

•137 

.187 

•293 

.408 

•531 

.66 

•95 

(  3X  not  exc'ding  5  ft. 
I  •$%  above  5  feet. 

i* 

.077 

.121 

.165 

.259 

•  360 

.469 

•59 

.84 

\2j4  not  exc'ding  4  ft. 
(  3      above  4  feet. 

i 

.068 

,105 

.143 

.224 

.312 

.406 

•5i 

•73 

2j^ 

Ib. 

Ib. 

Ib. 

ib. 

ib. 

Ib. 

ib. 

ib. 

# 

6 

9 

13 

22 

32 

43 

55 

84 

2 

< 

6 

13 

22 

32 

43 

55 

84 

2 

% 

5 

8 

12 

20 

3i 

44 

— 

— 

IX 

x 

6 

10 

15 

27 

42 

— 

— 

— 

I# 

DIAMETERS  IN  FEET. 

Pitch. 

5 

6 

7 

8 

9 

10 

II 

12 

inches. 

cwt. 

cwt. 

cwt. 

cwt. 

cwt. 

cwt. 

cwt. 

cwt. 

6 

3.98 

5.09 

6.30 

7.63 

9.06 

1  0.60 

12.24 

14.00 

5 

3.38 

4.32 

5.36 

6.48 

7.70 

9.00 

10.40 

11.88 

4/2 

3.08 

3-94 

4.88 

5.90 

7.01 

8.20 

9-47 

10.82 

4 

2.78 

3-55 

4.40 

5-33 

6-33 

740 

8.55 

9-77 

3^ 

2.48 

3-i7 

3-93 

4-75 

5.64 

6.60 

7.62 

8.71 

3X 

2-33 

2.98 

3-69 

4.46 

5-30 

6.20 

7.16 

8.18 

3 

2.18 

2.78 

3-45 

4.18 

4.96 

5.80 

6.70 

7.66 

2X 

2.03 

2.59 

3.21 

3-89 

4.62 

5.40 

— 

— 

2K 

1.88 

2.40 

2.98 

3.60 

4.28 

5.00 

— 

— 

2X 

1-73 

2.21 

2-74 

3-31 

3-93 

4.60 

— 

— 

2 

1.58 

2.02 

2.50 

3.02 

3-59 

— 

— 

— 

I* 

1-43 

1.82 

2.26 

2.74 

— 

— 

— 

IX 

1.28 

1.63 

2.02 

— 

— 

— 

— 

— 

iX 

i-i3 

1.44 

— 

— 

— 

— 

— 

— 

I 

.98 

— 

— 

— 

— 

— 

— 

— 

For  pitches  of  i  inch  and  upwards,  the  weight  per  inch  of  breadth,  for  a 
given  diameter,  increases  directly  as  the  pitch. 


FRICTIONAL  WHEEL-GEARING.  741 

The  following  formulas  are  deduced  from  the  actual  weights  of  spur- 
wheels  of  from  fa  inch  to  4  inches  pitch,  and  from  6  inches  to  1  2  feet  in 
diameter. 

Weight  of  cast-iron  spur-wheels  of  from  i  inch  to  6  inches  pitch,  per 
inch  of  breadth  :  — 


x  (i+.io//)  ...................  (24) 

Weight  of  cast-iron  spur-wheels  of  pitches  less  than  i  inch  :  — 

For  y%  inch  pitch,  W  =  .  0935  ^+.0235  d*  .........  (  25  ) 

For  ^         „          W  =  .0935  </+.  0235  ^2  .........  (26) 

For  YZ         „          W  =   .  069^+  .0345  d*  .........  (27) 

For  fa         „          W  =    .o8o^+.o53o^2  .........  (28) 


W  =  the  weight  of  the  wheel  in  hundredweights  per  inch  of  breadth. 
d=  the  diameter  in  feet. 
/  =  the  pitch  in  inches. 

The  first  formula  would  probably  be  suitable  for  finding  the  weights  of 
wheels  up  to  20  feet  in  diameter. 

The  results  given  by  the  above  formulas  are  average  results. 

Mortise-wheels.  —  The  weight  of  mortise-wheel  castings  is  the  same  as  that 
of  spur-wheels,  having  the  same  leading  dimensions. 

B  evil-wheels  and  Mitre-wheels.  —  The  weight  is  less  than  that  of  spur-wheels 
of  the  same  leading  dimensions,  varying  from  two-thirds  or  three-fourths 
of  the  weight  of  spur-wheels,  for  the  larger  diameters,  to  about  seven-eighths 
for  the  smaller  diameters. 

The  table  No.  259  of  the  weight  of  spur-wheels  is  calculated  by  means 
of  formulas  (24)  to  (28). 


FRICTIONAL   WHEEL-GEARING. 

When  one  smooth  cast-iron  wheel  is  employed  to  drive  another  by  direct 
contact  at  the  circumferences,  the  adhesion  or  driving  force  is  produced  by 
interpressure  between  the  wheels,  and  may  be  taken  as  one-sixth  of  the 
pressure. 

Robertson's  grooved-surface  frictional-gearing  consists  of  wheels  or 
pulleys  geared  together  by  frictional  contact,  in  which  the  driving  surfaces 
are  grooved  or  serrated  annularly,  the  ridges  of  one  surface  entering  the 
grooves  of  the  other.  A  lateral  wedging  action  is  obtained,  which  aug- 
ments the  adhesion  of  the  surfaces,  as  compared  with  flat  friction-surfaces, 
in  the  ratio  of  9  to  i.  That  is,  the  grooved  wheels  require  a  force  of 
3  Ibs.  acting  at  their  circumference  to  make  them  slip,  for  every  2  Ibs. 
applied  on  the  axis;  whereas,  two  flat  surface-wheels  would  require 
(2x9-)  1 8  Ibs.  of  pressure  on  the  axis,  to  enable  them  to  resist  a  force  of 
3  Ibs.  acting  at  the  circumference. 

Compared  with  leather  belts,  frictional  gearing,  worked  under  a  pressure 
equal  to  the  tension  of  the  belts,  has  been  proved  to  have  greater  adhesive 
force: — in  one  experiment,  about  30  per  cent.  more. 


742  MILL-GEARING. 

The  grooves  are  made  of  V  shape,  for  which  50°  is  the  most  suitable 
angle.  The  pitch  of  the  grooves  is  varied  according  to  the  velocity  and 
the  power  to  be  transmitted : — from  ^  inch  to  3^  inch;  the  ordinary  pitch  is 
Y%  inch.  The  general  laws  of  friction  are  found  to  apply  to  the  action  of 
frictional  gearing.1 


BELT-PULLEYS   AND    BELTS. 

The  acceleration  or  reduction  of  the  angular  velocity,  or  speed,  of  shafts 
driven  by  means  of  belts  and  drum-pulleys,  is,  like  that  of  shafts  driven  by 
toothed  gearing,  in  the  inverse  ratio  of  the  diameters  of  the  pulleys;  and, 
when  speed  is  brought  up  by  means  of  successive  shafts  and  small  and 
large  pulleys,  the  ratio  of  the  initial  to  the  final  speed  is  the  product  of 


Fig.  324. — Belt-pulleys  and  Belts : — Multiplication  of  Speed. 

the  ratios  of  the  successive  accelerations  of  speed;  and  these  may  be 
expressed  in  terms  of  the  diameters  of  the  successive  pairs  of  pulleys. 
For  example,  the  driving  pulley  a,  36  inches  in  diameter,  on  the  shaft  A, 
makes  60  revolutions  per  minute,  and  drives  by  a  belt  the  1 2-inch  pulley  b 
on  the  shaft  B;  which  carries  the  36-inch  pulley  c,  which  drives  the  8-inch 
pulley  d  on  the  third  shaft  C.  The  speeds  are  calculated  thus : — 

Turns  per  Minute. 

Shaft  A 60 

Shaft  B  6ox^,  or  60x3  =  180 

12 

Shaft  C  60 x^x  ^,  or  60x3x4.5- 810 

12      8 

In  these  calculations,  it  is  assumed  that  there  is  no  slip  of  the  belt  on 
the  pulley ;  and  M.  Morin  supports  this  assumption.  Mr.  R.  H.  Buel 2  deduced 
from  direct  observation  that  the  reduction  of  speed  by  the  slip  or  "  creep," 
of  five  belts  employed  to  communicate  motion  from  an  engine-shaft  to  a 
distant  shaft,  through  intermediate  shafts  and  pulleys,  varied  from  just  one- 
half  to  one-quarter  per  cent.;  whilst  M.  Krest3  deduced  a  slip  for  one 
belt  only,  amounting  to  2  per  cent.  The  size  of  the  belts  observed  by 
Mr.  Buel  were  greatly  in  excess  of  those  actually  required  for  the  transmis- 
sion of  the  power. 

TENSILE  STRENGTH  OF  DRIVING  BELTS. 

Several  particulars  of  the  strength  of  belts  are  given  at  pages  679,  680.  It 
has  there  been  noticed  that  Messrs.  Briggs  and  Towne4  tested  the  strength  of 

1  See  a  paper  by  Mr.  James  Robertson  on  "Grooved-Surface  Frictional-Gearing,"  in 
the  Proceedings  of  the  Institution  of  Mechanical  Engineers,  1856. 
*  Journal  of  the  Franklin  Institute,  vol.  Ixviii.,  1874,  page  256. 
8  Annales  des  Mines,  1862.  4  Journal  of  the  Franklin  Institute,  January,  1868. 


BELT-PULLEYS   AND   BELTS.  743 

3-inch  leather  belts,  ?/32  inch  or  .219  inch  thick.  They  found  that  the 
weakest  parts  of  an  ordinary  belt  are  the  ends  through  which  the  lacing 
holes  are  punched,  and  that  the  belt  is  usually  weaker  than  the  lacing  itself. 
The  rivetted  splices  are  the  next  weakest  points.  The  strengths  of  new 
and  partially  used  belts  were  found  to  be  almost  identical. 

Ultimate  Tensile  Strength. 

T  ..  ,  Per  inch  of  Per  square  inch 

Width.  of  Section. 

At  lacings  ..................     629  Ibs.   ...   210  Ibs.  ...     958  Ibs. 

At  splicings  ................    1146    „     ...  382    „      ...    1744   „ 

At  solid  part  ...............   2025    „     ...  675    „     ...  3086   „ 

Messrs.  Briggs  and  Towne  adopt  200  Ibs.  per  inch  wide,  as  the  ultimate 
strength  of  laced  belts  .22  inch  thick;  and  they  take  a  third  of  this,  or 
66  z/z  Ibs.  per  inch  wide,  as  the  working  strength.  This  is  just  one-sixth  of 
the  working  stress,  400  Ibs.  per  inch  of  width,  adopted  for  toothed  wheels. 
It  is  equivalent  to  304  Ibs.  per  square  inch  of  section, 

M.  Morin  adopts  */$  kilogramme,  and  M.  Claudel  ^  kilogramme,  per 
square  millimetre;  equivalent  to  284  Ibs.,  and  355  Ibs.,  per  square  inch  of 
section,  for  the  working  strength  of  leather  belts. 

Dr.  Hartig  found,  from  the  results  of  experiments  made  by  him  in  a 
woollen  mill,  that  the  tension  on  the  driving  belts  varied  from  30  Ibs.  to 
532  Ibs.  per  square  inch  of  section,  and  that  it  averaged  273  Ibs.  per  square 
inch.1 

An  average  working  strength  of  300  Ibs.  per  square  inch  of  section  of 
leather  belts  may  be  accepted  for  purposes  of  calculation. 

CALCULATION  FOR  HORSE-POWER  TRANSMITTED  BY  LEATHER  BELTS. 

The  formulas  for  the  power  of  toothed  wheels,  (15)  to  (21),  pages 
737  and  738,  may  be  adapted  for  the  calculation  of  the  power  of  belts. 
Let 

H  =  the  horse-power. 

W  =  the  working  stress  transmitted  per  inch  wide,  in  pounds. 

b  =  the  breadth  of  the  belt,  in  inches. 
b'  =         do.  do.       in  feet. 

W  b  =  the  total  working  stress  transmitted,  in  pounds. 

v  =  the  velocity  of  the  belt,  in  feet  "per  second. 
v'  =         do.  do.       in  feet  per  minute. 

D  =  the  diameter  of  the  pulley,  in  feet. 

S  =  the  speed  of  the  pulley,  in  turns  per  minute. 

In  Terms  of  Horse-Power,  Working  Stress,  and  Velocity  of  Belt. 


550  v 

\ 

1  "Essais  Dynamometrique  "  (la  Laine  Cardee),  by  Dr.  Ernest  Hartig,  translated  by 
M.  E.  Simon.     Annales  du  Conservatoire  des  Arts  et  Metiers,  vol.  viii.,  page  6n. 


744  MILL-GEARING. 

In  Terms  of  the  Velocity  of  the  Belt,  and  the  Diameter  and  the  Speed 

of  the  Pulley. 


H-W**DS  ................................  (6) 

10,500 

io25op_H 
W£xD 

The  performances  of  belts  may  be  compared  by  calculating  the  number 
of  square  feet  of  belt-surface  passed  over  either  pulley  per  minute  per 
horse-power  :  —  involving  the  elements  of  working  stress  and  velocity.  It  is 
found  by  multiplying  the  velocity  in  feet  per  minute  by  the  breadth  of  the 
belt  in  feet,  and  dividing  the  product  by  the  horse-power  transmitted  :  — 

Belt-surface  described  per  Minute  per  Horse-Power  Transmitted. 


Belt-area  in  square  feet  =  =-;  .........  (8) 

H  H 

In  the  second  expression  of  value,  the  velocity  is  expressed  in  feet  per 
second,  and  the  breadth  is  in  inches. 

ADHESION  AND  POWER  OF  LEATHER  BELTS. 

The  normal  pressure  per  square  inch  of  a  belt  on  the  surface  of  a  pulley 
is  equal  to  the  quotient  of  the  tension  on  a  belt  i  inch  wide  by  the  radius 
of  the  pulley  in  inches.  Otherwise,  the  tension  on  a  belt  i  inch  wide,  is 
equal  to  the  product  of  the  normal  pressure  on  the  pulley  per  square  inch 
by  the  radius  in  inches.  This  is  the  same  equation  as  is  used  to  find, 
reversely,  the  action  of  steam  within  a  boiler  :  —  the  transverse  stress  on  each 
side  of  the  shell  of  the  boiler,  per  inch  of  length,  is  equal  to  the  product  of 
the  internal  pressure  per  square  inch  by  the  radius  in  inches. 

M.  Moriris  Experiments. 

M.  Morin1  ascertained  that,  when  one  pulley  is  driven  by  another,  by 
means  of  a  leather  belt,  the  sum  of  the  tensions  in  the  two  sides  of  the 
belt,  is  the  same  when  in  motion  and  at  rest.  The  tension  on  the  pulling 
side,  when  in  motion,  is  therefore  increased  by  as  much  as  the  tension  on 
the  return  side  is  reduced.  When  the  belt  just  slides  on  the  pulley  by 
tension,  the  relation  of  the  sliding  and  the  slack  tensions  is  expressed  by 
the  formula  — 

Log  T  =  log/  +0.434^  ..................   (9) 

T  =  the  greater  tension. 

/  =  the  slack  tension. 

c  =  the  coefficient  of  friction  between  the  band  and  the  pulley. 
C  =  the  length  of  the  arc  of  the  circumference  embraced  by  the  belt. 
R  =  the  radius  of  the  pulley. 

1  Aide-Memoire  de  Mecanique  Pratique,  1864;  page  289. 


BELT-PULLEYS   AND   BELTS. 


745 


RULE. — To  find  the  tension  just  sufficient  to  cause  a  leather  belt  to  slide  over 
a  pulley  with  a  given  slack  tension.  Divide  the  length  of  the  arc  embraced 
by  the  belt,  by  the  radius  of  the  pulley,  and  multiply  the  quotient  by  the 
coefficient  of  friction,  and  by  0.434;  add  the  product  to  the  logarithm  of 
the  slack  tension.  The  sum  is  the  logarithm  of  the  sliding  tension.1 

The  coefficients  of  friction  deduced  by  M.  Morin  are  as  follows : — 

For  leather  belts  in  ordinary  condition  on  wooden  pulleys 0.47 

For  new  belts  on  wooden  pulleys 0.50 

For  belts  in  ordinary  condition  on  cast-iron  pulleys,  either  turned  or  rough  0.28 

For  wet  belts  on  cast-iron  pulleys 0.38 

For  hemp  ropes  on  wooden  pulleys 0.50 

To  facilitate  calculation,  M.  Morin  gives  the  following  tablet  of  the 
ratios  of  the  sliding  tension  to  the  slack  tension,  for  various  proportions  of 
the  arc  of  the  circumference  embraced  by  the  band,  and  for  the  several 
coefficients  of  friction: — 

Tablet  A. 


RATIO  OF  SLIDING  TO  SLACK  TENSION. 

K. 

Ratio  of 

the  Arc 
Embraced, 
to  the 

New 

Belts  in  Ordinary 
Condition. 

Wet 

Cords  on  Wooden  Pulleys 
or  Winches. 

Circumfer- 
ence. 

Belts  on 
Wooden 
Pulleys. 

Wooden 
Pulleys. 

Cast-iron 
Pulleys. 

Belts  on 
Cast-iron 
Pulleys. 

Rough. 

Polished. 

.20 

1.87 

1.  80 

1.42 

I.6l 

1.87 

I.5I 

•30 

2.57 

243 

1.69 

2.05 

2-57 

1.86 

.40 

3-51 

3-26 

2.02 

2.6o 

3-51 

2.29 

.50 

4.8l 

4.33 

2.41 

3-30 

4.8l 

2.82 

.60 

6.59 

5.88 

2.87 

4.19 

6.58 

347 

.70 

9-00 

7.90 

343 

5.32 

9-01 

4.27 

.80 

12.34 

10.62 

4.09 

6.75 

12.34 

5.25 

.90 

16.90 

14.27 

4.87 

8.57 

16.90 

6.46 

1.  00 

23.14 

I9.l6 

5.81 

10.89 

23.90 

7-95 

1.50 

— 

— 

— 

— 

III.3I 

22.42 

2.00 

— 

— 

— 

— 

53547 

63.23 

2.50 

— 

— 

— 

— 

2575.80 

178.52 

Example. — The  slack  tension  of  a  belt  in  ordinary  condition,  on  a 
wooden  pulley,  is  100  Ibs.;  and  the  belt  embraces  half  the  circumference. 
Opposite  .50  in  column  i,  find  the  multiplier  4.38  in  column  3.  Then, 
100  x  4.38  =  438  Ibs.,  the  sliding  tension. 

It  is  found  by  experience  that  the  resistance  of  pulleys  to  sliding  of  belts 
is  independent  of  the  diameter. 

When  a  rope  is  wound  once  round  a  wooden  barrel,  it  is  seen  by  the 
table  that  the  resistance  is  24  times  the  pull  at  the  slack  end,  on  a  rough 
barrel;  and  that,  when  wound  2^  times  round  the  barrel,  the  resistance  is 
2575  times  the  pull. 

Calculation  of  the  Power  of  Belts,  by  M.  Morin' s  Data. — The  pull  available 
for  driving  is  equal  to  the  difference  of  the  sliding  and  slack  tensions,  or 

1  Messrs.  Briggs  and  Towne  show  clearly  how  this  formula  is  arrived  at ;  Journal  of  the 
Franklin  Institute^  January,  1 868,  page  1 8. 


746  MILL-GEARING. 

T-t  =  Kt-t  =  t  (K-  i),  in  which  K  is  the  ratio  of  T  to  /,  as  in  tablet  A; 
and, 


The  value  of  the  driving  pull,  or  the  difference  of  tensions  (T  -  /),  is  equal 

to  ^^  —  ,  in  which  H  is  the  horse-power,  and  v  the  velocity  of  the  belt  in 

v 

feet  per  second.     Substituting  this  value  for  (T  -  /)  in  equation  (  a  ), 


550 

That  is  to  say,  the  minimum  slack  tension  is  equal  to  550  times  the  horse- 
power, divided  by  the  velocity,  and  by  the  ratio  of  the  sliding  to  the  slack 
tension  minus  i. 

And,  the  available  horse-power  is  equal  to  the  product  of  the  slack  tension 
by  the  velocity  of  the  belt,  and  by  the  ratio  of  the  tensions  minus  i  ;  divided 

by  55°- 

M.  Morin  recommends  that  the  value  of  the  slack  tension,  formula  (10), 
should  be  increased  by  Vioth,  to  cover  the  friction  of  the  journals. 

M.  Claudel's  Data  for  Belts. 

M.  Claudel  gives  the  following  empirical  formula,  in  common  use,  for 
finding  the  breadth  of  a  leather  belt  enveloping  half  the  circumference  of 
a  pulley.  Altering  the  measures  :  — 


in  which  #  =  the  breadth  in  inches,  H  =  the  horse-power,  ^  =  the  speed  of 
the  belt  in  feet  per  second;  and  c  a  constant,  26  for  upright  shafts,  and  20 
for  horizontal  shafts. 

This  formula  gives  values  for  the  breadth,  averaging  about  double  what 
would  be  given  by  the  table  No.  261,  when  the  belt  laps  half  round  the 
pulley.  The  belt  then  works  to  half  its  power;  and  M.  Claudel  instances 
the  common  experience  that  a  belt  3  ^  inches  broad,  moving  at  a  velocity 
of  9  feet  per  second,  can  very  well  transmit  i  horse-power,  with  ordinary 
tension,  and  without  overstraining,  working  on  turned  and  smooth  pulleys 
of  equal  diameter.  This  example,  if  adopted  as  a  basis,  would  give  a 
coefficient  of  29  in  formula  (12).  The  working  tension  is  only  about 
20  Ibs.  per  inch  wide. 

At  the  same  time,  the  values  given  by  the  empirical  formula  (12)  are 
little  more  than  those  deducible  from  the  data  of  M.  Morin. 

Mr.  Evan  Leigh's  Rules  for  Belting}- 

Mr.  Leigh  is  of  opinion  that  a  main  driving-belt,  to  be  rightly  applied, 
should  pass  through  3000  or  4000  lineal  feet  per  minute;  and  should  be 
of  sufficient  width  to  drive  all  the  machinery  and  shafting  to  be  driven,  quite 

1  The  Science  of  Modern  Cotton  Spinning,  1873,  PaSe  37- 


BELT-PULLEYS  AND   BELTS.  747 

easily,  running  in  a  slack  condition.     The  belt  should  work  from  the  peri- 
phery of  the  fly-wheels  of  quick-running  engines. 

RULE  i. — To  find  the  horse-power  of  a  main  driving  double  belt,  working 
slackly  and  easily.  Multiply  the  number  of  square  inches  covered  by  the  belt, 
on  the  surface  of  the  driven  pulley,  by  half  the  speed  of  the  belt  in  feet  per 
minute;  and  divide  the  product  by  33,000.  The  quotient  is  the  horse-power. 

RULE  2. —  To  find  the  proper  width  of  a  main  driving  double  belt  for  a  given 
horse-power.  Multiply  the  horse-power  by  33,000,  and  divide  the  product 
by  the  length  in  inches  of  periphery  of  driven  pulley  covered  by  the  belt, 
and  by  half  the  speed  of  the  belt  in  feet  per  minute.  The  quotient  is  the 
width  in  inches. 

For  existing  establishments,  where  it  is  desired  not  to  disturb  actual 
arrangements,  the  following  rule,  for  single  belts,  approaches  nearer  to 
ordinary  practice : — 

RULE  3. — To  find  the  width  of  a  single  belt  for  any  given  horse-power 
[actual  practice].  Multiply  the  horse-power  by  33,000,  and  divide  the 
product  by  the  length  in  inches  of  periphery  of  the  smaller  pulley  covered 
by  the  belt,  and  by  the  speed  of  the  belt  in  feet  per  minute.  The  quotient 
is  the  width  in  inches. 

By  this  rule,  which  is  based  on  ordinary  practice,  single  belts  are  calcu- 
lated to  do  twice  as  much  duty  as  double  belts  of  the  same  width  by  Rule  2 ; 
and,  comparatively,  the  stronger  double  belts,  calculated  by  Rules  i  and  2, 
have  exceedingly  easy  work.  Hence  their  great  durability.  Applying  Rule  i 
to  the  second  example  of  wide  double  belts  quoted  from  Mr.  Cooper, 
below : — there  are  two  driven  pulleys,  7  feet  in  diameter,  lapped  by  the  belt 
for  2/5  ths  of  their  circumference,  equal  to  8.8  feet,  or  105.6  inches  on  each; 
the  width  of  each  belt  is  14^  inches,  and  105.6  x  14.5  x  2  =  3062  square 
inches  covered.  The  velocity  of  the  belts  is  3498  feet  per  minute;  and 
3062x3498^2^33,000=162.3  horse-power.  The  horse-power  actually 
transmitted  by  the  two  belts  together  is  250,  or  fully  a  half  more  than  is 
allowed  by  Mr.  Leigh's  rule ;  yet  those  belts  are  said  to  have  lasted  upwards 
of  2  2  years. 

Examples  of  Very  Wide  Belts. 

Mr.  J.  H.  Cooper1  details  some  examples  of  the  performance  of  belts  in 
taking  off  the  power  of  steam  engines  from  the  fly-wheel  to  one,  two,  or 
three  line-shafts — 


Fig.  325. — Main  Driving  Belts,  with  intermediate  gearing. 

i.  Horizontal  condensing  Corliss  engine,  with  intermediate  gearing,  at 
Conestoga  Mills,  No.  2,  Lancaster,  Pa.     Three  belts.     Fig.  325. 


Journal  of  the  Franklin  Institute,  vol.  Ixviii.,  1874,  page  256. 


748  MILL-GEARING. 

2.  Engine  of  the  same  class;   belts  driven  by  fly-wheel.     Two  belts. 

Fig.  326. 

3.  Horizontal  Corliss  engine;  belt  driven  by  fly-wheel.     One  belt. 


Fig.  326.  Fig.  327. 

Main  Driving  Belts,  off  the  fly-wheel. 

4.  Horizontal  Corliss  engine  at  Manayunk;   belts  driven  by  fly-wheel. 

Three  belts.     Fig.  327. 

5.  Corliss  engine,  at  Great  Bend,  Indiana.     One  belt. 

The  chief  particulars  and  results  are  reduced  in  table  No.  260. 

No.  i  belts  have  been  at  work  upwards  of  twenty  years;  No.  2,  upwards 
of  twenty-two  years ;  No  3  has  been  eight  or  nine  years  at  work.  Nos.  i 
to  4,  double- thickness  belts,  transmit  a  tension  of  from  50  to  86  Ibs.  per 
inch  wide,  or  from  25  to  43  Ibs.  per  inch  wide  for  one  thickness.  The 
single-thickness  belt,  No.  5,  averages  126  Ibs.  per  inch  wide;  and  Mr. 
Cooper  mentions  a  6-inch  belt  employed  to  drive  a  wheel-forcing  press 
on  a  24-inch  pulley,  transmitting  a  stress  of  104  Ibs.  per  inch  wide. 

Messrs.  Briggs  and  Townees  Experiments. 

Messrs.  Briggs  and  Towne's  experiments  on  the  friction  or  adhesion  of 
leather  belts,  were  made  with  3-inch  belts,  and  the  arc  of  contact  equal  to 
1 80°,  on  ordinary  cast-iron  pulleys.  The  average  results  of  168  experiments, 
under  tensions  of  from  7  to  no  Ibs.  per  inch  of  width  of  belt,  gave  a 
sliding  tension  6.294  times  the  slack  tension,  and  a  friction  co-efficient  of 
0.5833.  To  cover  the  contingencies  of  temperature  and  moisture  of  the 
atmosphere,  they  adopt  only  6/i0ths  of  6.294,  or  3.776  as  the  maximum 
practical  value,  giving  a  friction  co-efficient  of  0.423.  These  values  are  by 
one-half  greater  than  those  of  M.  Morin. 

Messrs.  Briggs  and  Towne  calculate  the  maximum  working  stress  per  inch 
of  width,  transmitted  by  belts  having  a  working  strength  of  66  2/3  Ibs.  per 
inch  of  width,  by  means  of  the  formula  deduced  from  Rankine's  formula 
(Applied  Mechanics),  and  based  upon  the  foregoing  data : — 


w  = 


(12) 


W  =  the  working  stress  transmitted  per  inch  of  width,  in  pounds. 
a  =  the  arc  of  contact,  in  degrees. 


BELT-PULLEYS   AND   BELTS. 


749 


Table  No.  260. — FIRST-MOTION  DRIVING  BELTS  IN  THE  UNITED  STATES. 

.     (Mr.  Cooper.) 


Cylinder. 

Diameter 
and 
Stroke. 

Revolu- 
tions 
per 
Minute. 

Fly-wheel. 
Diameter. 

Intermediate 
Gearing. 

Driven  Pulleys. 

Number  and 
Diameter. 

Speed 
per 

Minute. 

ins.      ft. 

turns. 

feet. 

ft.     ins. 

turns. 

I 

30X6 

^2  ]/2 

spur,  22; 

spur,...         Q  75^) 

25  tons 

3  pulleys,     9  6      > 

3  pulleys,  5  ft. 

228 

speed,,.  ..120         ; 

2 

28x5 

SO/2 

pulley,  22;  ) 
17  tons    ) 

2     „         7  » 

159 

3 

16x4 

6S 

14 

T                         r 

182 

(  i               6 

2o8 

4 

30x5 

52 

24 

(  2       I            8  V 

156 

5 

18x4 

65 

12 

i      ,,3  ft.  6  in 

223 

Table  continued. 

Velocity  of  Belt. 

Tension  on  Belt. 

Belt 

Thickness  and  Width 
of  Belt. 

Horse- 
power 
Trans- 
mitted. 

Surface 
per  Min- 
ute per 

H.P. 

Feet  per 
Minute. 

Feet  per 
Second. 

Total. 

Per  inch 
Wide. 

feet. 

feet. 

H.  P. 

Ibs. 

Ibs. 

sq.  feet. 

,   f 

double,  23  yz  ins. 

3582 

59-7 

125 

1152 

49 

54-9 

1   / 

»      29       „ 

3582 

59-7 

175 

1612 

56 

49-5 

•1 

„       I4K    „ 
»       HX 

3498 
3498 

58.3 
58.3 

125 

125 

1180 
1180 

81 

33-6 
33-6 

3 

»       12 

2859 

47-7 

90 

1038 

86.5 

31-8 

f 

»       17 

*J 

„       26^ 

. 

total, 

3920 

65.3 

457 

3849 

59 

46.5 

65 

5 

single,   22       „ 

2453 

2555 

116 
136 

23.6 

20.2 

Concluding  Table  of  the  Driving  Power  of  Leather  Belts. 

On  the  whole,  it  may  be  concluded  that  Messrs.  Briggs  and  Towne's  data 
afford  a  satisfactory  basis  for  the  application  of  general  practical  rules. 
Table  No.  261  gives  particulars  of  the  practical  driving-power  of  leather 
belts  .22  inch  thick,  per  inch  wide,  based  on  their  data  for  arcs  of  contact, 
of  from  90°  to  270°,  assuming  66^5  Ibs.  per  inch  wide,  as  the  maximum 
working  strength.  The  maximum  transmitted  working  stress,  calculated 
by  them  by  means  of  formula  (12),  is  given  in  column  2 ;  the  3d  and 
4th  columns  were  calculated  by  means  of  formulas  (  i  )  and  (  6  ) ;  and  the 
horse-power  in  column  4,  multiplied  by  33,000,  gives  column  5.  The  sum 
of  the  tensions,  in  column  6,  is  calculated  by  adding  to  the  transmitted 
stress  in  column  2,  twice  the  difference  between  it  and  66^.  Thus,  for  the 
first  case,  66 2/3 -32.33  =  34.33,  which  is  the  slack  tension,  and  32.33  + 


750 


MILL-GEARING. 


(34.33  x  2)=  101.00,  in  column  6.  The  last  column  gives  the  resultant 
stress,  by  the  parallelogram  of  forces,  caused  by  the  tensions  of  the  belt,  on 
the  bearings  of  the  shaft. 

Messrs.  Briggs  and  Towne  give  many  instances  in  practice,  in  corrobora- 
tion  of  their  deductions. 

Table  No.  261. — DRIVING-POWER  OF  LEATHER  BELTS. 

Maximum  Working  Strength  66%  Ibs.  per  inch  wide,  single  thickness,  .22  inch. 
(Based  on  Messrs.  Briggs  and  Towne's  data. ) 


A  „_.. 

Maximum 

"\\7rtvlrtMrr 

Power  transmitted,  per  inch  wide. 

Sum  of  the 

Resultant 

Arcs 
of 
Contact. 

VvorKing 
Stress  trans- 
mitted, per 
inch  wide. 

At  i  foot 
per  second, 
Velocity  of  Belt. 

Per  foot  of  Diameter 
of  Pulley,  and 
per  Turn  per  Minute. 

on  both  sides 
of  a  Belt, 
per  inch  wide. 

x  ressure  on 
the  Journals, 
per  inch 
width  of  Belt. 

degrees. 

Ibs. 

horse-power. 

horse-power. 

foot-lbs. 

Ibs. 

Ibs. 

90° 

32.33 

.059 

.00308 

102 

101.00 

71.42 

100 

34.80 

.063 

.00331 

109 

98.53 

7547 

no 

37-07 

.067 

•00353 

116 

96.26 

78.85 

120 

39.18 

.071 

.00373 

123 

94.15 

81-53 

135 

42.06 

.076 

.00400 

132 

91.27 

84.32 

150 

44.64 

.O8l 

.00425 

140 

88.69 

85.67 

180 

49.01 

.089 

.00467 

154 

84.32 

84.32 

210 

52.52 

.095 

.00500 

165 

80.8  1 

78.05 

240 

55-33 

.100 

.00527 

174 

78.00 

67.59 

270 

57-58 

.105 

.00548 

181 

75-75 

53-56 

Note. — The  thickness  of  belt  is  .22  inch  for  a  maximum  working  strength  of  66%  Ibs. 
per  inch  wide.  For  any  other  thickness  the  data  in  the  table  are  to  be  altered  in  the  ratio 
of .  22  to  the  thickness. 

INDIA- RUBBER  BELTING. 

Driving  Belts1  manufactured  from  American  cotton  canvas,  cemented  in 
layers  by  vulcanized  india-rubber,  and  coated  with  the  same  material,  have 
been  tested  for  strength  and  adhesion.  It  is  stated  that  a  strip  i  inch 
wide  bears  a  tensile  stress  of  200  Ibs.,  and  that  the  india-rubber  belt 
possesses  about  three  times  the  surplus  or  effective  adhesion  of  leather 
belts. 

WEIGHT  OF  BELT-PULLEYS. 

The  weight  of  pulleys  of  the  same  diameter,  varies  within  much  wider 
limits  than  that  of  spur-wheels,  not  merely  because  the  breadth  varies  very 
much,  but  also  that  there  is  greater  variation  per  inch  of  breadth.  The 
following  formulas  for  the  weight  of  drum-pulleys  are,  therefore,  the  expres- 
sion of  average  weights  for  pulleys  of  medium  proportions.  For  pulleys 
designedly  strong  and  heavy,  up  to  30  inches  in  diameter,  the  weight  per 
inch  of  breadth  may  be  as  much  as  25  per  cent,  more  than  the  average,  or, 
for  particularly  light  pulleys,  as  much  lighter.  As  the  diameter  increases, 
the  percentage  of  variation  diminishes;  and  for  6-feet  or  y-feet  pulleys, 
it  may  never  exceed  10  per  cent,  either  way. 


1  Manufactured  by  the  North  British  Rubber  Company. 


BELT-PULLEYS   AND   BELTS. 


751 


The  author  is  indebted  to  Mr.  R.  Heber  Radford  for  the  examples  of  the 
actual  weights  of  pulleys  as  used  in  the  Sheffield,  Manchester,  and  Bradford 
districts,  given  in  tables  No.  262,  263,  and  264.  The  averaged  weights  per 
inch  wide  of  the  Sheffield  and  the  Manchester  finished  pulleys,  in  terms  of  the 
diameter  ranging  from  i  foot  to  4  feet,  are  found,  by  plotting,  to  be  expressed 
by  the  same  formula.  For  the  pulleys  of  the  Bradford  district,  the  formula 
is  slightly  different  from  that;  but  the  chief  interest  of  the  Bradford  ex- 
amples, consists  in  the  data  they  afford  of  the  reduction  of  the  weight  of 
the  rough  castings,  by  the  operations  of  turning,  boring,  and  slotting. 
From  these  data,  the  following  formulas  have  been  deduced,  showing  that 
the  increase  of  weight  per  inch  wide,  is  simply  as  the  increase  of  diameter : — 


Table  No.  262. — WEIGHT  OF  FINISHED  CAST-IRON  PULLEYS, 
SHEFFIELD  DISTRICT. 

(Examples  contributed  by  Mr.  R.  Heber  Radford.) 


WEIGHT, 

WEIGHT, 

•tir-  j.  r 

turned,  bored,  and  slotted. 

TIT;  J*.L 

turned,  bored,  and  slotted. 

Diameter. 

Width. 

Total. 

Per  inch 
wide. 

Diameter. 

VYiutn. 

Total. 

Per  inch 

wide. 

feet.     ins. 

inches. 

Ibs. 

Ibs. 

feet.     ins. 

inches. 

Ibs. 

Ibs. 

O 

6 

28 

4-7 

2      6 

10 

224 

22.4 

3 

6 

63 

10.5 

2      6 

12 

232 

19-3 

3 

10 

86 

8.6 

2      8 

9 

110 

12.2 

3 

12 

102 

8.5 

2  \\y2 

6 

140 

23-3 

5 

7 

66 

94 

3    o 

8 

120 

15.0 

6 

9 

75 

8.3 

3    o 

12 

1  08 

9.0 

1      9 

9 

80 

9.0 

3    o 

H 

136 

97 

1      9 

10 

84 

8.4 

3    5 

8 

160 

20.0 

2        0 

6/2 

114 

17.5 

3    6 

6 

160 

26.7 

2        O 

10 

120 

12.0 

3    7 

12 

175 

14.6 

2        0 

10 

I58 

I5.8 

4    o 

8 

212 

26.5 

2        0 

16 

170 

10.6 

4    o 

8 

225 

28.1 

2        5 

6 

110 

18.3 

4     i# 

12 

338 

28.2 

Table  No.  263. — WEIGHT  OF  FINISHED  CAST-IRON  PULLEYS, 
MANCHESTER  DISTRICT. 

(Examples  contributed  by  Mr.  R.  Heber  Radford.) 


Dia- 

WEIGHT  —  turned, 
bored,  and  slotted. 

Dia- 

WEIGHT— turned, 
bored,  and  slotted. 

Diameter. 

Hole. 

Total 

Per  inch 
wide. 

Diameter. 

Width. 

meter  of 
Hole. 

Total. 

Per  inch 
wide. 

feet.    ins. 

inches. 

inches. 

Ibs. 

Ibs. 

feet.   ins. 

inches. 

inches. 

Ibs. 

Ibs. 

i     4 

3 

2^ 

28 

9-3 

2      6 

6 

2^ 

105 

17-5 

2      0 

6 

2^ 

74 

12.3 

2      6 

8 

2 

116 

14.5 

2      0 

6 

2 

*    76 

12.7 

3    o 

6 

2^ 

129 

21.5 

2      0 

8 

2^ 

84 

10.5 

3     o 

6 

2 

134 

22.3 

2      2 

6 

2 

87 

14.5 

3    6 

6 

2^ 

137 

22.8 

2      3 

6 

2^ 

93 

15-5 

3    6 

6 

2 

144 

24.0 

752 


MILL-GEARING. 


Table  No.  264. — WEIGHT   OF   ROUGH   CASTINGS,  AND  FINISHED  CAST- 
IRON  PULLEYS,  BRADFORD  DISTRICT. 

(Examples  contributed  by  Mr.  R.  Heber  Radford.) 


WEIGHT, 

WEIGHT, 

Diameter. 

Width. 

Diameter 
of  Hole. 

Rough  Castings. 

turned,  bored,  &  slotted, 

Reduction 
of  Weight. 

Total. 

Per  inch 
wide. 

Total. 

Per  inch 
wide. 

feet,     inches. 

inches. 

inches. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

per  cent. 

10 

3 

I# 

16 

5-3 

13 

4-3 

19 

0 

4X 

2 

21 

5.0 

18 

4.0 

14 

2 

4 

2 

31 

775 

27 

6.75 

13 

4 

4^ 

2X 

44 

9.26 

38 

8.0 

13-6 

6 

5X 

2^ 

63 

11.4 

53 

9.6 

16 

8 

9 

2^ 

104 

1  1.6 

92 

10.2 

11.5 

10 

10 

2^ 

132 

13.2 

118 

ii.  8 

10.6 

Weight  of  Pulleys  per  inch  wide,  in  the  Lancashire  and  Yorkshire  Districts, 
from  i  foot  to  4  feet  in  Diameter. 

Rough  Castings, W  =  7.625^- 1.5   (13) 

Turned  and  Finished  Pulleys, W=         7^-1.75   (J4) 

W=the  weight  of  the  pulley  in  pounds  per  inch  wide. 
d=  the  diameter  in  feet. 

Note. — These  formulas  are  probably  applicable  for  pulleys  of  from 
10  inches  to  10  feet  in  diameter. 

From  the  weights  of  a  very  large  number  of  rough  castings  of  pulleys, 
ranging  from  i  foot  to  7  feet  in  diameter,  as  used  in  the  London  district, 
supplied  by  Mr.  Charles  Mackintosh,  the  following  formulas  have  been 
deduced.  For  rough  castings  above  2  feet  in  diameter,  the  weight  in- 
creases simply  as  the  diameter  increases.  For  diameters  less  than  2  feet, 
the  weight  increases  with  the  square  of  the  diameter.  The  same  propor- 
tional reduction  for  the  finished  weight  may  be  applied  to  the  London 
pulleys  as  was  done  to  the  Lancashire  pulleys : — 

Weight  of  Pulleys  per  inch  wide,  in  the  London  District,  from  i  foot  to 
7  feet  in  diameter. 


-o       ,  (  not  exceeding  2  feet  in  diameter, W  =  3*/2  +  3 

ngs  \  2  feet  in  diameter  and  upwards,....  W=  12^^-9.5 


(15) 
(16) 

Turned  and  (  not  exceeding  2  feet  in  diameter,  W  =  3  dz  -  .625  d+  2.75  ( 17 ; 
finished  pulleys  (  2  feet  in  diameter  and  upwards,  W  =  1 1.625  d-  9.25 ( 18  ) 

The  weights  of  pulleys,  rough  as  cast,  and  turned  and  finished,  have  been 
calculated  by  means  of  the  foregoing  formulas,  for  diameters  increasing 
from  10  inches  to  8  feet;  given  in  table  No.  265.  It  is  apparent  that  the 
London  pulleys  are  much  heavier  than  the  country  pulleys.  The  reduc- 
tion of  the  weight  of  the  rough  castings  by  turning  and  finishing,  varies 
from  13  per  cent,  for  1 2-inch  pulleys,  to  10^  per  cent  for  2-feet  pulleys, 
and  9  per  cent,  for  8-feet  pulleys,  for  the  country  pulleys ;  and  from  1 5  to 
7  per  cent,  for  the  London  pulleys. 


BELT-PULLEYS  AND  BELTS. 


753 


Table  No.  265. — BELT  PULLEYS — CALCULATED  WEIGHTS. 


Lancashire 
and  Yorkshire. 

London. 

Lancashire 
and  Yorkshire. 

London. 

Diameter. 

Weight 
per  inch  wide. 

Weight 
per  inch  wide. 

Diameter. 

Weight 
per  inch  wide. 

Weight 
per  inch  wide. 

Rough 
Castings. 

Turned 
and 
Finished. 

Rough 
Castings. 

Turned 
and 
Finished. 

Rough 
Castings. 

Turned 
and 

Finished. 

Rough 
Castings. 

Turned 
and 
Finished. 

inches. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

feet. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

10 

4.8 

4.08 

5-i 

4-33 

2.75 

19.5 

17.50 

24.2 

22.2 

II 

5-5 

4.67 

5-5 

4.68 

3 

21.4 

19.25 

27.2 

25.1 

12 

6.1 

5-25 

6.0 

5-13 

3-25 

23-3 

21.00 

30.2 

27-9 

!3 

6-7 

5-83 

6.6 

5-67 

3-5 

25.2 

22.75 

33-2 

30.1 

H 

7-5 

6.42 

7-i 

6.12 

4 

29.0 

26.25 

39-5 

36.8 

15 

8.0 

7.00 

7-7 

6.67 

4-5 

32-8 

29.75 

45-5 

42.4 

16 

8.7 

7-58 

8.3 

7.22 

5 

36.6 

33-25 

51-5 

48.1 

18 

9-9 

8.75 

9-7 

8.51 

5-5 

40.4 

36.75 

57-5 

53-8 

20 

II.2 

9.91 

n-3 

10.00 

6 

44.2 

40.25 

64.0 

60.0 

21 

11.6 

10.50 

12.2 

10.9 

6.5 

48.1 

43-75 

70.0 

65.7 

feet. 

2 

13-7 

12.25 

15.0 

13-50 

7 

51.9 

47-25 

76-3 

71.7 

2.25 

15-7 

14.00 

18.1 

16.4 

7-5 

55-7 

50-75 

82.5 

77-3 

2-5 

17.6 

15-75 

21.  1 

19-3 

8 

59-5 

54.25 

90.5 

84-3 

ROPE-GEARING.1 

Round  hemp-ropes  working  in  grooved  wheels  are  occasionally  employed 
instead  of  belts  or  toothed  wheels  for  transmitting  power  from  the  engine. 
The  fly-wheel  is  made  considerably  wider  than  a  spur  fly-wheel  would  be, 
but  rather  less  than  a  belt-wheel  would  be,  and  V  grooves  are  turned  out 
of  the  circumference,  the  sides  of  which  are  at  an  angle  of  40°,  and  the 
number  and  size  of  which  are  regulated  by  the  quantity  of  power  to  be 
taken  off.  The  ropes  are  usually  5^  and  6^  inches  in  circumference  for 
larger  powers,  and  4^  inches  for  smaller  powers.  To  prevent  wear  and 
tear  of  rope,  the  circumference  or  the  diameter  of  a  pulley  should  be  at 
least  30  times  that  of  the  rope,  and  the  shafts  should  be  at  a  distance  apart 
of  from  20  to  60  feet. 

The  number  of  ropes  required  for  the  transmission  of  a  given  power  is 
determined  from  the  circumferential  velocity  of  the  fly-wheel,  which  is  gener- 
ally between  3000  and  6000  feet  per  minute.  Mr.  Durie  instances  the 
rope-gearing  of  Messrs.  Nicoll's  factory  at  Dundee,  which  was  erected  in 
1870.  The  power  of  the  engine  varies  from  400  to  425  indicator  horse- 
power. The  fly-wheel,  22  feet  in  diameter,  makes  43  turns  per  minute, 
with  a  surface  velocity  of  2967  feet  per  minute;  it  is  4  feet  10  inches  wide, 
and  has  18  grooves,  each  of  which  is  occupied  by  a  6}^-inch  rope,  trans- 
mitting the  power  of  say  23  indicator  horse-power.  Five  ropes  are 
employed  to  transmit  the  power  to  the  ground  floor,  over  a  7^-feet  pulley; 

1  See  a  paper  read  by  Mr.  James  Durie,  at  the  Institution  of  Mechanical  Engineers, 
published  in  Engineering,  November  3,  1876,  page  394. 

48 


754  MILL-GEARING. 

four  ropes  drive  a  5^-feet  pulley  on  the  first  floor;  and  six  ropes  drive  a 
5  ^-feet  pulley  on  the  second  floor.  Lastly,  on  the  other  side  of  the  engine- 
shaft,  the  power  is  transmitted  by  three  ropes  to  a  weaving-shed,  on  a  7%- 

feet  pulley.     For  23  horse-power,  the  stress  on  each  rope  is  (  33°°°  x  23  _  ^ 

256  Ibs.,  less  the  resistance  of  the  engine.  When  the  ropes  become  too  slack, 
they  are  cut  and  re-spliced,  and  the  work  of  a  rope  under  such  treatment 
is  temporarily  performed  by  the  other  ropes  driving  the  same  pulley. 

In  another  example  40  indicator  horse-power  is  disposed  of,  for  each 
6^2 -inch  rope,  at  a  velocity  of  3784  feet  per  minute;  and  the  equivalent 

stress  is  (33°°°  x  4°  _  ^  ^49  Ibs.  for  each  rope. 
3784 

Taking  the  ultimate  strength  of  a  6^-inch  rope  at  10  tons,  or  22,400  Ibs., 
it  would  appear  that  the  working  stress  is  only  about  i  yz  per  cent,  of  the 
ultimate  strength;  giving  a  factor  of  safety,  67. 

It  is  believed  that  an  economy  of  power  is  effected  by  the  substitution 
of  rope-gearing  for  toothed-gearing.  A  6^-inch  rope  is  equivalent,  accord- 
ing to  Mr.  Durie,  to  a  leather  belt  4  inches  wide,  for  the  transmission  of 
work,  at  say  3000  feet  per  minute.  From  some  comparative  experiments 
made  by  Mr.  W.  A.  Pearce,  Dundee,  it  appears  that  a  6-inch  rope  in 
a  grooved  pulley  possessed  four  times  the  adhesive  resistance  to  slipping 
exhibited  by  a  half-worn  ungreased  4-inch  single  belt. 

The  ropes  used  for  gearing  are  made  of  carefully  selected  hemp : — the 
fibres  very  long,  well  twisted  and  laid,  yet  soft  and  elastic.  The  splice 
should  be  uniform,  of  the  same  diameter  as  the  rope,  and  9  or  10  feet  long. 

TRANSMISSION  OF  POWER  BY  ROPE  TO  GREAT  DISTANCES. 

Wire-Ropes. — M.  Him,  in  1850,  made  many  trials  with  endless  bands 
of  steely  iron  passing  over  pulleys  for  the  conveyance  of  power  to  great 
distances;  but  he  finally  adopted  iron  wire-ropes,  unannealed,  working  over 
grooved  pulleys  of  large  diameter.  M.  Umber,1  in  1859,  described  the 
apparatus.  The  pulleys  may  be  of  hard  wood;  they  are  formed  with  a 
groove  slightly  rounded,  about  2  inches  deep  and  i  yz  inches  wide,  lined  at 
the  bottom  with  leather  or  gutta-percha.  They  should  be  at  least  i  metre, 
or  3.28  feet,  in  diameter,  and  should  be  driven  at  the  greatest  practicable 
speed.  A  diameter  equal  to  200  times  that  of  the  cable,  is  the  most  suitable 
proportion.  The  distance  apart  of  the  driving  and  the  driven  pulleys 
should  be  not  less  than  from  130  to  160  feet,  and  the  pulleys  may  be  placed 
at  any  greater  distance  apart.  The  greater  the  distance  apart,  the  steadier 
the  movement.  The  velocity  of  the  rope  is  about  50  feet  per  second,  or 
3000  feet  per  minute.  At  this  rate,  a  force  of  n  Ibs.  would  be  equi- 
valent to  i  horse-power. 

The  most  common  sizes  of  wire-rope  employed  are  as  follows  : — 

Diameter.  Weight  per  Metre.  Weight  per  Yard. 

4  millimetres,  or  .16  inch.  .10  kilogramme.  .20  pound. 

6  „          or .24     „  .17          „  .34     „ 

9  »          or  .35     „  .31          „  .62     „ 

12  „  or  .47     „  .45  „  .90     „ 

At  Colmar,  a  force  of  47  horse-power  is  transmitted  a  distance  of  250 
1  Annales  des  Fonts  et  Chaussees,  1859. 


TRANSMISSION   OF   POWER  BY   ROPE.  755 

yards  by  a  ^-inch  wire-rope,  over  two  pulleys  of  3  metres,  or  about  10  feet  in 
diameter,  making  95  turns  per  minute.  The  rope  is  supported  at  the  middle 
of  the  span  by  pulleys  of  i  metre  in  diameter.  The  frictional  resistance  is 
less  than  3  per  cent.  The  ropes  receive  a  coat  of  a  mixture  of  oil  and  tar 
twice  per  month,  and  they  wear  well. 

The  ropes,  in  all  cases,  consist  of  36  wires,  in  six  strands  of  6  wires  each, 
on  a  core  of  hemp.  Each  strand  likewise  is  formed  on  a  hempen  core.  The 
hempen  cores  are  favourable  for  flexibility. 

From  another  account,  it  appears  that  100  horse-power  can  be  transmitted 
1 20  yards  without  any  intermediate  support,  by  an  endless  wire-rope  of 
.40  inch  in  diameter,  over  pulleys  from  13  to  14  feet  in  diameter,  making 
100  turns  per  minute,  equivalent  to  a  velocity  of  rope  of  upwards  of  4000 
feet  per  minute.  For  longer  distances,  the  rope  is  supported  at  intervals  of 
1 60  yards  by  y-feet  pulleys.  The  calculated  loss  of  power  in  transmitting  120 
horse-power  is  17^  per  cent,  or  21  horse-power.  The  sources  of  loss  are: 
— i  st.  The  resistance  of  the  air  to  the  arms  of  the  wheels.  2d.  The  resistance 
of  rigidity  of  the  rope  in  passing  over  the  wheels.  3d.  Axle-friction: — fixed 
loss  2^  per  cent,  for  the  large  pulleys,  and  i  per  cent,  for  every  1000  yards. 

In  an  excellent  illustrated  account  of  M.  Hirn's  rope-transmitter,  by  Mr.  H. 
M.  Morrison,1  he  states  that  soft  willow  wood  succeeds  best  as  lining  for  the 
large  pulleys.  The  pulleys  were  constructed  successively  of  copper,  hardwood, 
and  polished  cast  iron,  and  were  also  faced  with  leather,  horn,  india-rubber, 
lignum-vitse,  and  boxwood ;  but  all  these  materials  failed,  as  the  facings  were 
soon  worn  out,  and  when  the  groove  was  of  metal  or  hardwood,  and  did  not 
itself  wear,  it  destroyed  the  rope.  The  tension  in  the  upper  rope,  he  says,  is 
just  double  that  in  the  lower  rope.  The  best  method  of  changing  the  direc- 
tion of  transmission  of  the  power,  at  any  point  in  its  course,  has  been  found 
by  experience  to  be  by  the  use  of  bevil-wheels.  Directing  pulleys  are  not  so 
good  for  the  purpose.  For  high  speeds,  the  pulleys  should  be  of  best  cast 
steel,  as  iron  pulleys  may  fly  to  pieces  by  centrifugal  force.  Mr.  Morrison 
states  that  the  fine  makes  of  ropes  are  constructed  of  6  strands  of  1 2  wires 
each— 72  wires  in  all;  and  that  in  America,  the  wires  are  still  finer  and 
closer,  and  as  many  as  135  in  number. 

Cotton  Ropes. — Mr.  Ramsbottom,  in  1863,  applied  cotton  ropes  or  cords, 
for  driving  the  traversing  cranes  at  Crewe  workshops.2  The  cords  are  made 
of  soft  white  cotton,  y%  inch  in  diameter  when  new,  and  weighing  i  yz  oz. 
per  foot;  they  soon  become  reduced  to  9/l6  inch  thick  by  stretching,  and  they 
last  about  eight  months.  They  are,  when  new,  rubbed  over  with  a  little 
tallow  and  wax.  The  total  lengths  of  each  of  the  two  cords,  in  three  different 
shops,  are  respectively  800,  320,  and  560  feet.  The  pulleys  over  which  they 
are  passed  are  not  less  than  18  inches  in  diameter,  or  32  diameters  of  the 
cord;  and  in  the  first  of  the  above  shops,  alone,  the  cord  makes  from  12 
to  20  bends  according  to  the  machinery  in  action.  The  groove  of  the 
driving-pulleys  is  V-shaped,  at  an  angle  of  30°,  and  the  cord  is  gripped 
between  the  inclined  sides.  The  cord  is  supported  at  intervals  of  12  or 
14  feet  by  flat  slippers  of  chilled  cast  iron. 

The  velocity  of  the  cord  is  5000  feet  per  minute;  and  as  some  of  the 
pulleys  make  1000  turns  per  minute,  they  require  to  be  perfectly  self-balanced, 

1  Proceedings  of  the  Institution  of  Mechanical  Engineers,  1874. 

2  See  his  paper  on  the  subject,  in  the  Proceedings  of  the  Institution  of  Mechanical 
Engineers,  1864. 


756  MILL-GEARING. 

in  order  to  run  with  steadiness  and  ease.  In  the  overhead  traversers,  the 
total  leverage  is  slightly  over  3000  to  i  ;  and  in  lifting  a  load  of  9  tons,  the 
actual  pull  on  the  rope  is  17  Ibs.  A  tightening  stress  on  the  cord  of  109  Ibs., 
applied  by  means  of  a  weighted  pulley,  is  found  to  keep  it  steady,  and  to 
give  the  required  degree  of  hold  on  the  main  driving-pulley. 

In  the  wheel-shop,  two  dozen  pairs  of  blocks  and  ropes  had  previously 
been  employed,  requiring  a  large  number  of  labourers  to  work  them;  whilst 
now  the  two  traversing  cranes,  with  two  men  to  work  them,  do  the  whole 
work  of  the  shop,  and  it  is  done  much  more  quickly  than  before. 


SHAFTING. 

Shafting  is  subject  to  two  kinds  of  stress:  —  transverse  and  torsional. 
The  dimensions  of  shafts  are  settled  by  conditions  of  stiffness,  or  resistance 
to  deflection,  under  the  action  of  either  kind  of  stress. 

TRANSVERSE  DEFLECTION  OF  SHAFTS. 

The  deflections  of  cast-iron,  wrought-iron,  and  steel  bars  or  shafts,  loaded 
at  the  middle,  are  given  by  formulas  (8)  and  (10),  page  564;  (5)  and  (6), 
page  590;  and  (3)  and  (4),  page  6  19.     The  deflection  under  the  same 
weight  uniformly  distributed,  is  $ths  of  that  under  the  weight  when  placed 
at  the  middle.     Altering  some  of  the  measures,  let 
D  =  the  deflection,  in  inches. 
W  —  the  weight,  in  pounds. 

/=  the  length  or  distance  between  centres  of  bearings,  in  feet. 
d=  the  diameter  of  the  round  shaft,  in  inches. 
b  —  the  side  of  the  square  shaft,  in  inches. 

The  modified  formulas,  adapted  for  uniformly  distributed  weight,  are 
given  in  two  series;  first,  for  shafts  simply  supported  at  the  ends;  second, 
for  shafts  fixed  at  both  ends,  as  are  continuous  shafts.  In  settling  the 
divisors  for  the  second  series,  it  is  assumed  that  the  deflections  are  one-half 
of  those  of  the  first  series.  The  deflections,  actually,  are  not  so  great  as 
one-half;  and  margin  is  thus  left  for  deflection  arising  from  the  mode  of 
application  of  the  torsional  force,  and  for  the  excess  of  deflection  at  the 
loose  end  of  the  line  of  shafting. 

Transverse  Deflection  of  Shafts,  under  Uniformly  Distributed  Weight. 

Supported  at  the  ends.     Fixed  at  the  ends. 
W   /3  W  /3 

Cast-iron  shafts  :—  Round,  D  =  —  —  __  :  D  =  _  —  _  ....   (  i  ) 

39,400  d^          79,000  d4 

W  /3  W   /3 

Square,  D  =    0W  /      ;  D  =  —  W  /    L  ...  (2) 
58,000  b^          116,000  £4 


Wrought-iron  shafts:-Round,D  =  i;  D=_4...  (3) 


Square,  D  =  ;  D  =  _  ...  (4) 

97,500  b^  195,000  b4 

Steel  shafts:—  Round,  D  =  -^-;  D  =    ^  l"    ......  (5) 

78,800  158,000 

0  T^  W  /3  ~  W  /3  ,  ,  \ 

Square,  D  =  —  --  ;  D  =  —       —  ......   (6) 

116,000  232,000 


TRANSVERSE   DEFLECTION   OF   SHAFTS.  757 

The  working  limit  of  deflection  is  taken  as  */IOO  inch  per  foot  of  length, 

or  of  the  distance  of  bearings;  and  the  limiting  value  of  D  in  inches  is 

100  ' 
By  substitution  of  this  value  for  D  ;  and  by  reduction  and  inversion  :  _ 

The  Diameter  and  the  Side  for  the  Limiting  Transverse  Deflection. 

Supported  at  the  ends.      Fixed  at  the  ends. 
W  /2  IV  /2 

Cast-iron  shafts  :—  Round,  d*  =  —  :  d*  =  —  —  ........   (  7  ) 

394  790 

W  /2  W  72 


W  /2  W  72 

Wrought-iron  shafts  :—  Round,  d*  =  —L  :  d  4  =  .  .         .  (  o  ) 

664  1330 


Square,  t<  =         ;  6*,         .........  (  IO  ) 

975  195° 

W  72  W  /2 

Steel  shafts:—  Round,  </4=-~;  d*=?LL  .........  (  „  ) 

788  1576 

W  /2  W  /2 

Square,  d*=^~i-',  d<  =  —  .........  (  I2  ) 

1160  2320 

The  Distributed  Weight  for  the  Limiting  Trans-verse  Deflection. 

Supported  at  both  ends.    Fixed  at  the  ends. 

Cast-iron  shafts  :—  Round,  W  =  394^  .  w  „  79°^4  .......  (  r  3  ) 

q8o  b*    AIr     n6o^4  /       x 

Square,  W=^—  ;  W  =  —  ^_  ......  (14) 

Wrought-iron  shafts  :—  Round,  W  =  664/';  W=133^4  ......  (  15  ) 


Square,  W=;W  =  ......  (  l6  ) 


Steel  shafts  :-Round,  W=-  W  =  ......  (  17  ) 


Square,  W  =  ;W=.  ......  (l8) 

\Overhung  shafts.  —  When  the  weight  is  overhung  on  a  length  /  from  the 
bearing,  the  deflection  is  approximately  16  or  20  times  the  deflection  as 
found  by  the  above  formulas,  for  the  same  weight,  on  the  same  length,  /, 
when  supported  at  both  ends.] 

Let  the  total  distributed  weight,  including  the  weight  of  the  shaft  with 
wheels  and  pulleys,  and  the  resultant  stresses  of  driving  bands,  be  taken  at 
i^  times  the  weight  of  the  shaft.  The  weight,  in  pounds  per  foot,  of  a 
wrought-iron  round  shaft,  is  3.33  Ibs.  per  square  inch  of  section;  and  with 
wheels,  &c.,  it  is  (3.33  x  i^  =  )  5.83  Ibs.  per  square  inch.  In  terms  of  the 
diameter  d,  for  a  length  in  feet  /,  the  gross  weight  W  for  wrought-iron  is 
.7854^  /x  5.83;  for  cast  iron,  in  the  same  way,  it  is  .j&s^d2/*  5.47; 
and  for  steel,  it  is  .7854  d2  /  x  5.95.  Whence:  — 


MILL-GEARING. 

Gross  Distributed  Weight  of  Roimd  Shafts  with  Mounting,  for  the 

Limiting  Span. 
(Shaft  fixed  at  both  ends.) 

Cast-iron  round  shafts, W  =  4.30  d2 1 (  19  ) 

Wrought-iron  round  shafts, W= 4.58  d*  I (  20  ) 

Steel  round  shafts, W  =  4.6y  d2 1 (  21  ) 

Substituting  these  values  of  W,  in  formulas  (13),  (15),  and  (17),  for  the 
shaft  fixed  at  the  ends,  and  reducing: — 

Length  of  Span  for  the  Limiting  Transverse  Deflection  under  the  Gross 

Distributed  Weight. 
(Shaft  fixed  at  both  ends. ) 

Cast-iron  round  shafts, /3=  184  d2;  and  /=  JY 184  d2...   (  22  ) 

Wrought-iron  round  shafts,  /3=  290  d2;  and  /=  ^7290  d2...   (  23  ) 
Steel  round  shafts, ^3  =  337^2;  and  /=  A/337  d2—   (  24  ) 

When  the  shaft  is  employed  to  transmit  power  without  giving  off  any, 
the  distributed  weight  is  only  that  of  the  shaft  itself.  The  formulas  (22), 
(23),  and  (24)  may  be  adapted  for  the  less  weight  by  increasing  the  coeffi- 
cients in  the  ratio  of  i  to  i^. 

Length  of  Span  for  the  Limiting  Transverse  Deflection  under  the  Net  Weight 

of  the  Shaft  only. 
(Shaft  fixed  at  both  ends.) 

Cast-iron  round  shafts, l*  =  $22d2;  and /=  A*/ 322  d2 ...  (  25  ) 

Wrought-iron  round  shafts,  /3  =  508  d2;  and  /=  A/  508  d2 ...  (  26  ) 

Steel  round  shafts, l*  =  $()id2;  and  /=  A/59  *  d* ...  (27) 

The  length  of  span  under  the  gross  distributed  weight,  is  to  that  under 
the  net  weight  of  the  shaft,  as  i  to  A/ 1.75,  or  as  i  to  1.205,  or  i  to  i  z/5 . 

THE  ULTIMATE  TORSIONAL  STRENGTH  OF  ROUND  SHAFTS. 

Modifying  formulas  (14),  page  566;  (9),  page  591;  and  (5),  page  620; 
to  express  the  relations  of  the  moment  W'  R,  in  statical  foot-pounds,  being 
the  diameter  in  inches. 

Torsional  Strength  of  Rotmd  Shafts. 
Forcastiron, W'R  =   373  </3;      </3=W/R...   (28) 

O  /  O 

For  wrought  iron, W'R=   933  </3;      rf**SJr.....  (29) 

For  steel,  tensile  strength  \W'^  =  II20^       d,= W^R       (       } 
30  tons  per  sq.  in j  II2o 


TORSIONAL   DEFLECTION   OF   ROUND   SHAFTS.  759 

TORSIONAL  DEFLECTION  OF  ROUND  SHAFTS. 

The  deflections  of  round  bars  or  shafts  of  cast  iron,  wrought  iron,  and 
steel,  under  torsional  stress,  within  elastic  limits,  are  given  by  formulas  (18), 
page  566;  (14),  page  592;  and  (8),  page  621.  Altering  some  of  the 
measures,  let 

D'  =  the  angular  deflection  in  parts  of  a  revolution. 
W'  =  the  twisting  force,  in  pounds. 
R  =  the  radius  of  the  force,  in  feet. 
W  R  =  the  moment  of  the  force,  in  statical  foot-pounds. 
/'  =  the  length  of  the  shaft,  in  feet. 
</=the  diameter  of  the  shaft,  in  inches. 

The  modified  formulas  are  as  follows  ;  the  coefficients  are  given  in  the 
nearest  round  numbers. 

Torsional  Deflection  of  Round  Shafts. 

Cast-iron  shafts,..  ..D'  =   W/R/  .......................   (3I) 

n,  100  d* 

Wrought-iron  shafts,  .........  D'  =   ^  R  ^  .......................   (32) 

16,600  a4 


steelshafts> 


A  torsional  deflection  of  i°,  in  a  length  equal  to  20  diameters  of  the  shaft, 
is  a  good  working  limit  of  deflection  ;  that  is,  J/36o  th  part  of  a  turn,  or 

.00278  turn,  for  20  diameters.  Now,  for  cast-iron,  W  R  =  —  —  -  —  —  ; 
wrought  iron,  W  R  =  l6'6o°  *D';  steel,  W  R  .34.30°*  P.  and>  substi. 

/  / 

tuting  .00278  for  D',  and  ^—  for  /,  in  these  equations,  and  reducing:  — 

The  Working  Moment  of  the  Force,  and  the  Diameter,  for  the  Limiting 
Torsional  Deflection. 

For  cast  iron,  .......  W'R=i8.5</3;     ^  =  W  R  ...............  (  34  ) 

18.5 

For  wrought  iron,    W'R  =  27.7^3;     d*  =  ^—^  ...............  (35) 

Forsteel,  ............  W'R  =  57.2^.     ^  =  ^^  ...............  (36) 

By  these  convenient  transformations,  the  diameter  is  reduced  to  the  third 
power;  and,  since  the  ultimate  torsional  strength  is  also  in  the  ratio  of  the 
third  power,  the  same  margin  of  strength,  or  factor  of  safety,  is  provided  by 
the  formulas  for  all  diameters.  Comparing  the  coefficients  in  the  foregoing 
formulas,  for  the  ultimate  and  the  working  moments,  the  factors  of  safety  are 
found  to  be,  — 


760  MILL-GEARING. 


For  cast-iron  round  shafts,  ......  -       =  20,  factor  of  safety. 

18.5 

For  wrought-iron  round  shafts,  -2^  =  34,  factor  of  safety. 

27.7 

For  steel  round  shafts,  ............  —  —  =  19.5,  factor  of  safety. 

The  formulas  (34),  (35),  and  (36)  are  reduced  to  rules  as  follows:  — 

RULE  i.  To  find  the  maximum  Torsional  Stress  that  may  be  transmitted 
by  a  shaft,  within  good  working  limits.  —  Multiply  the  cube  of  the  diameter 
in  inches,  by  18.5  for  cast  iron;  by  27.7  for  wrought  iron;  or  by  57.2  for 
steel.  The  product  is  the  torsional  stress  in  statical  foot-pounds. 

RULE  2.  To  find  the  Diameter  of  a  shaft  capable  of  transmitting  a  given 
torsional  stress,  within  good  working  limits.  —  Divide  the  torsional  stress  in 
statical  foot-pounds,  by  18.5  for  cast  iron;  by  27.7  for  wrought  iron;  or  by 
57.2  for  steel.  The  cube-root  of  the  quotient  is  the  diameter  in  inches. 

Note.  —  The  torsional  stress  is  expressed  by  the  product  of  the  actual 
torsional  force  in  pounds,  by  the  radial  distance  in  feet  at  which  it  is  applied. 

POWER  THAT  MAY  BE  TRANSMITTED  BY  ROUND  SHAFTS,  WITHIN 
GOOD  WORKING  LIMITS. 

The  working  moments  of  the  force,  in  statical  foot-pounds,  formulas  (34), 
(35),  and  (36),  are  equivalent  to  as  many  pounds  acting  at  a  radius  of 
i  foot;  and  for  one  turn  of  the  shaft,  the  work  done  is  equivalent  to  the 
product  of  the  moment  by  (2  feet  x  3.1416  =  )  6.28:  — 

The  Work  for  One  Turn  of  a  Round  Shaft. 

Cast  iron,  ......  U=  n6d3  ........................  (  37  ) 

Wrought  iron,  U  =  1  74  d*  ........................   (  38  ) 

Steel,  ...........  U  =  359</*  ........................  (39) 


The  horse-power  developed  is  equal  to  the  work  done  in  one  turn  multi- 
plied by  the  speed,  or  number  of  turns  per  minute,  divided  by  33,000:  — 

Horse-power  of  a  Round  Shaft. 


Castiron,..    ..H=  =        .  .  (40) 

33,000       285 


Wrought  iron,  H  =  =        .......................  (       } 

33,000       190 

Steel,  ............  H  =  359^_S=^  .......................  (      } 

33,000       92 

in  which  S  =  the  speed  in  turns  per  minute,  and  H  =  the  horse-power. 

RULE  3.  To  find  the  maximum  Horse-power  of  a  shaft,  within  good  work- 
ing limits.  —  Multiply  the  cube  of  the  diameter  in  inches,  by  the  speed  in 
turns  per  minute;  and  divide  by  285  for  cast  iron,  by  190  for  wrought  iron, 
or  by  92  for  steel.  The  quotient  is  the  horse-power. 

The  following  additional  rules  are  obtained  by  inversion  of  the  formulas 
(40),  (41),  and  (42):— 


WEIGHT  AND   STRENGTH   OF   WROUGHT-IRON    SHAFTING.       /6l 


RULE  4.  To  find  the  Diameter  of  a  shaft  capable,  within  good  working 
limits,  of  transmitting  a  given  horse-power. — Multiply  the  horse-power  by  285 
for  cast  iron,  by  190  for  wrought  iron,  or  by  92  for  steel;  and  divide  by 
the  speed  in  turns  per  minute.  The  cube-root  of  the  quotient  is  the  diameter 
in  inches. 

RULE  5.  To  find  the  Speed  required  for  transmitting  a  given  horse-power, 
within  good  working  limits. — Multiply  the  horse-power  by  285  for  cast  iron, 
by  190  for  wrought  iron,  or  by  92  for  steel;  and  divide  the  product  by  the 
cube  of  the  diameter  in  inches.  The  quotient  is  the  speed  in  turns  per 
minute. 

The  table  No.  266  shows  the  net  weight  of  round  wrought-iron  shafting 
per  lineal  foot,  extracted  from  table  No.  76,  page  240;  and  the  gross  weight 
per  lineal  foot,  comprising  weight  of  pulleys,  stress  of  belts,  &c.,  taken  at 
i  24  times  the  net  weight  of  the  shafting. 

Table  No.  266. — WEIGHT  OF  ROUND  WROUGHT-!RON  SHAFTING. 

Gross  weight  =  1^  times  net  weight  of  shaft. 


Diameter 

of  Shaft. 

Weight 
per  lineal  foot. 

Diameter 
of  Shaft. 

Weight 
per  lineal  foot. 

Diameter 
of  Shaft. 

Weight 
per  lineal  foot. 

Net. 

Gross. 

Net. 

Gross. 

Net. 

Gross. 

inches. 

Ibs. 

Ibs. 

inches. 

Ibs. 

Ibs. 

inches. 

Ibs. 

Ibs. 

I 

2.62 

4.58 

3X 

33-5 

58.8 

9 

212 

371 

iX 

4.09 

7-15 

4 

41.9 

73-5 

9X 

236 

413 

IJB; 

5.89 

10.3 

4X 

47-3 

82.8 

10 

262 

458 

i# 

6.91 

1  2.  1 

4X 

53-o 

92.8 

ii 

317 

555 

\K 

8.02 

14.0 

4^ 

59.1 

104 

12 

377 

660 

t# 

9.20 

16.1 

5 

65.5 

H5 

13 

398 

697 

2 

10.5 

I8.3 

5X 

72.2 

126 

H 

462 

808 

2^ 

11.8 

20.6 

SK 

79.2 

139 

15 

53° 

928 

2X 

13-3 

23-3 

$x 

86.6 

152 

16 

670 

1176 

2^ 

14.8 

25.9 

6 

94.2 

165 

17 

759 

1330 

2^ 

16.4 

28.7 

6/2 

in 

196 

18 

848 

1484 

2% 

19.8 

34-6 

7 

128 

224 

19 

945 

1652 

3 

23.6 

41-3 

7% 

H7 

257 

20 

1040 

1834 

3X 

27.7 

48.4 

8 

1  68 

294 

3^ 

32.1 

56.2 

8^ 

189 

33i 

The  table  No.  267  gives  the  torsional  strength  and  horse-power  of  round 
shafts  of  wrought  iron,  within  the  good  working  limits  already  denned,  from 
i  inch  to  20  inches  in  diameter.  The  2d  column,  ultimate  torsional 
resistance  reduced  to  statical  foot-tons,  is  calculated  by  means  of  formula 
(29),  page  758;  the  3d  column,  working  torsional  stress,  is  calculated 
with  formula  (35),  page  759;  the  4th  column,  work  done  for  one  turn,  with 
formula  (38),  page  760;  the  5th  column,  horse-power  at  the  rate  of  one  turn 
per  minute,  with  formula  (41),  page  760;  the  6th  column,  speed  required 
tor  one  horse-power,  contains  the  reciprocals  of  the  values  in  column  5 ;  they 
may  be  calculated  by  rule  5,  above;  the  7th  and  8th  columns,  distance  of 
bearings  and  distributed  weight,  are  calculated  with  formulas  (23)  and  (20), 
page  758;  and  the  9th  column,  distance  of  bearings  under  net  weight,  with 
formula  (26).  Multipliers  for  shafts  of  cast  iron  and  of  steel,  are  subjoined 
to  the  table. 


762 


MILL-GEARING. 


Table  No.  267. — STRENGTH  OF  ROUND  WROUGHT-! RON  SHAFTING. 


Diameter 
of  Shaft. 

TORSIONAL  ACTION. 

TRANSVERSE  ACTION. 

Ulti- 
mate 
Resist- 
ance. 

Working 
Stress. 

Work 
for 
One  Turn 
per 

Minute. 

Horse- 
Power 
at  the 
rate  of 
One  Turn 
per 
Minute. 

^  Speed 
in  Turns 
per 
Minute 
for  One 
Horse- 
Power. 

Under  the  Gross 
Distributed  Weight. 

Under  the 
net  Weight 
of  Shaft. 

Distance 
of  Bearings 
for  the 
Limiting 
Defl'tion. 

Gross 
Weight 
for  the 
Span. 

Distance 
of  Bearings 
for  the 
Limiting 
Deflection. 

w 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

(9) 

inches. 

statical 
ft.-tons. 

statical 
ft.  -pounds. 

ft.  -pounds. 

H.  P. 

turns. 

feet. 

Ibs. 

feet. 

I 

.42 

27.7 

174 

.00526 

190 

6.6 

30 

7-9 

1% 

.82 

54-1 

340 

.OIO28 

97-3 

7-7 

55 

9.2 

'j* 

1.42 

93-5 

587 

.01779 

56.2 

8.6 

89 

10.3 

ify 

1.  80 

118.9 

746 

.02259 

44-3 

9-2 

112 

II.  0 

1^ 

2.25 

148.4 

932 

.02820 

35-4 

9-6 

134 

n-5 

i^i 

2.77 

182.6 

1147 

.03469 

28.8 

IO.  I 

163 

12.  1 

2 

3.36 

221.6 

1391 

.04211 

23-7 

10.5 

193 

12.7 

2^ 

4.00 

265.8 

1669 

.05062 

19.8 

II.  O 

227 

13.2 

2% 

4.80 

315.5 

1981 

.05995 

16.7 

11.4 

264 

13-7 

2H 

5.62 

37I-I 

2330 

•07051 

14.2 

ii.  8 

305 

14.2 

2^2 

6.56 

432.8 

2718 

.08224 

12.2 

12.5 

359 

15.0 

2% 

8-73 

576.1 

3618 

.1094 

9.14 

13.0 

45° 

15.6 

3 

II-3 

747.9 

4697 

.1421 

7-04 

13-7 

566 

I6.5 

3X 

14-4 

951-0 

5972 

.1807 

5-54 

14.5 

701 

17.4 

3/^ 

18.0 

1188 

7458 

.2257 

4-43 

15.2 

854 

I8.3 

3%" 

22.1 

1461 

9173 

•2775 

3-6o 

16.0 

1029 

19.2 

4 

26.9 

1773 

II,I3O 

.3368 

2-97 

16.7 

1225 

2O.  I 

4/^ 

32.2 

2127 

13,360 

.4040 

2.48 

17.4 

1439 

20.9 

4.1/2 

38.2 

2524 

15,850     .4796 

2.09 

1679 

21.7 

4^ 

45-o 

2969 

18,650     .5642 

•77 

18.8 

1943 

22.6 

k 

60'.  7 

3463 
4008 

21,740 
25,170 

•6579 
.7616 

•52 
•31 

19.4 

20.0 

222O 
2525 

24.0 

69.8 

4609 

28,950 

.8758 

.14 

20.  6 

2854 

24.7 

5^ 

79-8 

5266 

33,070 

1.  000 

.00 

21.2 

3210 

25.4 

6 

90.6 

5983 

37,570 

1.137 

.880 

21.8 

3600 

26.2 

6/4 

117 

7606 

47,770 

1-445 

.692 

22.9 

4421 

27-5 

7 

H4 

9501 

59,670 

1.805 

•554 

24.2 

5426 

29.0 

7K 

177 

1  1,  680       73,390 

2.  22O 

•450 

25.3 

6518 

30.4 

8 

215 

14,180       89,070 

2.694 

•371 

26.5 

7774 

31-8 

8/^ 

258 

17,010      106,800 

3.232 

•309 

27.6 

9133 

33-  ! 

9 

3°6 

20,190 

1  26,  SCO 

3.837 

.261 

28.7 

10,650 

34-4 

9/^5 

360 

23,75° 

149,200 

4.512 

.222 

29.8 

12,320 

35-7 

10 

420 

27,700 

174,000 

5.260 

.190 

30.8 

14,100 

36.9 

ii 

559 

36,870 

231,500 

7.005 

•143 

32-8 

18,180 

39-4 

12 

725 

47,860 

300,600 

9-095 

.IIO 

34-7 

22,880 

41.7 

13 

922 

60,860 

382,200 

11.83 

.0865 

36-6 

28,330 

44.0 

14 

1152 

76,010 

477,300 

14-44 

.0693 

38.5 

34,56o 

46.2 

15 

1417 

93,490 

587,100 

17.76 

•0563 

40.3 

4i,53o 

48.4 

16 

1720 

113,50° 

712,500 

21.56 

.0464 

42.1 

49,330 

50.5 

17 

2062 

136,100 

854,800 

25.86 

.0387 

43-8 

57,970 

52.6 

18 

2447 

161,500 

1,015,000 

30.69 

.0326 

45-5 

67,490 

54-6 

19 

2880 

190,000 

1,193,000 

36.10 

.0277 

47-2 

78,040 

56.6 

20 

3360 

221,600  1,391,000 

42.11 

.0237 

48.8 

80,660 

58.5 

Note. — To  find  the  corresponding  values  for  shafts  of  cast  iron  and  of  steel,  multiply  the  tabular  values 
by  the  following  multipliers  : — 


Cast  Iron, 
Steel, 


1.2 


2/3       I  »/3 

5.06         2.06 


.86 
[.05 


.81 

[.07 


.86 
1.05 


FRICTIONAL   RESISTANCE   OF   SHAFTING  763 

FRICTIONAL  RESISTANCE  OF  SHAFTING. 

The  frictional  resistance  of  horizontal  shafting  running  on  cylindrical 
journals,  is  calculable  by  means  of  formulas  (  2  )  and  (  7  ),  pages  725,  726; 
where  the  coefficient  of  friction,/,  is  determined  by  experiment.  M.  Morin's 
data,  page  722,  show  that  the  coefficient  is  .075  with  ordinary  oiling,  and 
.042  with  continuous  oiling. 

The  table  No.  268,  next  page,  is  based  on  the  results  of  extensive  obser- 
vations on  the  resistance  of  mill-shafting  in  America,  by  Mr.  S.  Webber,1 
of  Manchester,  N.H.  Take  the  average  of  his  frictional  coefficients  with 
those  of  M.  Morin : — 

Ordinary  Oiling.  Continuous  Oiling. 

M.  Morin's  coefficients,...  .075   042 

Mr.  Webber's  coefficients,  .066  044 


Means,  .............  070  or,  '/I4th    .......  043  or,  V23d 

Substituting  these  values  of  the  coefficient  /  in  formulas  (  2  )  and  (  7  ), 
pages  725,  726:— 

Work  Absorbed  by  Friction  for  One  Turn  of  a  Horizontal  Shaft. 

Ordinary  oiling,  ...............   U  =  .oi82  W^/  ...................   (43) 

Continuous  oiling,  ...........   U  =  .oii2W</  ...................   (44) 

Horse-power  Absorbed  by  Friction  of  a  Horizontal  Shaft. 


Ordinary  oiling,  ......  H^01*,  .  (  45  ) 

33,000          1,800,000 

Contmuousoiling,...H  =  -0112  W  ^S=     W  ^S      .  .  (  46  ) 

33,000         2,950,000 

U  =  work  absorbed,  in  foot-pounds. 

W  =  total  weight  of  shafting  and  pulleys,  plus  the  resultant  stress  of  belts, 

in  pounds. 
H  =  horse-power  absorbed. 

d=  diameter  of  journals,  in  inches. 

S  =  the  number  of  turns  per  minute. 

The  resistance  of  upright  shafting  is  probably  about  three-fourths  of  that 
of  horizontal  shafting  :  —  in  the  ratio  of  the  resistance  of  a  cylindrical  pivot 
to  that  of  a  journal. 

Ordinary  Data  for  the  Resistance  of  Shafting.  —  Mr.  R.  H.  Tweddell  gives 
the  following  results  of  observations  :  — 

Indicator  horse-power,  driving  the  shafting  of  a  tool-shop  alone,  6.65 
Do.  do.  resistance  of  steam-engine  alone,  ...........  3.51 

Do.  do.  net  power  absorbed  by  shafting  alone,  .....  3.14 

The  shafting  was  300  feet  long,  and  so  consumed  i  horse-power  per  100 
feet  of  length  in  turning  it.  The  speed  is  not  stated.  This  resistance  was 
equal  to  from  15  to  17  per  cent,  of  the  total  indicator  horse-power,  for 
ordinary  full  work. 

1  Journal  of  the  Franklin  Institute^  vol.  Ixviii.,  1874,  p.  261. 


764 


MILL-GEARING. 


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/66  MILL-GEARING. 

Mr.  Westmacott  states  that  1200  feet  of  shafting,  having  an  average 
diameter,  2^  inches,  and  that  had  been  running  for  years,  absorbed 
i  indicator  horse-power  per  100  feet  of  length  to  drive  it  alone — all  belts 
off — at  120  turns  per  minute. 

Mr.  B.  Walker  states  that  the  resistance  of  the  shafting  at  the  flax  mills 
of  Messrs.  Marshall,  Leeds,  with  belts,  absorbed  less  than  10  per  cent,  of 
the  total  indicator  power  of  the  engine.1 

M.  E.  Cornut  found,  by  careful  experiments,  that,  in  the  flax  mills  at 
Hamegicourt,  when  all  the  machines  were  at  work,  the  total  power 
required  to  drive  the  mill  was  150  indicator  horse-power;  and  the  power 
required  to  drive  the  engine  and  shafting  alone,  was  about  30  indicator 
horse-power,  or  20  per  cent,  of  the  total  power.  - 

Mr.  R,  Davison,  about  1842,  tested  the  power  absorbed  by  shafting  at 
Truman,  Hanbury,  and  Buxton's  brewery.  There  were  190  feet  of  hori- 
zontal shafting,  and  80  feet  of  upright  shafting;  total  length,  270  feet,  on 
thirty-four  bearings  having  3300  square  inches  of  area,  together  with  eleven 
pairs  of  spur  and  bevil  wheels,  from  2  to  9  feet  in  diameter.  They  absorbed 
7.65  indicator  horse-power.  The  shafting  had  probably  an  average  diameter 
of  4^  or  5  inches;  and  the  resistance  was  at  the  rate  of  2.73  horse-power 
per  100  feet.  The  speeds  were  not  given.3 

Mr.  Webber,  table  No.  268,  shows  that,  taking  great  lengths  only,  from 
0.33  to  0.78  horse-power  per  100  feet  is  absorbed,  with  constant  oiling; 
and  that  from  0.40  horse-power  to  nearly  i^  horse-power  per  100  feet, — 
averaging  about  i  horse-power  per  100  feet, — is  absorbed  with  ordinary 
oiling. 

JOURNALS  OF  SHAFTS. 

The  journals  or  bearings  of  shafts  should  be  proportioned  with  reference 
to  the  pressure  or  load  to  be  sustained  by  the  journal  and  its  pedestal.  The 
simplest  measure  of  the  bearing  capacity  of  a  journal  is  the  product  of  its 
length  by  its  diameter,  in  square  inches;  and  the  axial  area  thus  obtained 
may  be  multiplied  by  a  proper  unit  of  pressure  per  square  inch,  to  give  the 
bearing  capacity.  Sir  Wm.  Fairbairn  and  Mr.  Box  give  instances  of  the 
weights  on  bearings  of  shafts,4  from  which  the  following  deductions  are 
made,  showing  the  pressure  per  square  inch  of  axial  section  of  journal: — 

in.  in.  Ibs. 

Fly-wheel  shafts; — journal,  18  x  14;  pressure  per  square  inch,  178 

ii  *    9^  „  225 

„       lo^xS^  „  222 

Average, 208 

Ibs.  Ibs.  Pressure. 

Link-bearings, 456  to  690;  average  per  square  inch, 573  Ibs. 

Crank-pins, 687  to  1152;  „  „         874 

Mr.  Box  says,  that  the  pressure  on  bearings,  in  most  cases,  should  not 
exceed  500  Ibs.  per  square  inch,  measured  on  the  circumference  of  the 
journal,  equivalent  to  750  Ibs.  per  square  inch  of  the  axial  section. 

1  Proceedings  of  the  Institution  of  Mechanical  Engineers,  1874. 

2  Essais  Dynamometriques ,  1873- 

3  Proceedings  of  the  Institution  of  Civil  Engineers,  1843. 

*  Mills  and  Milhuork,  part  ii.  page  73;  Mill-Gearing,  page  52. 


JOURNALS   OF  SHAFTS.  767 

Dr.  Rankine l  gives  the  following  as  ordinary  values  of  the  intensity  of 
pressure  between  a  pair  of  greased  surfaces : — 

Per  square  inch.  Per  square  inch. 

Ibs.  Ibs.  Ibs.  Ibs. 

For  journals,  450  to  150  circumferentially,  or  675  to  225  axially 
For  flat  pivots, 2240 

whilst  Sir  Wm.  Fairbairn  limited  the  pressure  on  pivots  to  240  Ibs.  per 
square  inch. 

The  length  of  the  journals  of  shafts  is  ordinarily  i^  times  the  diameter. 
With  this  proportion,  the  gross  weight  on  the  journals  of  shaftings,  as 
tabulated  in  table  No.  267,  varies  from  20  Ibs.  per  square  inch  axially,  for 
i-inch  shafting,  to  about  40  Ibs.  for  2-inch  shafting,  and  134  Ibs.  per 
square  inch  for  a  20-inch  shaft. 

Journals  of  Railway  Axles. — The  journals  of  the  axles  of  railway  carriages 
and  waggons  are  usually  made  with  a  length  equal  to  more  than  twice  the 
diameter.  A  common  size  is  3^  inches  diameter  by  8  inches  or  9  inches 
long.  With  a  brass  bearing  having  a  width  of  2  ^  inches  measured  on  the 
chord,  the  horizontal  area  of  bearing-surface  is,  for  a  length  of  8  inches 
(8x2^  =  )  20  square  inches;  and  a  load  of  6000  Ibs.  on  each  journal  is 
equivalent  to  a  pressure  of  300  Ibs.  per  square  inch  of  horizontal  area  of 
bearing  surface :  a  satisfactory  proportion. 

Again,  the  proportion  of  the  load  to  the  horizontal  area  of  the  journal 
itself,  say  (8x3^  =  )  28  square  inches,  or,  for  a  smaller  diameter,  say, 
(8  x  3*^  =  )  26  square  inches,  averages,  say,  224  Ibs.,  or  2  cwts.  per  square 
inch  of  horizontal  section,  or  10  square  inches  per  ton  of  load.  This  is  an 
ordinary  working  proportion,  both  for  carrying  and  for  locomotive  stock.2 

1  Steam- Engine  and  other  Prime  Movers,  page  16. 

2  See,  on  this  subject,  papers  "  On  the  Construction  of  Railway  Waggons,"  by  Mr.  W. 
R.  Browne,  and  on  "  Railway  Rolling  Stock  Capacity,"  by  Mr.  W.  A.  Adams,  in  the 
Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xlvi.,  1875-76,  pages  81,  loo. 


EVAPORATIVE    PERFORMANCE 
OF    STEAM-BOILERS. 


NORMAL   STANDARDS. 

The  evaporative  efficiency,  or  the  efficiency  p,  of  a  steam-boiler  is  measured  by 
the  proportional  quantity  of  the  whole  heat  of  combustion  of  a  given  fuel, 
absorbed  into  the  boiler  and  applied  to  the  conversion  of  water  into  steam. 
Efficiency  is  also  expressed  by  the  weight  of  water  evaporated  by  one  pound 
of  the  fuel,  in  the  sense  of  the  French  word  rendement  —  yield  ;  and,  for  the 
purpose  of  directly  comparing  performances  effected  at  various  pressures 
and  temperatures,  it  is  customary  to  reduce  them  to  a  normal  standard  of 
efficiency,  expressed  by  the  equivalent  weight  of  water  which  would  be  con- 
verted into  steam  if  it  were  supplied  to  the  boiler  at  212°  F.  and  evaporated 
at  2  12°,  and  of  course  under  one  atmosphere  of  pressure  :  —  briefly,  evaporated 
from  and  at  212°  F. 

The  standard  temperature  of  the  water  as  supplied  to  the  boiler,  is 
sometimes  taken  at  62°  F.,  the  average  natural  temperature  of  cold  water; 
or  at  100°  F.,  which  is  about  the  temperature  of  the  condensing  water  of 
steam-engines. 

The  uniform  standard,  of  water  evaporated  from  and  at  212°,  is  adopted 
in  the  following  discussions. 

Evaporative  rapidity,  or  evaporative  power,  is  expressed  by  the  quantity  of 
water  evaporated  per  hour  by  a  steam-boiler.  It  may  be  the  total  quantity 
of  water,  or  it  may  be  the  quantity  of  water  per  square  foot  of  grate-area, 
or  per  square  foot  or  per  square  yard  of  heating  surface. 

Evaporative  performance  comprises  both  the  elements,  efficiency  and 
rapidity;  though  it  is  also  used  to  express  simply  the  evaporative  efficiency 
of  the  boiler,  or  of  the  fuel. 

Let  w  =  the  weight  of  water  evaporated  per  pound  of  a  fuel,  from  water 
supplied  at  the  temperature  /,  into  steam  of  the  total  heat  H,  measured  from 
32°  F.  Let  w',  t',  and  H',  be  any  other  corresponding  values  for  the  same 
expenditure  of  heat.  Then,  the  total  heat  expended  in  evaporating  T  Ib. 
of  water  is  H  +  32  -  /,  or  H'  +  32  -  /',  and 


Let  H'be  the   total  heat  of  steam  generated  at  212°  F.,  or  1146°,  and 
/'  =  212°  F.;  and,  by  substitution  in  formula  (i),  and  reduction, 

,  H+  7.2  -t  ,        N 

w=w*      966     '  ....................  (2) 


HEATING   POWER   OF  FUELS. 


769 


in  which  w'  is  the  equivalent  weight  of  water  as  evaporated  from  and  at 
212°  F. 

RULE. — To  find  the  equivalent  weight  of  water  evaporated  from  and  at  2 1 2°  F., 
when  a  given  weight  of  water  is  stipptied  at  a  given  temperature,  and  evaporated 
at  a  given  pressure.  Find,  in  table  No.  128,  page  387,  the  total  heat  of  the 
steam  generated  at  the  given  absolute  pressure;  add  32°  to  it,  and  from  the 
sum  subtract  the  temperature  of  the  feed- water;  and  divide  the  remainder 
by  966.  Multiply  the  given  weight  of  water  by  the  quotient  The  product 
is  the  equivalent  weight  of  water  evaporated  from  and  at  212°  F. 

When  the  water  is  to  be  taken  as  evaporated  at  212°,  but  supplied 

at  /'=  100°  F.,  use  the  divisor  1078,  in  formula  (2) 
at/'  =   62°  F.,  „  1116, 

HEATING  POWER  OF  FUELS. 

The  heating  powers  of  fuels,  treated  in  detail,  in  pages  409  to  458,  are 
here  collected  for  ready  reference,  in  table  No.  269. 


Table  No.  269. — HEATING  POWER  OF  FUELS. 


No. 

FUEL. 

Heating  Power  of  a  Pound 
of  Fuel. 

Units  of 
Heat. 

Water  Eva- 
porated per 
Pound  of  Fuel, 
from  and  at 

212°  F. 

2 

3 

4 

6 

8 
9 

10 

ii 

12 
13 
H 
15 

16 

17 
18 

19 

Warlich's  Fuel  

units. 
16,495 

15,567 
15,502 

14,133 
13,55° 
11,678 
16,655 
7,792 
S.fS 
12,696 

9,951 
7,156 
12,325 

6,100 
4,284 

5,231 
20,240 

27,53i 
34,292 

Ibs. 

17.07 

i6.n 
16.04 
14.62 
14.02 

12.10 

17.24 
8.07 
5.80 

13.13 
10.30 

7<4A 
12.76 

6.3I 

4-44 
5-44 
20.33 
28.50 
35-50 

units.            Ibs. 

Coal:  —  Ebbw  Vale   1848  16,221     16.79 

Powell's  Duffryn,  1848  ...15,715     16.25 
Llangennech,  1848-71  ....14,765     15.28 

Average  (best  Welsh)  15,567     16.11 
Has  well  Wallsend  (Newcastle)  

British  coals,  average      ..        

Coke 

Lignite,  perfect 

Asphalte  

Wood  perfectly  dry 

Do     25  per  cent  moisture 

Wood-charcoal,  dry  

Peat,  perfectly  dry  

Do.  25  per  cent,  moisture  

Peat-charcoal  85  per  cent  carbon  dry 

Tan  perfectly  dry,  1  5  per  cent  of  ash        .... 

Do.  30  per  cent,  moisture  

Straw    I5/^  per  cent,  moisture                   

Petroleum  

Petroleum,  oils                                            .        .  . 

Cosl-sfcis  (mean  of  Ross  and  Harcourt) 

49 

77O  EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 

EVAPORATIVE   PERFORMANCE   OF   STATIONARY  AND 
MARINE   STEAM-BOILERS,  WITH   COAL. 

The  chemical  history  of  the  combustion  of  coal  in  furnaces  has  been 
briefly  outlined,  page  426.  Regarded  mechanically,  there  are  three  modes 
of  supplying  coal  to  ordinary  furnaces  by  hand-firing,  namely, — first,  spread- 
ing-firing,  in  which  the  charge  of  coal  is  scattered  evenly  over  the  whole 
surface  of  the  grate;  second,  alternate  firing,  in  which  the  coal  is  laid  evenly 
along  half  the  width  of  the  grate  at  a  time,  each  side  alternately;  third, 
coking-firing,  in  which  the  coal  is  thrown  on  to  the  dead-plate  in  front  of  the 
bars  and  left  there  for  a  time,  in  order  that  the  mass  may  become  coked, 
after  which  the  mass  is  pushed  towards  the  bridge,  and  another  charge  is 
thrown  on  to  the  front  of  the  fire  in  its  place. 

The  proportion  of  surplus  air,  the  presence  of  which  is  required  for  the 
combustion  of  coal,  in  ordinary  furnaces,  in  excess  of  the  quantity  which  is 
chemically  consumed,  is  diminished  as  the  rate  of  combustion  is  increased ; 
and  the  diminution  of  the  excess  is  one  of  the  reasons  why  the  temperature 
in  the  furnace  rises  as  the  rapidity  of  combustion  is  increased.  The  follow- 
ing are  the  results  of  observations  on  the  proportion  of  surplus  air  admitted 
into  the  furnace,  in  parts  of  the  air  that  was  chemically  consumed : — 

RATE  OF  COMBUSTION.  SURPLUS  AIR. 

Coal  per  Square  Foot  of 
Grate  per  Hour. 

R.  Hunt,  Cornish  Boilers, ...   2  to  4    Ibs 100      percent. 

Professor  Johnson,  America,        7         „          100            „ 

Delabeche  and  Playfair, 10  to  16  „          25  to  50        „ 

J.  A.  Longridge,  Newcastle  (20  Ibs.  and  ) 

trials, (       upwards.  J      " 

The  evidence  is  not  sufficient  to  settle  the  question;  and  it  is  doubtful 
whether  Mr.  Longridge's  deduction  is  applicable  to  boilers  in  general.  It 
is  very  probable  that  surplus  air  only  ceases  to  be  present  when  the  rates 
of  combustion  are  very  much  higher  than  20  Ibs.  per  square  foot. 

With  very  slow  and  uniform  rates  of  combustion,  all,  or  nearly  all  the 
air  required,  may  be  drawn  through  the  grate.  If  the  combustion  be 
rapid,  a  considerable  proportion  of  the  air  must  be  introduced  directly  above 
the  fuel,  to  consume  the  gases.  It  was  seen,  page  428,  that,  allowing  a 
total  of  140  cubic  feet  of  air,  chemically  consumed  for  the  combustion  of 
one  pound  of  coal  of  average  composition,  36  per  cent,  is  consumed  by  the 
volatilized  portions,  and  64  per  cent,  by  the  fixed  portion,  or  coke.  Mr. 
Longridge  mentions  an  instance  in  which,  with  ordinary  stoking  and  a 
closed  doorway,  dense  smoke  was  given  off,  the  quantity  of  air  that  passed 
through  the  furnace,  exclusively  through  the  grate,  only  amounted  to  100 
cubic  feet  per  pound  of  coal.  This  quantity  was  little  more  than  equal  to 
what  was  sufficient  to  burn  the  fixed  portion  of  the  coal.  The  smoke  was 
prevented  when  an  additional  supply  of  air  was  admitted  from  the  doorway, 
above  the  fuel. 

EXPERIMENTS  ON  THE  EVAPORATIVE  POWER  OF  COALS,  CONDUCTED  BY 
MESSRS.  DELABECHE  AND  PLAYFAIR,  1847-50. 

Referring  to  the  averaged  results  of  performance  of  these  coals,  page  414, 
it  is  well  understood  that  they  were  not  burned  under  the  conditions  most 


LANCASHIRE  STATIONARY  BOILERS  AT  WIGAN.  77 1 

favourable  for  each  variety  of  coal.  Yet  they  throw  light  on  the  conditions 
of  composition,  which  appear  to  control  the  heat-producing  power  of  coals, 
and  probably  of  wood  and  other  fuels  too.  Neither  the  variations  of  the 
quantity  of  constituent  hydrogen,  nor  those  of  the  carbon,  are  commen- 
surate with  the  wide  range  of  performance;  but  it  is  evident  that  the 
evaporative  performance  decreases  regularly  as  the  oxygen  increases,  thus, — 

Water  Evaporated  per 
Oxygen.  pound  of  Coal 

from  and  at  212°. 

Patent  fuels, 2.79  per  cent 9.2o-lbs. 

Welsh  coals, 4.15       „  9.05    „ 

Newcastle  coals, 5.69       „  8.37    „ 

Lancashire  coals, 9.53       „  7.94    „ 

Scotch  coals, 9.69       „  7.70    „ 

Derbyshire  and  Yorkshire  coals, 10.28       „  7.58    „ 

Taking  averages,  it  is  seen  that  the  evaporative  efficiency  of  coal  varies 
directly  with  the  quantity  of  constituent  carbon,  and  inversely  with  the 
quantity  of  constituent  oxygen;  and  that  it  varies,  not  so  much  because 
there  is  more  or  less  carbon,  as,  chiefly,  because  there  is  less  or  more 
oxygen.  The  percentages  of  constituent  hydrogen,  nitrogen,  sulphur,  and 
ash,  taking  averages,  are  nearly  constant,  though  there  are  individual  excep- 
tions, and  their  united  effect,  as  a  whole,  appears  to  be  nearly  constant  also. 

EVAPORATIVE  PERFORMANCE  OF  LANCASHIRE  STATIONARY  BOILERS, 
AT  WIGAN,  I866-68.1 

The  coal  selected  for  trial  was  Hindley  Yard  coal,  from  Trafford  Pit, 
which  ranks  with  the  best  coals  of  the  district.  Three  stationary  boilers  were 
selected;  ist,  an  ordinary  double-flue  Lancashire  boiler,  7  feet  in  diameter, 
and  28  feet  long;  the  flue-tubes  were  2  feet  7%  inches  in  diameter  inside, 
of  ^-inch  plate.  2d,  Another  Lancashire  boiler  of  the  same  dimensions, 
in  which  the  tubes  were  of  s/l6-inch  steel  plate.  3d,  A  Galloway  or  water- 
tube  boiler,  26  feet  long,  and  6  feet  6  inches  in  diameter;  with  two  furnace- 
tubes  2  feet  7^6  inches  in  diameter,  opening  into  an  oval  flue  5  feet  wide 
by  2  feet  6^  inches  high,  containing  24  vertical  conical  water-tubes.  The 
first  and  second  boilers  were  new  and  specially  constructed  for  the  trials; 
the  third  boiler  was  a  second-hand  one.  These  three  boilers  were  set  side 
by  side,  on  side  walls  and  with  two  dampers.  The  flame  passed  through 
the  flue-tubes,  back  under  the  boiler,  then  along  the  sides  to  the  chimney. 
The  chimney  was  105  feet  high,  above  the  floor;  octagonal,  6  feet  10  inches 
wide  at  the  base,  and  5  feet  wide  at  the  top,  where  the  sectional  area  was 
2 1  square  feet. 

Total  grate-area  in  each  boiler: — 6  feet  long;  31.5  square  feet. 
»  »  4         „  21.0         » 

1  The  Author  is  indebted  for  the  particulars  of  these  trials  to  "The  South  Lancashire  and 
Cheshire  Coal  Association's  "  Report  on  the  Boiler  and  Smoke- Prevention  Trials,  condiicted 
at  Wigan,  1869.  The  experiments  were  conducted,  and  the  report,  excellently  reasoned, 
was  written  by  Mr.  Lavington  E.  Fletcher. 


772  EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 

LANCASHIRE.  GALLOWAY. 

Heating  surface,  in  flue-tubes, 464.34  square  feet.   431.12  square  feet. 

In  external  flues... 303. 08          „  288.24         „ 


Total  surface,  ...767.42          „  7J9-36 

Ratio  of  grate-area,  6  feet  long,  to  )  g 

heating  surface,  .................  j 

Ratio  of  grate-area,  4  feet  long,  to  )  6 

heating  surface,  .................  f  34'3 

Circuit,  or  length  of  heating  sur-  } 

face,  traversed  by  the  draught  >   80  feet.  74  feet. 

from  the  centre  of  the  grate,   J 
Total  distance  from  centre  of  grate  ) 

to  base  of  chimney,  ............  j       7     " 

Height  of  chimney  above  level  of  )  r    . 

floor,  ..............................  J 

96  feet  9  inches' 


A  Green's  fuel-economizer  was  placed  in  the  main  flue;  it  had  12  rows 
of  4^-inch  cast-iron  pipes,  8  feet  9  inches  long,  placed  vertically  —  84  tubes 
in  all  —  having  a  collective  heating  surface  of  850  square  feet,  exclusive  of  the 
connecting  pipes  at  top  and  bottom.  The  feed-water  was  passed  through 
the  economizer  on  its  way  to  the  boiler,  and  absorbed  a  portion  of  the 
waste  heat. 

The  fire-grates  were  tried  at  2  lengths,  6  feet  and  4  feet.  The  shorter 
grate  gave  the  more  economical  result,  but  it  generated  steam  less  rapidly. 
Three  modes  of  firing  were  tried;  —  spreading,  coking,  and  alternate  firing. 
With  round  coal,  on  the  whole,  the  greatest  duty  was  obtained  by  coking 
firing,  with  the  least  smoke.  With  slack,  alternate-side  firing  had  the 
advantage. 

Fires  of  different  thicknesses  were  tried  :  6  inches,  9  inches,  and  1  2  inches. 
It  was  found  that  9  inches  was  better  than  6  inches,  and  12  inches  better 
than  9  inches.  Air  admitted  at  the  bridge  gave  a  slightly  better  result 
than  by  the  door;  and  the  admission  of  air  in  small  quantity  on  the  coking 
system,  prevented  smoke.  The  doors  were  double,  slotted  on  the  outside, 
and  pierced  with  holes  on  the  inner  side.  The  maximum  area  of  opening 
was  31^  square  inches  for  each  door,  being  at  the  rate  of  2  square  inches 
per  square  foot  for  the  6-feet  grates,  and  3  square  inches  for  the  4-feet 
grates.  The  amount  of  opening  was  regulated  by  a  slide. 

The  standard  fire  adopted  for  trial  was  1  2  inches  thick,  of  round  coal, 
treated  on  the  coking  system,  with  a  little  air  admitted  above  the  grate,  for 
a  minute  or  so  after  charging. 

The  water  was  evaporated  under  atmospheric  pressure. 

The  quantity  of  refuse  from  the  Hindley  Yard  coal,  averaged  in  the  trials 
with  the  marine  boiler,  to  be  afterwards  described,  2.8  per  cent,  of  clinker, 
2.8  per  cent,  of  ash,  and  .8  per  cent,  of  soot;  in  all,  6.4  per  cent.  Making 
allowance  for  the  difference  of  soot,  the  total  refuse  may  be  taken,  in  the 
trials  of  the  stationary  boilers,  at  6  per  cent. 

General  Deductions.  —  The  advantage  of  the  4-feet  grate  over  the  6-feet 
grate,  was  manifested  by  comparative  trials  with  round  coal  12  inches  thick, 


LANCASHIRE   STATIONARY   BOILERS   AT   WIGAN.  773 

and  slack  9  inches  thick.  With  the  4-feet  grate,  the  evaporative  efficiency, 
taking  averages,  was  9  per  cent,  greater  than  with  the  6-feet  grate;  though  the 
rapidity  of  evaporation  was  15  per  cent,  less,  at  the  same  time  that  19^  per 
cent,  more  coal  was  burned  per  square  foot  per  hour. 

When  equal  quantities  of  coal  were  burned  per  hour,  the  fires  being 
12  inches  thick,  8  per  cent,  more  efficiency  and  12  per  cent,  greater 
rapidity  of  evaporation  were  obtained  from  the  shorter  grate.  Thus : — 

Coking  Firing. 

Length  of  grate, 6  feet.  4  feet. 

State  of  damper, two-thirds  closed,    fully  open. 

Coal  per  hour, 4.0  cwts.  4.14  cwts. 

Coal  per  square  foot  of  grate  per  hour, 14  Ibs.  23  Ibs. 

Water  at  100°  evaporated  per  hour, 65  cubic  feet.  7  2. 6  cubic  feet. 

Water  at  212°  per  pound  of  coal, 10.10  Ibs.  10.91  Ibs. 

Smoke  per  hour: — 

Very  light, 4.3  minutes.  4.  i  minutes. 

Brown, 0.4         „  0.3         „ 

Black, o.o         ,,  o.o         „ 

To  compare  the  performances  with  coking  and  spreading  firing,  having 
i2-inch  fires  for  round  coal,  and  9-inch  fires  with  slack: — Whilst,  with  round 
coal,  the  rapidity  of  evaporation  was  the  same  with  both  modes  of  firing, 
the  efficiency  was  from  3  to  4  per  cent,  greater  with  coking.  With  slack, 
on  the  contrary,  the  spreading  fire  evaporated  a  fourth  more  water  per 
hour  than  the  coking  fire,  though  with  4^  per  cent,  less  efficiency. 

With  thicknesses  of  coking  fire,  6  inches,  9  inches,  and  12  inches,  for 
round  coal;  and  6  inches  and  9  inches  for  slack;  the  results  were  in  all 
respects  decidedly  in  favour  of  the  thicker  fires  rather  than  the  thinner  fires. 
Comparing  the  thinnest  and  the  thickest  fires,  from  5^  to  20  per  cent, 
more  water  was  evaporated  per  hour  by  the  thickest  fires,  and  from  1 1  to 
1 8  per  cent,  more  per  pound  of  fuel. 

The  effect  of  the  admission  of  air  above  the  grate,  continuously  or 
intermittently,  for  the  prevention  of  smoke,  as  compared  with  that  of  its 
non-admission,  was  ascertained  with  round  coal,  and  with  slack.  The 
averaged  results  showed  that  by  admitting  the  air  above,  the  evaporative 
efficiency  was  increased  7  per  cent. ;  but  that  the  rapidity  of  evaporation 
was  diminished  3}^  per  cent. 

Comparing  the  admission  of  air  above  the  fuel  at  the  door,  and  at  the 
bridge  through  a  perforated  cast-iron  plate;  it  was  found  that  the  admission 
at  the  bridge  made  a  better  performance,  by  about  2^  per  cent.,  than  at 
the  door. 

To  try  the  effect  of  increasing  the  supply  of  air  above  the  fuel,  the  door- 
frame was  perforated  to  give  an  additional  square  inch  of  air-way  per  foot 
of  grate,  making  up  3  square  inches;  an  allowance  of  i  square  inch  was 
also  provided  at  the  bridge.  Round  coal  was  burned  on  the  coking 
system,  1 2  inches  thick,  on  6-feet  grates,  with  a  constant  admission  of  air 
above  the  fuel.  When  the  supply. by  the  door  was  increased  from  2  inches 
to  3  inches  per  square  foot  of  grate,  the  evaporative  efficiency  fell  oft 
8^  per  cent.,  and  the  rapidity  3  per  cent.  When  an  extra  inch  was 
supplied  at  the  bridge,  making  up  4  square  inches  per  foot  of  grate,  the 
evaporative  efficiency  only  fell  off  0.65  per  cent.,  and  the  rapidity  i^  per 


774          EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 

cent.  The  effect  of  this  evidence  is,  that  the  bridge  is  the  better  place  for 
the  admission  of  air,  and  that  if  the  air  be  admitted  by  the  bridge  alone,  the 
area  of  supply  may  be  beneficially  raised  to  4  square  inches  per  square  foot 
of  grate. 

Comparing  the  effect  of  the  admission  of  air  in  a  body,  undivided,  with 
that  of  its  admission  in  streams,  on  a  6-feet  grate,  with  coking  fires 
12  inches  thick  of  round  coal;  there  was  6j^  per  cent,  of  loss  of 
efficiency  by  the  admission  in  a  body,  though  the  smoke  was  equally 
well  prevented. 

Mr.  Fletcher  concludes  that  the  greatest  rapidity  of  evaporation  was 
obtained,  when  the  passages  for  the  admission  of  air  above  the  fuel  were 
constantly  closed;  that  the  next  degree  of  rapidity  was  obtained  when  they 
were  open  only  for  a  short  time  after  charging,  and  the  lowest  when  they 
were  kept  open  continuously.  He  also  concludes  that,  whilst,  in  realizing 
the  highest  power  of  a  free-burning  and  gaseous  coal,  smoke  is  prevented ; 
yet,  in  realizing  the  highest  power  of  the  boiler,  smoke  is  made. 

In  burning  slack,  smoke  was  prevented  as  successfully  as  in  burning  round 
coal,  though  its  evaporative  efficiency  was  from  i  to  i  y%  Ibs.  of  water  per 
pound  of  fuel,  less  than  with  round  coal. 

To  work  out  the  problem  of  firing  slack  without  smoke,  and  without  loss 
of  rapidity  of  evaporation;  trials  were  made  at  the  boilers  of  16  mills,  when 
the  slack  was  fired  on  the  alternate-side  system.  No  alterations  were 
made  in  the  furnaces  in  preparation  for  these  trials;  in  many  instances,  the 
fire-doors  had  no  air-passages  through  them.  The  grates  were  from  3  feet 
7  inches  to  7  feet  long;  they  averaged  6  feet  in  length. 

Number  of  boilers  fired, 65  boilers. 

Slack  burned  per  boiler  per  week  of  60  hours, 17-35  tons. 

Slack  per  square  foot  of  grate  per  hour, 1 9. 2  5  Ibs. 

Smoke  per  hour : — 

Very  light, 11.5  minutes. 

Brown, 2.3 

Black, 0.3 


14.1 

In  1 2  instances,  no  black  smoke  whatever  was  made.  It  is  said  that  the 
steam  was  as  well  kept  up,  and  the  speed  of  the  engines  as  well  maintained, 
as  before  the  trials  were  made. 

COMPARATIVE  PERFORMANCE  OF  THE  STATIONARY  BOILERS  AT  WIGAN. 

There  were  made  altogether  about  two  hundred  and  ninety  trials  with 
the  three  boilers,  of  which  sixty  may  be  regarded  as  comparative  trials 
of  the  boilers.  The  results  of  these  sixty  trials  are  embodied  in  the  table 
No.  270,  page  776.  The  second  part  gives  the  best  results  that  had  been 
obtained  from  each  boiler,  supplied  with  round  coal,  on  the  coking  system; 
and  with  air  admitted  through  the  doors  for  a  few  minutes  after  charging. 
Suffice  it,  meantime,  to  remark  that  the  performance  of  the  double-flue  boilers 
amounted  practically  to  the  same  as  that  of  the  water-tube  boiler.  Thus, 


LANCASHIRE   STATIONARY  BOILERS  AT  WIGAN.  775 

the  means  of  the  double-flue  boilers  compare  as  follows  with  the  results  of 
the  conical  water-tube  boiler : — 

AVERAGES  OF  SIXTY  TRIALS,  WITHOUT  ECONOMIZER — 

Water  at  100°  consumed  Water  at  212° 

per  hour.  per  Ib.  of  coal. 

Double-flue  boilers, 79.65  cubic  feet  10.31  Ibs. 

Conical  water-tube  boiler, 78.95         „  10.34,, 

BEST  RESULTS  OBTAINED  : — WITHOUT  ECONOMIZER — 

Double-flue  boilers, 81.92         „  10.86,, 

Conical  water-tube  boiler, 79. 1 7         „  10.58,, 

WITH  ECONOMIZER — 

Double-flue  boilers, 90.72         „          11.56,, 

Conical  water-tube  boiler, 86.31         „  11.82  „ 

In  doing  the  same  work,  it  is  to  be  noted  that  the  water-tube  boiler  was 
2  feet  shorter,  and  6  inches  less  in  diameter,  than  the  double-flue;  and  that 
it  had  48  square  feet,  or  6  per  cent,  less  area  of  heating  surface. 

A  trial  was  made  with  the  object  of  testing  the  comparative  merits  of  the 
plain  double-flue  and  the  water-tube  flue,  by  shutting  off  the  draught  from  the 
external  flues,  and  leading  it  direct  from  the  internal  flues  to  the  chimney, 
with  the  following  results  (grates  6  feet  long,  coking  firing,  12  inches 
thick):— 

WITHOUT  ECONOMIZER—  Water  at  100°  consumed  Water  at  212° 

per  hour.  per  Ib.  of  coal. 

Iron  double-flues, 82.97  cubic  feet  8.23  Ibs. 

Water-tube  flue, 80.00         „          8.50,, 

WITH  ECONOMIZER — 

Iron  double-flues, 98.85         „  10.08,, 

Water-tube  flue, 89.08         „          10.16  „ 

Showing  that  the  double  flues,  having  33  square  feet,  or  nearly  8  per  cent, 
more  heating  surface  than  the  water-tube  flue,  evaporated  more  water  per 
hour,  but  with  rather  less  efficiency  than  the  water-tube  flue. 

The  evaporative  power  of  the  boilers  was  rather  increased  than  diminished 
by  the  closing  of  the  external  flues,  though  there  was  a  sacrifice  of  evaporative 
efficiency. 

Water-tubes. — Four  water-tubes  were  inserted  in  each  flue  of  the  iron-flue 
boiler,  5  ^  inches  in  diameter  inside,  and  2  feet  7  ^  inches  long,  making  an 
addition  of  30  square  feet,  or  6^  per  cent.,  to  the  flue-heating  surface,  or 
4  per  cent,  of  the  total  heating  surface.  The  result  of  the  insertion  showed 
equal  rapidity  of  evaporation,  and  a  gain  of  3  per  cent,  in  efficiency ;  as 
follows : — 

Water  at  100°  Water  at  212°  F. 

consumed  per  hour.  per  pound  of  coal. 

Without  water-tubes, 9I-I5  cubic  feet i o. 43  Ibs. 

With  water-tubes, 91.12         „  io-77  » 


776 


EVAPORATIVE  PERFORMANCE  OF  STEAM-BOILERS. 


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WITHOUT  ECONOMIZER. 
Double-flue  boiler,  with  iron  tubes.... 
Double-flue  boiler,  with  steel  tubes.... 
Conical  water-tube  boiler  

Double-flue  boiler,  with  iron  tubes.... 
Double-flue  boiler,  with  steel  tubes.... 
Conical  water-tube  boiler  

SPREADING  FIRING,  12  inches  thi< 
Double-flue  boiler,  with  iron  tubes.... 
Double-  flue  boiler,  with  steel  tubes  (no 
Conical  water-tube  boiler  

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Conical  water-tube  boiler  

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LANCASHIRE   STATIONARY  BOILERS   AT  WIGAN. 


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WITH  ECONOMIZER. 
Double-flue  boiler,  with  iron  tubes  
Double-flue  boiler,  with  steel  tubes  
Conical  water-tube  boiler  

Double-flue  boiler,  with  iron  tubes  (no  trials) 
Double-flue  boiler,  with  steel  tubes  
Conical  water-tube  boiler  

Means  of  the  Two  Sizes  of  Grate. 
Double-flue  boiler,  with  iron  tubes  
Double-flue  boiler,  with  steel  tubes  
Conical  water-tube  boiler  
Average  of  the  three  boilers  

"^                                                     •                                                                                                                   - 

%     i  ;  i   ;  ;     •§  ;  ;    % 

ROUND  COAL,  COKING  FIRING,  12 
WITHOUT  ECONOMIZER. 
Double-flue  boiler,  with  iron  tubes 
Double-flue  boiler,  with  steel  tubes 
Conical  water-tube  boiler  

Double-flue  boiler,  with  iron  tubes 
Double-flue  boiler,  with  steel  tubes 
Conical  water-tube  boiler  

Means  of  the  Two  Sizes  of  Gr 
Double-flue  boiler,  with  iron  tubes 
Double-flue  boiler,  with  steel  tubes 
Conical  water-tube  boiler  
Average  of  the  three  boile 

7/8  EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 

Green's  Fuel-Economizer. — From  the  average  results  of  various  compara- 
tive trials,  burning  round  coal  and  slack,  and  with  coking  firing,  on  6-feet 
and  4-feet  grates,  it  appeared  that,  burning  equal  quantities  of  coal  per  hour, 
the  rapidity  of  evaporation  was  increased  9.3  per  cent.,  and  the  efficiency 
10  per  cent,  by  the  addition  of  the  economizer. 

Temperature  of  the  Products  of  Combustion,  and  of  the  Feed-water,  when 
the  water  is  passed  through  the  Economizer.  Average  results : — 

With  6-feet  Grate.  With  4-feet  Grate. 

Before.  After.  Before.  After. 

Temperature  of  gases  in  the  flues  before  )  ,     0  o  0  o 

and  after  traversing  the  economizer,  j    49         34  3 

Temperature  of  the  feed-water, 47         157  41         137 

Whence  it  follows  that,  to  raise  the  temperature  of  the  feed-water  through 
100°  F.,  the  gases  were  cooled  down  through  an  average  of  250°  F. 

Temperature  of  the  Products  of  Combustion,  without  the  Economizer. — The 
variations  to  which  the  temperature  of  the  escaping  gases  is  subject,  are 
illustrated  in  the  annexed  statement,  showing  the  temperature  with  different 
thicknesses  of  fire,  burning  round  coal  with  coking  firing,  without  the 
economizer. 


ROUND  COAL.  With  6-feet  Grate. 

Thickness  of  fires, inches  12  9  6 

Coal  per  foot  of  grate  per  hour,... pounds  19  20  20 

Water  at  1 00°  evaporated  per  hour,  cu.  feet  85.7  85.5  81.2 

Water  at  212°  per  pound  of  coal,  pounds  10.12      9.79      9.16 

Temperature  in  chimney-flue, 630°  556°  539° 

Smoke  per  hour — 

Very  light, minutes  2.0        o.o        0.5 

Brown, minutes  0.4        o.o        o.i 

Black,..  ...minutes  o.o       o.o       o.o 


With  4-feet  Grate. 
12          9  6 

23          24         24 
72.8      70.7     6l.7 
10.90      9.95     9.21 

505°    451°    445° 
2.8      0.4     o.o 

O.I          0.0        0.0 
O.O         0.0        O.O 


It  is  shown  that  the  temperature  in  the  chimney  flue  is  lower  with  the  4-feet 
grate,  than  with  the  6-feet  grate;  it  averages  107°  lower,  and  correspond- 
ingly, the  evaporative  efficiency  averages  higher.  But,  with  the  same  grate, 
both  the  evaporative  efficiency  and  the  temperature  become  less  with  the 
thinner  fire,  due,  no  doubt,  as  Mr.  Fletcher  points  out,  to  the  passage  of  a 
greater  surplus  of  air  through  the  thinner  fire. 

Volume  of  Air  Supply  and  Products  of  Combustion. — The  volume  of  air 
entering  the  ash-pit  and  passing  through  the  grate,  when  the  doors  were 
closed,  was  found,  by  means  of  Biram's  anemometer,  to  be,  for  grates 
4  feet  long,  with  fires  9  inches  thick,  from  245  to  250  cubic  feet  per  pound 
of  coal  burned ;  the  average  velocity  of  entrance  into  the  ash-pit,  which  was 
2  feet  square,  having  been  observed  to  be  9.3  feet  per  second.  As  the 
composition  of  the  coals  has  not  been  given,  it  may  only  be  assumed 
roughly,  that  the  coal  chemically  consumed  140  cubic  feet  of  air  for  the 
combustion  of  one  pound ;  and,  if  the  above-noted  quantities  of  air  supplied 
be  exact,  it  would  follow  that  a  surplus  of  air  amounting  to  from  75  to 
80  per  cent,  was  present.  This  is  questionable,  and  it  is  probable,  in  the 
scarcity  of  data,  that  the  observations  for  velocity  were  made  at  the  centre 
of  the  draught- way,  where  the  velocity  was  a  maximum,  and  that  no  correction 
was  made  for  the  inferior  velocities  at  other  parts  of  the  section. 


LANCASHIRE   STATIONARY  BOILERS  AT  WIGAN.  779 

From  an  analysis  of  the  products  of  combustion  in  the  chimney,  it 
appeared  that  there  was  no  appreciable  quantity  of  carbonic  oxide  present. 

Trials  under  Steam  of  more  than  one  Atmosphere  of  Pressure. — As  the 
experiments  at  Wigan  were  made  under  one  atmosphere  of  pressure,  a  few 
trials  were  made  under  an  effective  pressure  of  40  Ibs.  per  square  inch,  with 
the  following  comparative  results : — 

At  atmospheric  At  40  Ibs. 

pressure.  per  square  inch. 

Water  at  100°  evaporated  per  hour,  cubic  feet,      83.6  80.4 

Water  at  212°  per  pound  of  coal,  pounds,  10.76  9.53 

showing  a  reduction  of  i  ^  pounds  of  water  in  evaporative  efficiency,  at  the 
higher  pressure,  which  is  more  or  less  accounted  for,  first,  by  the  greater 
total  heat  of  steam  at  the  higher  pressure,  requiring  more  fuel-heat  for  its 
formation;  secondly,  by  the  higher  temperature  of  the  water  in  the  boiler 
at  the  higher  pressure,  which  would  to  some  extent  check  the  absorption  of 
the  last  portions  of  heat  from  the  gases  before  they  escaped  into  the  chimney- 
flue.  Still,  the  difference  is  excessive. 

Trials  with  D.  K.  ClarKs  Steam-induction  Apparatus  for  the  Prevention 
of  Smoke. — In  Clark's  smoke-preventer,  the  air  was  admitted  through  the 
door,  regulated  in  quantity  by  a  flap-valve,  and  deflected  upwards  upon  an 
air-plate  placed  across  the  furnace  above  the  dead-plate,  and  against  the 
furnace-front.  Steam  from  an  auxiliary  boiler  was  conducted  by  a  pipe 
above  the  air-plate,  and  was  discharged  in  four  jets  over  the  fire,  towards 
the  bridge.  In  passing  over  and  beyond  the  edge  of  the  air-plate,  the 
steam  induced  the  air  which  passed  forward  from  the  door  under  the  air- 
plate,  and  carried  it  onward  above  the  fire — thus  forcibly  mingling  it  with 
the  combustible  gases,  and  at  the  same  time  increasing  the  draught. 

The  trials  were  made  in  three  ways — ist,  with  the  jets  and  the  air-valves 
constantly  open;  2d,  with  the  jets  and  the  air-valves  open  for  a  minute  or 
so  only,  after  each  charge;  3d,  with  the  jets  constantly  open,  while  the  air- 
valves  were  closed.  It  was  found  that,  when  the  jets  were  constantly  open, 
the  quantity  of  steam  consumed  from  the  auxiliary  boiler  to  supply  them 
amounted  to  one-thirtieth  of  the  quantity  of  water  evaporated. 

The  following  are  the  comparative  results  of  performance  on  6-feet 
grates,  with  the  steam-inductor,  and  with  the  ordinary  fire-door  and  the  split 
bridge.  The  jets  and  air-valves  of  the  steam-inductor  were  open  for  a 
minute  or  so  only  after  each  charge;  and,  taking  the  interval  between  the 
charges  at  fifteen  minutes,  it  is  evident  that  the  quantity  of  steam  consumed 
by  the  nozzles  was  insignificant : — 


Without 
Economizer. 


ROUND  COAL,  6-feet  Grate  ;  Firing,  12  inches  thick.  Coking.  Spreading. 

Coal  per  sq.  foot  of  grate  per  hour,  steam  inductor,  pounds,  18.77     23.86 
Water  at  .oo'perhour,  { 


Smoke  per  hour,  ordinary  door  — 

Very  light,  ..........................................  minutes,  3.1  5.3 

Brown,  ................................................      „  0.8  4.9 

Black,  ................................................       „  o.o  3.3 


With 

Economizer. 
Coking. 

18.20 

91.77 


;8o 


EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 


Without 
Economizer. 
SLACK,  6  feet  Grate;  Firing,  9  inches  thick.                             Coking.  Spreading. 

Coalpersq.  foot  of  grate  per  hour,  steam-inductor,  pounds,  18.7        — 

With 
Economizer. 
Coking. 
20.7 
IOI  OQ 

Waterat  100°  per  hour,  j  g^^^  CUblC  feet'  76*fs 

71   C8 

Wfltprat2T2°nernoundroal   \  steam-inductor,  pounds,    9.17       — 
Water  at  212  per  pound  coal,  |  gplit  bridge                          3  gg 

10.65 

9-7  "1 

Smoke  per  hour,  ordinary  door  — 
Very  light                                                           minutes     o  "> 

'"J 

O  O 

Brown                                            .        ...         ...                   oo      — 

o  o 

Black.  .                                                                                 o.o      — 

o.o 

It  is  shown  that,  with  round  coal  and  coking  firing,  there  was  no  advantage 
by  the  steam-inductor,  except  in  reducing  the  smoke,  whilst  the  evaporation 
was  rather  less  rapid  than  with  the  ordinary  door;  but  that,  with  spreading 
firing,  the  evaporation  was  more  rapid  by  17  percent.  With  slack,  the 
evaporation  was  decidedly  superior,  both  in  rapidity  and  efficiency,  with  the 
steam-inductor. 

Trials  with  Self-feeding  Fire-grates  (  Vicar?  System).  —  The  fire-bars  are 
impressed  with  a  slow  reciprocating  movement,  the  effect  of  which  is  to 
cause  the  fuel  to  travel  gradually  and  steadily  from  the  front  to  the  back. 

The  comparative  performances  of  Vicars'  grate  and  the  ordinary  grate, 
are  shown  by  the  subjoined  results:  — 


Waterat  .oo'perhour, 


ROUND  COAL,  4-feet  Grate. 

*"' 


Waterat  3I2-  per  pound  coal, 


£9! 


6  feet. 

77-94 
67.56 

9.52 


With 
Economizer. 

78.97 
71.58 

10.56 
9-23 


SLACK,  length  of  Grate,  4  feet. 

S  Vicars'  grate,.... cubic  feet,  64.95 
coking  firing,....         „          60.72 
spreading  „     ...         „          77.72 
(  Vicars'  grate,  pounds,    9.82 

Waterat  212°  per  pound  coal,  <  coking  firing,      „         9.58 

( spreading  „         „         8.94 

It  is  seen  that,  with  slack,  Vicars'  grate  had  the  advantage  both  in  rapidity 
and  efficiency  of  evaporation  over  hand-firing  coking,  and  that  it  also 
evaporated  more  rapidly  with  round  coal,  but  less  efficiently;  though,  if  the 
rapidity  had  been  the  same,  the  efficiency  would  probably  also  have  been 
the  same.  Compared  with  spreading  firing,  Vicars'  grate  was  superior  in 
evaporative  efficiency  as  well  as  in  the  prevention  of  smoke,  though  it  did 
not  evaporate  so  rapidly. 

In  burning  large  quantities  of  coal  continuously  on  Vicars'  grate,  the 
rapidity  of  evaporation  fell  off  in  the  longer  trials,  and  to  some  extent  also 
the  efficiency.  The  6-feet  grates  were  very  little  behind  the  4-feet  grates  in 
efficiency. 

Comparative  Performance  in  Calm  and  Windy  Weather. — A  high  wind 
invariably  increased  the  performance.  The  average  results  under  all  con- 
ditions showed  that  10  per  cent,  more  coal  and  12  per  cent,  more  water 
were  consumed,  and  that  the  evaporative  efficiency  was  increased  4.4  per 
cent. 


MARINE   BOILER  AT  WIGAN.  78 1 

Comparative  Performance  when  the  Natural  Draught  was  increased  by  the 
aid  of  an  Auxiliary  Furnace. — An  auxiliary  furnace  was  put  in  action  at  the 
bottom  of  the  chimney,  so  as  to  increase  the  draught.  The  effect,  taking  the 
mean  of  a  number  of  trials,  was  to  raise  the  rapidity  of  evaporation  from  72.96 
to  84.09  cubic  feet  of  water  at  100°,  per  hour,  whilst  the  water  evaporated 
per  pound  of  fuel  was  raised  from  10.77  to  10.81  pounds.  The  mean 
efficiency,  thus  slightly  raised,  was  in  fact  an  average  of  two  opposite  effects ; 
for,  with  round  coal,  the  efficiency  was  reduced,  whilst  with  slack  it  was 
increased,  by  the  additional  draught. 

Mr.  Fletcher's  Conclusions. — Mr.  Fletcher  draws  the  following  conclusions 
from  the  experiments  on  stationary  boilers  at  Wigan: — ist.  That  the  coals  of 
the  South  Lancashire  and  Cheshire  district,  though  of  a  bituminous  and 
free-burning  character,  can  be  economically  burned  in  the  ordinary  class  of 
mill-boiler,  without  smoke.  2d.  That  the  double-flue  Lancashire  boiler, 
whether  with  steel  or.- iron  flues,  and  the  Galloway,  or  water-tube  boiler,  are 
practically  equal  in  performance;  and  that  both  of  them  develop,  when 
suitably  set  and  fired,  high  economic  results.  3d.  That  external  brickwork 
flues,  though  adding  but  little  to  the  yield  of  steam,  save  fuel.  4th.  That 
the  addition  of  a  feed-water  heater  or  economizer  is  a  decided  advantage, 
not  only  in  increasing  the  yield  of  steam,  but  also  in  diminishing  the  annual 
cost  of  boiler  repairs  and  coal. 

EVAPORATIVE  PERFORMANCE  OF  SOUTH  LANCASHIRE  AND  CHESHIRE 
COALS,  IN  A  MARINE  BOILER,  AT  WIGAN.     I866-68.1 

The  marine-boiler  was  a  copy  of  the  test-boiler  at  Keyham  Dockyard. 
The  shell  was  rectangular,  5  feet  wide,  for  two  furnaces,  7  feet  8  inches 
long,  and  8  feet  10  inches  high.  The  furnaces  were  i  foot  8^5  inches  wide, 
2  feet  83/8  inches  high  at  the  front,  rising  to  3  feet  high  at  the  back,  and 
6  feet  deep  from  front  to  back  or  tube  plate.  There  were  124  flue-tubes, 
2%  inches  in  diameter  inside,  and  5  feet  long,  placed  at  a  pitch  of  3^3 
inches  from  centre  to  centre.  The  chimney  was  18  inches  in  diameter, 
and  52  feet  8  inches  high  above  the  boiler,  or  59  feet  8  inches  above  the 
level  of  the  grates.  The  proportions  of  furnaces  which  were  finally  adopted, 
after  many  preliminary  trials,  were  as  follows : — Dead-plate,  i  o  inches  long, 
1 6  inches  below  the  crown  of  the  furnace;  grates,  3  feet  long,  inclined 
24  inch  to  a  foot;  bars,  ^  inch  thick,  air-spaces  yz  inch;  bridge  built  up 
to  a  level  9  inches  below  the  crown,  and  9^  inches  above  the  grate.  The 
fire-doors  were  fitted  with  a  sliding  grid  for  the  admission  of  air  into  a 
perforated  box  inside  the  door.  In  the  first  instance,  there  were  730  perfora- 
tions, giving  an  area  of  33  square  inches,  or  3.2  inches  per  square  foot  of  grate. 
They  were  afterwards  reduced  to  342  in  number,  16^  square  inches  in 
area,  or  1.6  inch  per  foot  of  grate. 

During  the  preliminary  experiments,  it  was  found  of  advantage  to  reduce 
the  length  of  grate  from  4  feet  to  3  feet,  to  adopt  a  blind  dead-plate  in 
preference  to  a  perforated  one,  and  to  slightly  lower  the  grate.  Fires 
of  6  inches,  9  inches,  12  inches,  and  14  inches  in  thickness  were  tried; 
the  greater  the  thickness  the  better  was  the  performance.  The  firing  was 
tried  on  the  spreading  and  on  the  coking  systems. 

1  The  author  is  indebted  for  the  particulars  of  these  trials  to  Mr.  Lavington  E.  Fletcher's 
Report.  See  note,  page  771. 


782  EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 

Coking-firing  was  adopted  as  the  standard  method,  with  fires  of  14  inches 
and  12  inches  thickness.  The  furnaces  were  charged  alternately,  and  the 
entrance  for  air  through  the  door  was  allowed  to  remain  open  for  a  few 
minutes  after  each  charge  was  delivered,  for  the  prevention  of  smoke.  For 
each  trial,  1000  Ibs.  of  round  coal  was  consumed,  lasting  3  hours  27 
minutes  as  an  average;  average  rate  of  consumption,  290  Ibs.  per  hour, 
or  28  Ibs.  per  square  foot  of  grate  per  hour.  The  feed-water  was  supplied 
at  ordinary  temperatures.  The  steam  was  generated  under  one  atmosphere 
of  pressure,  and  escaped  direct  into  the  air. 

Total  grate-area, 10.3  square  feet. 

Total  heating  surface : — 

Plate,  above  the  grate,....     95  square  feet. 

Tubes,  outside  surface, ...  413         „  508  „ 


Ratio  of  grate-area  to  heating  surface,  say  i  to  50. 

For  some  trials,  an  inverted  bridge  was  added  at  the  back  of  the  furnace, 
9  inches  clear  of  the  first  bridge.  By  thwarting  the  current,  it  was  instru- 
mental in  preventing  smoke,  and  in  slightly  increasing  the  evaporative 
efficiency, — by  1.7  per  cent. ;  though  at  a  loss  of  7^  per  cent,  of  evaporative 
rapidity. 

The  i4-inch  fire  excelled  the  9-inch  fire,  burning  coal  at  the  rate  of  27  Ibs. 
per  square  foot,  by  7  ^  per  cent,  for  rapidity  and  efficiency  of  evaporation. 

Comparing  the  coking  and  the  spreading  systems,  there  was  6%  per  cent, 
gain  by  the  coking  system  in  efficiency,  with  a  loss  of  10  per  cent,  in 
rapidity.  When  air  was  shut  off  at  the  doorway,  the  smoke-making  was 
accompanied  by  4  per  cent,  loss  of  efficiency,  with  a  small  advance  of  i  ^ 
per  cent,  of  rapidity. 

When  the  trials  were  prolonged,  to  burn  1500  Ibs.  of  coal,  as  against  the 
standard  of  1000  Ibs.,  the  rate  of  consumption  of  fuel  was  i  Ib.  more  per 
square  foot  per  hour,  and  of  water  2^  per  cent,  less;  the  average  efficiency 
was  reduced  5  per  cent. 

Table  No.  271  shows  the  general  results  arrived  at  by  Messrs.  Richardson 
and  Fletcher,  compiled  from  the  Report  of  Mr.  Fletcher.  The  vacuum 
in  the  chimney  [at  the  base,  probably]  was  observed  to  vary  from  ^  inch 
to  fully  7/j6  inch  of  water;  and  in  the  flame-box  from  %  inch  to  fully 
s  /I6  inch.  The  fires  were  maintained  at  14  inches  thick,  and  the  coal  was 
stoked,  on  the  coking  plan,  in  charges  of  from  29  to  38  Ibs.,  at  intervals 
of  from  ii  to  17  minutes.  The  perforations  in  the  fire-doors  were  opened 
intermittently,  and  the  doors  were  opened  a  little,  occasionally,  after  firing. 
Each  trial  lasted  for  from  3  to  4  hours.  The  quantity  of  ash  varied  from 
1^/2,  to  7  per  cent.,  and  of  clinker  from  0.6  to  3  per  cent. 

From  the  table,  it  appears  that  the  quantity  of  water  evaporated  varied 
from  44.12  to  51.63  cubic  feet  per  hour,  at  the  rate  of  from  10.37  to 
12.54  Ibs.,  at  2 1 2°, per  pound  of  coal,  averaging  11.54  Ibs.;  and  that  the 
coal  was  burned  at  the  rate  of  from  25^  to  31^  Ibs.  per  square  foot 
of  grate  per  hour.  The  duration  of  the  smoke,  which  was  very  light,  varied 
from  0.2  to  6  minutes  per  hour;  the  mean  duration  was  2.4  minutes  in 
the  hour. 

A  mixture  of  Hindley  Yard  coal  and  Welsh  coal-dust,  in  the  proportion 


MARINE  BOILER  AT   WIGAN. 


of  2  to  i,  was  tried,  and  the  results  are  given  in  the  table,  showing  an 
evaporation  of  11.83  Ibs.  of  water  per  pound  of  the  fuel,  and  at  the  rate  of 
41.38  cubic  feet  per  hour. 

Table  No.  271. — SOUTH  LANCASHIRE  AND  CHESHIRE  COALS — RESULTS 
OF  TRIALS  IN  A  MARINE  BOILER  AT  WIGAN.     1866-68. 

(Compiled  from  the  Report  of  Mr.  Lavington  E.  Fletcher  to  the  Association 
for  the  Prevention  of  Steam-Boiler  Explosions.) 

Total  area  of  fire-grates,  10.3  square  feet. 


COAL. 

Coal  Con- 
sumed 

iCur. 

Coal  per 
Square 
Foot  of 
Grate  per 
Hour. 

Water 
Con- 
sumed 
from  100° 
per  Hour. 

Water  per 
Square 
Foot  of 
Grate  per 
Hour. 

Water 
Evapor- 
ated from 
212°  per 
Pound  of 
Coal. 

Smoke 
Hour. 

FIRST  SERIES  OF  TRIALS. 
Hindley  Yard. 

cwt. 
2  ?2 

Ibs. 
2t  24 

cub.  feet. 
4.6  17 

cub.  feet. 
4.  4.8 

Ibs. 
12  7Q 

minutes, 
very  light 
O  2 

Worsley  Top  Four  Feet        

2*88 

•2  I    36 

4.8  SO 

•2  O4. 

^•jy 

IO  77 

4.  O 

Upper  Crumbouke 

2  64 

28  74. 

48  17 

6-vq 
A  57 

II   71 

1  7 

Lower  Crumbouke  

2  47 

26  4.1 

q.0.  1^ 

48  60 

4.  72 

11.^1 

12  4.^ 

i'8 

Upper  Three  Yards 

2  48 

27  OO 

46  26 

4.  4Q 

1^-4i 
1  1  60 

a1? 

Six  Feet  Rams 

2  44. 

26  50 

44.  7S 

4-71 

1  1   74. 

•o 

2  O 

Great  Seven  Feet  

2  77 

2Q  71 

"\1.74 

*H  Jx 
4  08 

II  71 

C.Q 

Blackrod  Yard    .     . 

2  71 

2C   14. 

AC  77 

4.  4.O 

12  l8 

2  4. 

Pemberton  Four  Feet  

2  64 

mym  ie\- 

•JQ  87 

ej  67 

SOI 
.we 

II  71 

2  Q 

Haigh  Yard. 

2  7S 

2C   C-J 

4.7  78 

4  oo 

12   C4. 

O  1J 

Furnace  Mine  

2  66 

28  07 

44  4Q 

4.  72 

10  40 

I  4. 

Bickerstaffe  Four  Feet.     . 

2  ^4. 

^u.yj 

27  67 

4S  28 

II  08 

O  O 

Rushy  Park  and  Little  Delf,  mixed 
Ince  mixed    .                         . 

2.£ 

2  6l 

30.29 
28  64 

50.67 
4.6  ?2 

4-92 
A   r  T 

II.2Q 
IO  QQ 

4-3 
i  6 

Arley  Mine  

2  26 

24.  46 

44.  12 

H-O1 
4.  28 

12  l8 

O  4 

Average  results  of  15  samples  ) 
of  coal                .     .                  ) 

2-55 

27.63 

47-25 

4-59 

H-54 

2.4 

Mixture  of  2  Hindley  Yard  coal  ) 
and  i  Welsh  coal-dust  \ 

2.21 

24.OO 

41.38 

4.02 

11.83 

0.0 

I 

2 

3 

4 

5 

6 

7 

NOTE. — The  quantities  in  column  6  have  been  recalculated. — D.  K.  C. 

The  effect  of  reducing  the  flue-surface  w^s  tried  by  plugging  up  one- half 
of  the  number  of  flue-tubes,  in  alternate  diagonal  rows,  so  that  the  tube- 
surface  was  reduced  by  206.5  square  feet.  The  comparative  results  obtained 
with  1 2-inch  fires  were  as  follows : — 

Flue-tubes    Half  the  Tubes 
all  open.          plugged  up. 

Coal  per  square  foot  of  grate  per  hour, Ibs.,  25  24 

Water  at  100°  evaporated  per  hour, cubic  feet,  45.78  43«oi 

Water  per  pound  of  coal,  as  supplied  at  212°,        Ibs.,  12.41  12.23 

Smoke  per  hour — very  light, minutes,     2.8  8.0 


Showing  that  with  half  the  tubes,  the  performance  was  nearly  as  good  as 
with  them  all  open. 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


Messrs.  Nicoll  &  Lynn,  for  the  Board  of  Admiralty,  made  two  inde- 
pendent series  of  trials  of  the  South  Lancashire  and  Cheshire  coals ;  in  the 
second  of  which  the  draught  was  increased  by  a  steam-jet  from  a  neighbouring 
boiler.  The  average  results  of  these  trials  are  placed  beside  those  of  Mr. 
Fletcher's  trials  in  the  annexed  table,  No.  272. 

Table  No.  272. — SOUTH  LANCASHIRE  AND  CHESHIRE  COALS — SUMMARY 
RESULTS  OF  PERFORMANCE  IN  THE  WIGAN  MARINE  BOILER, 

In  three  series  of  trials,  by  Messrs.  Richardson  &  Fletcher,  and  by 
Messrs.  Nicoll  &  Lynn.       __ — 


Particulars. 

Trials  by 
Messrs. 
Richardson 
and 
Fletcher. 

Trials  of  Messrs.  Nicoll 
and  Lynn. 

Without  Jet. 

With  Jet. 

Area  of  fire-grate                              square  feet 

10.3 

10.3 

10.3 

Coal  per  hour,  cwt. 

215 
27.63 

2-53 
27.50 

370 
41.25 

Coal  per  sq.  foot  of  grate  per  hour,  Ibs. 

\Vater  from  100°  per  hour                 cubic  feet 

47.25 

4-59 
11.54 

48.30 
4.69 
11.92 

69.13 
6.7I 
11.36 

Water  per  sq.  foot  of  grate  per  hour,       „ 
Water  from  212°,  per  pound  of  coal,  Ibs. 

Duration  of  smoke,  in  the  hour,  )        „,,-„„ 

i  •     i    .                                                               /              IIHIlUlCo 

verv  lisriit                                  \ 

2.4 

I.I 

0.0 

TRIALS  OF  NEWCASTLE  AND  WELSH  COALS  IN  THE  WIGAN 
MARINE  BOILER. 

A  mixture  of  Davidson's  Hartley  and  Hasting's  Hartley  (Newcastle 
coals),  and  a  mixture  of  Powell's  Duffryn,  Nixon's  Navigation,  and  Davis's 
Abercwomboy  (Welsh),  were  tried  in  the  Wigan  boiler — being  the  same  as 
some  coals  that  had  been  tried  at  Keyham  Dockyard  in  1863.  The  general 
results  of  the  trials  are,  for  comparison,  placed  together  with  those  of  the 
South  Lancashire  and  Cheshire  coals,  thus : — 

Table  No.  273. — NEWCASTLE  AND  OTHER  COALS: — COMPARATIVE 
RESULTS  OF  EVAPORATIVE  PERFORMANCE. 


COALS. 

Coal  per  Square 
Foot  of  Grate 
per  Hour. 

Water  at  100° 
per  Hour. 

Water  at  112° 
per  Ib.  of  Coal. 

Newcastle,  

Ibs. 
28.83 

cubic  feet. 

Ci.4? 

Ibs. 
II.  (K 

Welsh,  

'    J 

26.2O 

4860 

12.4.4. 

South  Lancashire  and  Cheshire  :  — 
Average, 

27  6^ 

4.7  2S 

I  I    CA 

Highest  evaporative  efficiency  ) 
(Haigh  Yard)                        \ 

25-53 

47.38 

12.54 

Lowest  evaporative  efficiency  ) 
(Worsley  Top  Four  Feet),  ) 

31.36 

48.50 

10.37 

MARINE   BOILER  AT   NEWCASTLE-ON-TYNE.  785 

Showing  that  the  average  of  the  South  Lancashire  and  Cheshire  coals  is 
inferior  in  rapidity  and  in  efficiency  of  evaporation  to  both  of  the  other 
coals,  and  that  though  the  best  of  the  South  Lancashire  coals  has  a  greater 
evaporative  efficiency  than  the  others,  the  rapidity  of  evaporation  was  less. 
This  comparison  is  corroborative  of  the  deductions  made  from  Delabeche 
and  Playfair's  analysis  and  trials  of  coals  from  the  several  districts  (see 
page  413). 

EVAPORATIVE  PERFORMANCE  OF  NEWCASTLE  COALS  IN  A  MARINE 
BOILER,  AT  NEWCASTLE-ON-TYNE,  185  7.* 

v  \  •  ^ ' 

These  experiments  were  made  to  test  the  evaporative  power  of  the 
steam-coal  of  the  Hartley  district  of  Northumberland.  The  experimental 
boiler  was  of  the  marine  type,  10  feet  3  inches  long,  7  feet  6  inches  wide, 
and  10  feet  high;  with  2  internal  furnaces,  3  feet  by  3  feet  3  inches  high, 
and  135  flue-tubes  above  the  furnaces,  in  9  rows  of  15  each,  3  inches  in 
diameter  inside,  5^  feet  long.  The  dead-plates  were  16  inches  long,  and 
2 1  inches  below  the  crown  of  the  furnace.  As  the  result  of  many  preliminary 
trials,  two  standard  lengths  of  fire-grates  were  fixed  upon — 4  feet  9  inches, 
and  3  feet  2^  inches,  with  a  fall  of  y2  inch  to  a  foot;  and  the  fire-bars  were 
cast  y?.  inch  thick,  with  air-spaces  from  ^  to  ^  inch  wide.  The  fire-doors 
were  made  with  slits  ^  inch  wide  and  14  inches  long,  for  the  admission 
of  air.  The  chimney  was  2  feet  6  inches  in  diameter.  A  water-heater  was 
applied  at  the  base  of  the  chimney,  in  the  thoroughfare;  it  contained 
76  vertical  tubes,  4  inches  in  diameter,  surrounded  by  the  feed-water. 

Total  area  of  fire-grates,  4  feet  9  inches  long,  28^  square  feet. 

PO-  do.         3    „    2%    „         „     191^ 

Heating  surface  of  boiler  (outside),  749  square  feet. 

Do.  water  heater, 320         „ 

Ratio  of  larger  grate-area  to  heating  surface  of  boiler,  i  to  26.28 
Do.     smaller       do.  do.  i  to  38.91 

Two  systems  of  firing  were  adopted,  as  "standards  of  practice:" — First, 
ordinary  or  spreading  firing,  in  which  the  fuel  was  charged  over  the  grate, 
and  the  whole  of  the  supply  of  air  was  admitted  through  the  grate. 
Second,  coking-firing,  in  which  the  fuel  was  charged,  i  cwt.  at  a  time, 
upon  the  dead-plate,  and  subsequently  pushed  on  to  the  grate,  making 
room  for  the  next  charge;  and  air  was  admitted  by  the  doorway  as  well  as 
by  the  grate.  Four  systems  of  furnace  were  tried,  of  which  Mr.  C.  W. 
Williams'  was  adjudged  by  the  experimentalists  to  have  rendered  the  best 
performance.  According  to  this  system,  air  was  admitted  above  the  fire 
at  the  front  of  the  furnace,  by  means  of  cast-iron  casings,  having  apertures 
on  the  outside,  with  slides,  and  perforated  through  the  inner  face,  next  the 
fire,  with  numerous  ^-inch  and  ^-inch  holes,  having  a  total  area  of  80  square 
inches,  or  5.33  square  inches  per  square  foot  of  grate.  Alternate  firing 
was  adopted  by  Mr.  Williams.  The  general  results  of  the  experiments  are 
given  in  table  No.  274. 

1  The  author  has  derived  the  particulars  of  those  trials  from  the  Report  of  Messrs.  Long- 
ridge,  Armstrong,  6° Richardson  to  the  Steam  Collieries  Association  of  Newcastle-on-Tyne. 

1857- 

50 


786 


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Remarks  on  the  Prevention  of 
Smoke,  &c. 

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ate  and  88  cubic  feet  through 
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MARINE   BOILER  AT   NEWCASTLE-ON-TYNE.  787 

The  experimentalists  reported  that  Mr.  Williams'  plan  gave  the  best 
results,  and  they  concluded:  "  ist.  That  by  an  easy  method  of  firing,  com- 
bined with  a  due  admission  of  air  in  front  of  the  furnace,  and  a  proper 
arrangement  of  fire-grate,  the  emission  of  smoke  may  be  effectually  prevented 
in  ordinary  marine  multitubular  boilers  whilst  using  the  steam-coals  of  the 
Hartley  district  of  Northumberland.  2d.  That  the  prevention  of  smoke 
increases  the  economic  value  of  the  fuel  and  the  evaporative  power  of  the 
boiler.  3d.  That  the  coals  from  the  Hartley  district  have  an  evaporative 
power  fully  equal  to  that  of  the  best  Welsh  steam-coals,  and  that,  practically, 
as  regards  steam  navigation,  they  are  decidedly  superior." 

These  gentlemen  made  a  trial  of  Aberaman  Welsh  coal,  and  they  found 
that  its  practical  evaporative  power,  when  it  was  hand-picked,  and  the 
small  coal  rejected,  was  at  the  rate  of  12.35  H>s.  of  water  per  pound  of 
coal,  evaporated  from  212°;  this  may  be  compared  with  the  best  result 
from  Hartleys  coal,  large  and  small  together,  in  table  No.  274,  which  was 
12.53  Ibs.  water  from  212°  per  pound  of  coal,  or  with  another  result  of 
experiment,  with  Hartley  coal,  not  given  in  the  table,  showing  12.91  Ibs. 
water  per  pound  of  coal.  As  a  check  on  these  results,  they  ascertained 
the  total  heat  of  combustion  of  the  two  coals  here  compared,  by  means 
of  an  apparatus  constructed  by  Mr.  Wright,  of  Westminster,  so  contrived 
that  a  portion  of  coal  is  burned  under  water,  and  the  products  of  combustion 
actually  passed  through  the  water,  which  absorbs  the  whole  heat  of  combus- 
tion. The  following  are  the  comparative  values : — 

Water  practically  Evaporated    Total  Heat  of  Combustion 
per  Pound  of  Coal.  in  Evaporative  Efficiency. 

Welsh  coal,  hand-picked, 12.35  Ibs 14.30  Ibs. 

Hartley  coal,  large  and  small, 12.91    „      14.63    „ 

The  experimentalists  also  point  out  the  "  elasticity  of  action "  of  the 
Hartley  coals:  they  burned  them  at  rates  varying  from  9  to  37^  Ibs. 
per  square  foot  of  grate  per  hour  without  difficulty,  and  without  smoke. 
The  Welsh  coal,  burned  at  the  rate  of  34^  Ibs.  per  foot  per  hour,  melted, 
it  is  said,  the  fire-bars  after  an  hour  and  a  half  s  work. 


TRIALS  OF  NEWCASTLE  AND  WELSH  COALS  IN  THE  MARINE  BOILER  AT 
NEWCASTLE,  FOR  THE  BOARD  OF  ADMIRALTY.  By  Messrs.  Miller 
&  Taplin.  1858. 

Messrs.  Miller  &  Taplin,  representing  the  Board  of  Admiralty,  conducted, 
in  1858,  a  series  of  trials  at  Newcastle,  with  the  same  marine  boiler  as  was 
employed  by  Messrs.  Longridge,  Armstrong,  &  Richardson,  the  object  of 
which  was  to  investigate  the  comparative  evaporative  power  and  other  pro- 
perties of  Hartley  coal  and  Welsh  steam-coal,  and  the  merits  of  Mr. 
Williams'  plan  of  smoke-prevention. 

The  fire-bars  were  i^f  inch  in  thickness,  and  had  ^j-inch  air-spaces. 
The  feed-water  was  passed  through  the  heater,  except  when  otherwise 
stated.  Mr.  Williams'  apparatus  was  constantly  in  action  when  Hartley 
coal  was  burned  without  smoke;  and  it  was  closed  when  this  coal  was  tried 
for  smoke  making,  also  when  Welsh  coal  was  burned. 


788 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


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Blaengwarn  Merthy 
Woolwich  Docky 


* 


790          EVAPORATIVE    PERFORMANCE  OF   STEAM-BOILERS. 

During  the  trials  of  Hartley  coal,  the  fires  were  maintained  at  from  1 2  to 
14  inches  in  thickness  on  the  grates,  the  coal  was  stoked  on  the  coking 
system,  the  fresh  charges  of  coal  having  been  delivered  at  the  front,  on 
each  side  of  each  grate  alternately;  and  the  incandescent  fuel  pushed 
forward  towards  the  bridge  before  charging. 

During  the  trials  of  Welsh  coal,  the  fires  were  maintained  at  from  8  to 
10  inches  in  thickness;  and,  in  charging,  the  fresh  coal  was  thrown  where 
it  was  required,  all  over  the  fire;  the  burning  fuel  never  being  touched  by 
any  firing  tool. 

The  cinders  that  fell  through  the  grates  were  constantly  raked  together 
and  thrown  upon  the  fires. 

The  results  of  the  trials  made  by  Messrs.  Miller  &  Taplin  have  been 
analyzed  and  compiled  into  the  table  No.  275,  in  which  the  results  of  the 
performances  of  the  West  Hartley  coals  are  grouped,  to  which  is  added  the 
results  obtained  from  Lambton's  Wallsend  house  coal,  as  a  bituminous  or 
highly  smoky  coal.  The  results  of  the  trials  of  the  South  Welsh  coal  are 
likewise  grouped  in  the  table.  Separate  trials  of  each  coal  were  made,  in 
which  the  feed-water  was  delivered  direct  into  the  boiler,  the  heater  having 
been  for  this  purpose  disconnected. 

Messrs.  Miller  &  Taplin  concluded  from  the  results  of  their  experiments : 
ist,  that  when  the  smoke  from  Hartleys  coal  is  consumed,  the  evaporative 
value  of  this  coal  is  nearly  equal  to  that  of  Welsh  coal,  whilst  its  rapidity  of 
combustion  and  evaporation  of  water  is  greater;  2d,  that  the  Hartleys  coal  is 
less  liable  to  be  broken  up  by  movement  than  Welsh  coal ;  and  that  it  is  less 
disintegrated  by  long  exposure  to  the  atmosphere  than  Welsh  coal;  3d, 
that  Hartleys  coal  may  be  burned  without  making  smoke,  by  the  use  of 
Mr.  C.  W.  Williams'  apparatus. 

TRIALS  OF  WELSH  AND  NEWCASTLE  COALS  IN  A  MARINE  BOILER 
AT  KEYHAM  FACTORY.     1863. 

Trials  of  Welsh  and  Newcastle  coals,  singly  and  in  combination,  were  con- 
ducted with  the  coal-testing  boiler  at  Keyham  Factory,  by  Mr.  T.  W.  Miller. 
The  boiler  is  the  pattern  from  which  the  Wigan  boiler,  described  at  page  781, 
was  made,  and  of  which  it  was  a  copy,  except  that  the  flue-tubes  are  2  inches. 
The  dead-plate  was  10  inches  below  the  crown  of  the  furnace,  and  6  inches 
in  length.  The  grate  was  made  of  two  different  lengths,  4  feet  and  3  feet, 
and  inclined  2  inches  per  foot.  The  bridge  was  8  inches  below  the  crown. 
Two  different  doors  to  each  furnace  were  employed  during  the  trials :  one,  a 
common  door,  with  a  few  small  perforations  for  air;  the  other  was  made  double, 
and  air  entered  from  the  bottom,  and  passed  through  numerous  ^-inch  holes 
into  the  furnace.  The  air-way  in  the  second  door  amounted  to  60  square 
inches,  equal  to  8.6  inches  per  square  foot  of  the  longer  grates,  and  11.4 
inches  for  the  shorter  grates.  The  charges  of  coal  were  from  16  to  19  Ibs. 

Total  grate-area, 4  feet  long, 13. 75  square  feet. 

,,  3        »        io-3  » 

Total  heating  surface,  plate, 72.5  „ 

„  „      tubes  (outside),    324.5       397.0 


Ratios  of  grate-areas  to  heating  surface :  larger  grates,. . .  i  to  29 

„          smaller    „     ...  i  to  38.5 


STATIONARY  BOILER  IN   AMERICA.  791 

Mr.  Miller  reported  that  "the  combinations,  in  equal  proportions,  of 
Welsh  and  Newcastle  coals,  while  they  produced,  on  the  average,  nearly 
equal  economical  results,  measured  by  the  quantity  of  water  evaporated  by 
i  Ib.  of  fuel,  they  produced  on  the  average  greater  rapidity  in  evaporationy 
and  that  they  on  the  average  produced  the  least  amount  of  smoke."  He 
also  found  that  the  small  Welsh  coal  could  be  burned  beneficially  in  mix- 
ture with  Newcastle  coal.  The  general  results  of  his  experiments  are  given 
in  table  No.  276. 

EVAPORATIVE  PERFORMANCE  OF  AMERICAN  COALS  IN  A  STATIONARY 

BOILER.     1843. 

The  American  coals,  of  which  the  composition  was  given  page  418,  were 
tried  for  their  evaporative  performance  by  Professor  Johnson,  with  a  flat- 
ended  cylindrical  boiler,  3*4  feet  in  diameter  and  30  feet  long,  with  two 
thorough  internal  flues,  10  inches  in  diameter.  The  grate  was  placed 
below  the  boiler  at  one  end,  and  was  5  feet  long  by  3  feet  wide ;  it  was 
9  inches  below  the  boiler  at  the  front,  and  10  inches  at  the  back.  The 
bars  were  ^  inch  thick,  with  ^-inch  air-spaces.  The  grate  could  be 
shortened  8  inches,  by  inserting  a  perforated  air-plate  at  the  bridge;  and 
n^  inches  at  the  front,  by  inserting  a  coking-plate.  The  air  for  com- 
bustion was  heated  in  a  chamber  under  the  ash-pit,  before  passing  through 
the  grate. 

The  gases  passed  under  the  boiler  to  the  back,  returned  through  the  inside 
flues,  and  made  another  circuit  of  the  boiler  by  a  wheel-draught  through  side 
flues.  The  chimney  was  18  inches  square,  and  61  feet  high  above  the 
grate. 

Area  of  grate : — 

Full  length, 16.25    square  feet. 

With  air-plate, i4-°7  „ 

With  air-plate  and  dead-plate, 1 1.375  „ 

Heating  surface : — 

Lower  flue, 130  „ 

Two  flue-tubes, 157  „ 

Two  side-flues, 90.5  „ 


Total, 377.5 

:ea 
heating  surface: 


Ratio  of  grate-area  to  J  Full  length) ,  tQ 


With  air-plate, i  to  26.8 

With  air-plate  and  dead-plate,     i  to  33.2 

The  water  was  evaporated  into  steam  of  from  6  Ibs.  to  7  Ibs.  per  square 
inch  above  the  atmosphere.  The  coal  was  delivered  in  charges  of  from 
100  Ibs.  to  1 10  Ibs.  The  condensed  results  of  performance  are  given  in  table 
No.  277.  The  average  results  were  that  in  burning  7  Ibs.  per  square  foot  of 
grate  per  hour,  9^  Ibs.  of  water  was  evaporated  from  2 1 2°  F.  per  pound  of  coal. 

The  inferior  evaporation  for  the  bituminous  caking  coals  is  accounted 
for  by  its  imperfect  combustion,  evidenced  by  the  smoke  which  escaped 
in  considerable  quantity. 


792 


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794 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


Table  No.  277. — AMERICAN  COALS: — RESULTS  OF  THEIR  EVAPORATIVE 
PERFORMANCE  WITH  A  CYLINDRICAL  STATIONARY  BOILER  AT  THE 
NAVY  YARD,  WASHINGTON.  1843. 

(Reduced  from  the  Report  of  Professor  W.  R.  Johnson.) 


COAL. 

Area  of 
Fire- 
grate. 

Coal 

Consumed 
per  Hour. 

Coal  per 
Square 
Foot  of 
Grate 
per  Hour. 

Water 
Evapor- 
ated from 
Ordinary 
Tempera- 
ture per 
Hour. 

Water 
per 
Square 
Foot  of 
Grate 
per  hour. 

Water 

Evapor- 
ated from 

212°  F. 

per  Ib.  of 
Coal. 

sq.  feet. 

Ibs. 

Ibs. 

cub.  feet. 

cub.  feet. 

Ibs. 

Anthracites  (7  samples),... 

14.30 

94.94 

6.64 

12.37 

0.87 

9-63 

Free-burning  bituminous 
coals  (n  samples),  

14.14 

99.16 

7.01 

13-73 

0.97 

9.68 

Bituminous  caking  coals 
(Virginian,  I  o  samples)  , 

14.15 

105.02 

742 

12.16 

0.86 

8.48 

Averages, 

14.20 

9971 

7.02 

12.75 

0.90 

9.26 

CONDITIONS  OF  SMOKE. 

Anthracites,  No  smoke. 

Free-burning  bituminous  coals, Little  smoke,  and  mostly  when  charging. 

Bituminous  caking  coals,  Smoke  considerable,  in  one  instance  constant. 

The  air-plate  was  tried  both  open  and  closed  for  each  coal,  with  various 
effect,  good  and  bad.  The  average  results  for  the  open  air-plate  proved  a 
gain  in  efficiency  and  a  loss  in  rapidity,  thus : — 

Gain  of  Efficiency.          Loss  of  Rapidity. 

COALS.  

per  cent.  per  cent. 

Anthracites, 0.43  14.9 

Free-burning  coals, 2. 13 2.68 

Caking  coals, 1.96  1.48 

Foreign  and  western  coals, 3.38  5.37 

Showing  that  the  open  air-plate  was  most  beneficial  for  efficiency  and  least 
injurious  for  rapidity  with  the  smoke-making  coals. 

Surplus  air  in  the  products  of  combustion.— By  analysis,  it  was  found  that 
twice  the  quantity  of  atmospheric  air  that  was  chemically  necessary,  passed 
through  the  furnace. 

Temperature  of  the  air  and  smoke. — The  average  temperatures  were  as 
follows : — 

External  air, 73°  F. 

Air  on  arriving  at  the  grate, 250°.     Heated  177°. 

Ga^s  on  arriving  at  the  chimney, 292°.     Excess  above  steam  65°. 

Draught-gauge, 307  inch  of  water. 

Influence  of  soot  in  the  flues. — It  was  observed  that  whilst,  in  the  perform- 
ance of  the  anthracites,  day  after  day,  the  temperature  at  the  chimney  and 
the  evaporative  efficiency  were  practically  constant,  with  the  smoky  coals 
the  temperature  rose  and  the  efficiency  fell  off.  In  three  instances  of 
caking  coal,  the  temperature  rose  75°  from  an  average  of  298°  F.,  on  the 


MARINE  BOILER   IN   AMERICA.  795 

first  day,  to  373°  F.  on  the  last  day;  and  the  efficiency  fell  off  i  lb.,  from 
8.66  to  7.68  Ibs.  These  effects  are  due,  of  course,  to  the  accumulation 
of  soot  on  the  surface  of  the  boiler,  and  the  impediment  thus  caused  to  the 
passage  of  heat. 

Level  of  the  grate. — The  grate  was  tried  at  7  inches  and  12  inches  below 
the  crown,  the  standard  level  having  been  9  inches.  The  trials  showed 
that  the  7-inch  level  was  5^  per  cent,  better  than  the  9-inch,  and  the 
9-inch  level  8  per  cent,  better  than  the  1 2-inch. 

Effect  of  cutting  off  the  two  side-flues. — Reducing  thus  the  heating  surface 
by  90.5  square  feet,  the  comparative  performance  with  and  without  the 
side-flues  was  as  follows : — 

ANTHRACITE : Water  per  lb.  of  Coal.    Water  per  hour. 

Without  side-flues, 9.96  Ibs.  ...   13.33  cubic  feet. 

With          do 10. 1 1    „     ...   14.03         „ 

CAKING  COAL: — 

Without  side-flues, 7.80    „     ...   11.72         „ 

With          do 8.52    „     ...   11.30         „ 


Showing  that,  whilst  the  average  rapidity  was  not  affected,  the  efficiency 
was  diminished  by  the  closing  of  the  side-flues. 

EVAPORATIVE  PERFORMANCE  OF  AN  EXPERIMENTAL  MARINE  BOILER, 
NAVY  YARD,  NEW  YORK,  U.S.1 

Mr.  Isherwood  made  trials  of  an  experimental  multitubular  marine 
boiler  under  cover,  on  land.  The  boiler  was  covered  with  felt  stitched  on 
canvas.  Steam  of  20  Ibs.  effective  pressure  per  square  inch  was  generated 
and  blown  off.  The  shell  was  7  feet  7  inches  deep,  3  feet  i  ^  inch  wide, 
and  6  feet  5  inches  high,  with  a  single  fire-flue  26  inches  in  diameter,  and 
24  flue-tubes  above  the  fire-flue,  3  inches  in  diameter  outside,  5  feet 
10^  inches  long.  With  a  4-inch  dead-plate,  the  grate  was  5  feet  long, 
inclined,  being  12  inches  below  the  crown  at  the  front,  and  15^2  inches  at 
the  bridge. 

Area  of  fire-grate, 10.8  square  feet. 

Heating  surface : — furnace, 1 9. 7  square  feet. 

smokebox, 25-75         » 

tubes, 100.78         „ 

uptake, 4.02         „         150.30     „ 

Ratio  of  grate-area  to  heating  surface, i  to  14. 

The  fires  were  5  inches  in  thickness,  using  Pennsylvanian  anthracite  of 
medium  quality.  The  refuse  averaged  about  20  per  cent,  of  the  fuel. 

In  the  following  selection  of  the  results  of  the  performance,  the  equiva- 
lent quantities  of  water  as  evaporated  from  and  at  212°  R,  are  substituted 
for  the  original  quantities: — 

1  Experimental  Researches  in  Steam  Engineering,  vol.  ii.  1865. 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


Table  No.  278. — EVAPORATIVE  PERFORMANCE  OF  EXPERIMENTAL 
MARINE  BOILER,  AT  THE  NAVY  YARD,  NEW  YORK,  U.S. 


ist 

Area  of  Grate. 

Series  of 

Heating 
Surface. 

Trials:—  Varyi 

Ratio  of  Heating 
Surface  to  Grate. 

ng  Rate  of  Comt 

Coal  per  Square 
Foot  of  Grate. 

*)ustion. 

Water  per  Pound  of 
Coal  from  and  at  212°. 

square  feet. 

square  feet. 

ratio. 

Ibs. 

Ibs. 

A 

10.8 

^o-S 

14 

5-57 

9.27 

B 

10.8 

150.3 

14 

10.99 

8.95 

C 

10.8 

150-3 

14 

16.57 

7-94 

D 

10.8 

IS°-3 

14 

22.10 

7.80 

E 

10.8 

150.3 

14 

27.76 

7.40 

^d  Series  of  Trials:  —  Varying  Area  of  Grate. 

I 

8.64 

149.0 

17.24 

15 

8.58 

J 

6.48 

148.0 

22.84 

15 

8.28 

K 

4.32 

147.0 

34.03 

15 

8.93 

Ajh  Series  of  Trials  :  —  Constant  Total  Quantity  of  Fuel  Consumed  per 
Hour,  with  Varying  Grates. 

L 

8.64 

149,0 

17.24 

20.73 

8.03 

M 

6.48 

148.0 

22.84 

27.42 

7-43 

N 

4-32 

147.0 

34.03 

27.58 

7.24 

%th  Series  of  Trials  :  —  Heating  Surface  of  Tubes  cut  off. 

V 

10.8 

45-5 

4.21 

16.57     '.; 

5.91 

w 

10.8 

45-5 

4.21 

16.58 

6.10 

X 

10.8 

45-5 

4.21 

11.77 

6.64 

EXPERIMENTS  ON  THE  COMPARATIVE  EVAPORATIVE  PERFORMANCE  OF 
STATIONARY  BOILERS  IN  FRANCE.     1874. 

A  commission  was  appointed  by  the  Sotiete  Industridle  de  Mulhouse  to 
test,  under  identical  conditions,  the  comparative  performance  of  a  French 
or  elephant  boiler,  a  double-flue  Lancashire  boiler,  and  a  so-called  "  Fairbairn" 
boiler.1 

Lancashire  boiler. — 6.56  feet  in  diameter,  25.75  feet  long;  flues  27.5  inches 
in  diameter,  with  internal  fire-place  28.5  inches  in  diameter.  Shell-plates 
..64  inch,  flue-plates  ^  inch  thick.  Grate  inclined;  mean  level  below  crown, 
1 6  inches.  Fire-bars  .6  inch  thick,  air-spaces  j^  inch. 

"Fairbairn"  boiler. — Two  cylinders  4.1  feet  in  diameter,  25.75  feet  long; 
central  fire-tube  27.5  inches  in  diameter,  enlarged  at  the  end  to  form  an 
internal  fire-place  28.5  inches  in  -diameter.  The  two  cylinders  were  united 
to  a  third  above  them,  3.75  feet  in  diameter,  23  feet  long,  by  three  neck- 
ings or  pipes,  14  inches  in  diameter,  from  each  lower  cylinder.  Plates 

1  Bidletin  de  la  Societe  Industrielle  de  Mulhouse,  June,  1875.  See  also  Proceedings  of  the 
Institution  of  Civil  Engineers,  vol.  xliii.  p.  377,  where  an  abstract  of  the  Report  is  pub- 
lished. 


STATIONARY   BOILERS   IN   FRANCE. 


797 


y2  inch.     Grate  inclined;  mean  level  below  crown,  16  inches.     Fire-bars 
.6  inch,  air-spaces  %  inch. 

French  boiler. — Body  3.74  feet  in  diameter,  29.5  feet  long.  Three 
heaters,  1.64  feet  by  32.8  feet  long,  united  to  the  body  by  three  neckings 
to  each.  Plates  of  body  %  inch;  of  heaters  .4  inch  thick.  Grate  hori- 
zontal; level  below  middle  heater,  18  inches,  and  below  side  heaters,  16 
inches. 


"  Fairbairn." 

Lancashire. 

French. 

Length  of  boiler,  feet 

23  &  2;.  75 

25  75 

29  5  &  32  8 

Total  heating  surface,  .  .  .           square  feet 

I  OI7 

^:>-/5 
612 

607 

Length  of  grates,  feet 

A.  cq 

At  ? 

U«J/ 

A  2  T 

Combined  width  of  grates,  „ 

A    C  -3 

^•O  J 
4C-? 

A  76 

Total  grate-area,  square  feet 

2O.5 

20  5 

2O  I 

Ratio  of  heating  surface  to  grate-area,.... 
Total  capacity                             cubic  feet 

i  to  49.5 

6A2  C 

i  to  29.8 

6"?7  5 

i  to  30.3 

C  or    T 

\Vater-space,  

CAA  7 

UJ/O 

AI  2  5 

5J1-1 

4O8   I 

Steam-space,  „ 

Q7  8 

225  o 

1  2"?  O 

Heating  surface  per  cubic  )    (            f 
foot  of  water,....  }    scluare  feet 

1.87 

1.48 

1.49 

Total  weight,  with  accessories,  tons 

IQ  6 

166 

IA  C 

Weight  per  square  foot  of  heat-  )      ,, 
ing  surface,  \ 

42.4 

59-7 

*4O 
52.5 

The  gases  in  the  Fairbairn  boiler  passed  from  the  flues  by  the  sides  of 
the  lower  cylinders,  and  returned  by  the  sides  of  the  upper  cylinder,  towards 
the  chimney.  In  the  Lancashire  boiler,  they  passed  from  the  inside  flues 
on  each  side  to  the  front,  and  thence  under  the  boiler  to  the  chimney.  In 
the  French  boiler,  the  current  was  not  divided,  but  after  heating  the  three 
heaters  it  wound  round  the  boiler.  The  flues  delivered  into  the  same  chim- 
ney. The  temperature  in  the  flues,  just  at  the  chimney,  about  4  inches 
above  the  bottom,  was  taken  every  five  minutes.  The  steam  was  maintained 
at  from  4.6  to  5  atmospheres.  The  feed-water  was  supplied  at  from  79°  to 
84°  F.  The  regular  daily  work  lasted  from  6  a.m.  to  6  p.m.,  with  i  ^  hour 
interval;  working  time,  10^  hours.  The  coal  consumed  in  getting  up 
steam  was  included  in  the  consumption.  Two  days  before  the  trial,  each 
boiler  was  emptied  and  was  thoroughly  cleaned  inside  and  outside.  Each 
trial  lasted  several  days  consecutively.  The  coals  consumed  were  Ronchamp 
and  Saarbrucken,  the  general  composition  of  which  is  indicated  by  the 
following  analysis  t1 — 

Gaseous  Elements  onfy,  or  "Pure  Fuel" 


Carbon. 

Hydrogen. 

Oxygen 
and 
Nitrogen. 

Actual  Heat  of 
Combustion  of 
One  Pound 
of  Pure  Fuel. 

Ronchamp 

per  cent. 

88  5o 

per  cent. 
A  60 

per  cent. 
672 

English  units. 

16  416 

Saarbrucken,  

°y«py 
81  10 

A  75 

HTC 

TC    -32O 

1  "  Calorimetric  Trials  and  Analysis  of  Coals  and  Lignites,"  by  A.  Scheurer-Kestner 
and  C.  Meunier-Dollfus.  See  Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xliii. 
p.  396. 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


Table  No.  279 — FRENCH  AND  ENGLISH  BOILERS. — RESULTS  OF 
EVAPORATIVE  PERFORMANCE. 

(Reduced  from  the  Report  of  the  Mulhouse  Commission.) 
FUEL. — RONCHAMP  AND  SAARBRUCKEN  COALS. 


BOILER,  AND  FUEL. 

Coal  Consumed  per 
Hour. 

Water  Evapor- 
ated 3$  Hour  from 
and  at  212°  F. 

Water 
per  Ib. 
of 
Entire 
Coal. 

Tem- 
pera- 
ture of 
Gases. 

Air 
drawn 
in  per 
Ib.  of 
Coal. 

Total. 

Per  Sq. 
Foot  of 
Grate. 

Ash. 

Total. 

PerSq. 
Foot  of 
Grate. 

RONCHAMP  COAL. 
Heavy  Firing. 
"Fairbairn,"  
Lancashire 

cwt. 

3.39 
3.50 
3-69 

Ibs. 

18.53 
I9-I5 
20-57 

pr.  c't. 

13-8 
I4.I 
I4.I 

cub.  ft. 
56.06 

53-45 
54-73 

cub.  ft. 

2.73 
2.61 

2.72 

Ibs. 

9.21 

8.50 
8.26 

Fahr. 

421° 
572 
562 

cu.  ft. 

226 

183 

I94 

French,  

RONCHAMP  COAL. 
Light  Firing. 
'  '  Fairbairn,  "  

1.96 
1.91 

2.04 

10.70 
10.41 
11.36 

13-5 

I4.6 
13.6 

3i-i4 

30-52 
31-38 

1.52 

1.49 

1.56 

8.86 
8.92 
8.58 

337° 
406 

425 

26l 
194 
193 

Lancashire.                

French,  

SAARBRUCKEN  COAL. 
"  Fairbairn  " 

3.04 
3.02 

3-n 

16.59 
16.50 
17.32 

10.6 

9-7 
9-4 

43-20 
40.69 
41.89 

2.  II 

i-99 
2.08 

7-93 
7-51 
7-51 

402° 
554 
544 

195 
1  80 
179 

Lancashire,  

French 

GENERAL    AVERAGES    OF 
THE    FOREGOING    PER- 
FORMANCES. 
'  '  Fairbairn,  "  

2.80 

2.81 

2.95 

I5-27 

15-35 
16.42 

12.6 
12.8 

12.4 

43-74 
41-55 
42.67 

2.12 
2.03 
2.12 

8.67 
8-33 

8.12 

3870 
5" 
5io 

227 

1  86 

189 

Lancashire 

French,  

AVERAGES  OF  3  DAYS'  PER- 
FORMANCE,  when  equal 
rates  of  evaporation  were 
effected. 
Lancashire.         .  • 

3.57 

3-57 

iq.50 
19.87 

— 

54.10 

54-32 

2.64 
2.70 

8.44 
8-49 

587° 
572 

165 
197 

French,  

EVAPORATIVE  PERFORMANCE  OF  LOCOMOTIVE  BOILERS. 

The  author  collected,  from  various  trustworthy  sources,  the  results  of 
the  performance  of  locomotive  boilers,  of  the  earliest  as  well  as  the  most 
recent  designs,  and  has  reduced  them  and  placed  them  together  with  the 
results  of  his  own  observations,  in  table  No.  2  So.1 

Boilers  of  nearly  every  size  and  variety  that  have  been  used  in  England, 
are  represented  in  the  table;  the  areas  of  grate  vary  from  6  to  24  square 
feet,  the  heating  surfaces  from  40  to  2000  square  feet,  and  the  ratios  of  sur- 
face to  grate,  or  the  surface-ratios,  from  40  to  i  to  100  to  i.  The  fuel  used 
was  coke,  except  in  a  few  specified  instances  of  boilers  designed  for  burning 
coal,  in  which  coal  was  used. 

1  These  data  are  derived  from  the  author's  work  on  Railway  Machinery,  1855,  page  156; 
and  Railway  Locomotives,  page  33*.  Reference  is  made  to  these  works  for  information  on 
the  details  of  the  boilers. 


LOCOMOTIVE  BOILERS. 


799 


Table  No.  280. — LOCOMOTIVE-BOILERS: — PROPORTIONS  AND 
RESULTS  OF  EVAPORATIVE  PERFORMANCE. 

The  Fuel  used  was  Coke,  except  when  Coal  is  specifically  stated. 


No. 

Name  of  Locomotive. 

Area  of 
Fire- 
grate. 

Heating 
Surface 
(Tubes 
measured 
on  the 
Outside). 

rlatio  of 
Heating 
Surface 
to 
Grate. 

Coke 
consumed 
>er  Square  Foot 
of  Grate 
per  hour. 

Water 
Consum- 
ed per 
Square 
Foot  of 
Grate 
per  hour. 

Water 
Evapor- 
ated per 
Pound 
of  Coke, 
from 
and  at 

212°  F. 

j 

EARLIEST  LOCOMOTIVES. 
Killingworth,  .        

sq.  ft. 
7.O 

sq.  ft. 
41.25 

ratio. 

6 

Ibs. 
44  (coal) 

cu.  ft. 
2.3 

Ibs. 
4.O2 

2 
3 

Do.          improved,  .  .  . 
Rocket,  .  .                   ..... 

10.9 

6 

124 

138 

11,4 

21 

57  (coal) 
35.5 

4 
3 

5.32 
6.27 

4 

Phoenix,  

6 

/• 
326 

55 

54 

5-7 

7.86 

Atlas  . 

Q.2O 

275 

30 

60 

5-  *4 

6.35 

£ 

Star,  

7.76 

359 

2 

92 

8.22 

~-jj 
6.53 

8 

Average  of  4  locomotives, 
•  Soho 

6.5 

8.44. 

348 
412 

53-5 

35 

90 

100 

130 

9.8 

10 
13  O3 

8.04 
7.42 
7.38 

10 
ii 

Hecla,  

8.34 

418 

49 

92 
125 

II 
II.  3 

6^65 

12 
13 

Bury's  goods  locomotives, 
Bury's  passenger       ,, 

GT.  WESTERN  RAILWAY. 
Ixion 

9.2 
9.2 

13.4. 

461 
387 

699 

50 

42 

52 

in 

112 
138 

9.24 
8.15 

1C 

6.15 
4-93 

8-33 

IiJ 

Hercules,            

13.6 

vyy 
699 

51.4 

IO5 

15 

10.70 

16 
17 

Etna,  Capricornus,  
Giraffe,  

11.4 

12.5 

467 
608 

41 

48.6 

97 
76 

10.7 
8.8 

8.21 

8.61 

18 

IQ 

Mentor,  Cyclops,  
Royal  Star,                      .  . 

136 

ii.  7 

699 
822 

51-4 
7O 

69 
91 

8 
10.8 

8.67 
8.85 

20 
21 

Pyracmon  Class,  
Aiax. 

18.44 

13.67 

1363 
IO67 

74 
78 

69 

84 

8-4 

II.  2 

9-09 
0.90 

22 

23 
24 

25 
26 

27 

28 
29 
30 

•21 

Great  Britain,  Iron  Duke, 
Great  Britain  Variety,  .... 
Courier  Variety,  

LONDON  AND  NORTH- 
WESTERN RAILWAY,  &c. 
A,  York  &  North-Mid-  ) 
land  Railway,  j 
Hercules,  York&North-  1 
Midland  Railway,  .  ..  ) 
Sphynx,  Man.,  Sheffield,  [ 
(      &  Lincoln  Railway,    j 
(Later  engines.) 
Heron,  L.  &  N.-W.  Ry., 
No.  291, 
No.  300,         ,,          ,, 

SOUTH-EASTERN  RAILWAY 
No    142, 

21 
21 

23.62 

9.6 

9.6 
10.56 

10.5 
19 

22 
14  7 

1938 
1938 
1866 

903 
828 
1056 

782 
1449 
1263 

1158  2 

92 
92 

79 

94 
86 

100 

74-5 
76.26 

57-41 
788 

8* 

90 

75 

132 

105 
i57 

90 
56-5 
5o-7 

62.25  (coal) 

II 
II 

8.6 

17 
15 

22.1 

II.  I 

6.2 

6.6 
8  77 

9-95 
9.17 
8.60 

10.52 
10.70 
10.41 

9.29 
9^28 

10.15 

32 

No.  118,  

26.2^ 

063.5 

/"• 
36.7 

30.  86  (coal) 

4.  ^4 

10.60 

•2-2 

No     58 

12.25 

7O5  7 

•j    i 

57  6 

61  22  (coal) 

8.60 

10  13 

34 

No. 

44.49  (coal) 

7.35 

11.91 

•2C 

No    142          .                 .  . 

14.7 

II58.2 

78.8 

55.71 

7.73 

q  77 

3* 

No.  105,  

IO.5 

623.1 

59.3 

55.91 

9.43 

11.68 

^7 

No       Q 

IO.  5 

-   J 
623.  1 

cq.  3 

66.19 

IO.OO 

10  06 

8oo 


EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 


Table  No.  280  (continued). 


No. 

Name  of  Locomotive. 

Area  of 
Fire- 
grate. 

Heating 
Surface 

(Tubes 
measured 
on  the 
Outside). 

Ratio  of 
Heating 
Surface 
to 
Grate. 

Coke  Consumed 
Per  Square  Foot 
of  Grate. 

Water 
Consum- 
ed per 
Square 
Foot  of 
Grate. 

Water 

Evapor- 
ated per 
Pound 
of  Coke, 
from 
and  at 

212°  F. 

sq.  ft. 

sq.  ft. 

ratio. 

Ibs. 

cu.  ft. 

Ibs. 

LONDON  AND  SOUTH- 

WESTERN RAILWAY. 

38 

Snake,  

12.4 

'" 

985 

79 

87 

12.26 

10  no 

<JW 

Canute  (coal  burning  loco- 

-7     J 

/  .7 

/ 

3-7 

motive)  :  — 

39 

Canute,  feed-  water  heat-  ) 
ed,  tiles,  j 

16 

87I 

54-4 

35  (coal) 

6.18 

11.02 

40 

Canute,  feed  -water  heat-  ( 
ed,  tiles,  f 

,, 

5> 

,, 

57  (coal) 

9.65 

10.57 

41 

Canute,  cold  feed-  water,  1 
tiles,  f 

„ 

,, 

„ 

49  (coal) 

7-77 

9.90 

42 

Canute,  feed-  water  heat-  > 
ed,  no  tiles,    .               ) 

,, 

If 

,, 

42  (coal) 

6.42 

9-54 

43 

Canute,  cold  water,  no  ) 
',      tiles                              } 

„ 

}) 

M 

58  (coal) 

8.89 

9-56 

44 

Canute,  feed-  water  heat-  ) 
ed   tiles                        \ 

I 

„ 

„ 

46  (coke) 

6.46 

8.76 

45 

Canute,  feed-  water  heat-  ) 
ed,  no  tiles,  j" 

,, 

,, 

„ 

49  (coke) 

7.17 

9-13 

46 

Canute,  cold  water,  no  ) 
tiles,  f 

» 

J  J 

» 

54  (coke) 

8.69 

10.04 

CALEDONIAN  RAILWAY,&C. 

47 
48 
49 

No.  33,  Caledonian  Ry., 
No.  42, 
No.  43, 

10.5 
10.5 
10.5 

i 

79 

75 
75 

42 

11 

9-2 

12.46 

IO.  II 

11.31 

50 

No.  51, 

10.5 

788 

75 

45 

6-7 

11.04 

51 

52 

}No.  13,           „               | 

10.5 
10.5 

788 

75 
75 

57 

11.6 

8.2 

8.09 
10.71 

53 

No.  13, 

9.0 

788 

87.6 

102 

14.7 

9-52 

54 

Nos.  125,  127,  „ 

n-37 

1050 

t2 

66 

8.66 

9.72 

55 

No.  102, 

ii.  8 

974 

2-5 

94 

10.3 

8.15 

56 

Orion,  Sirius,  E.  &G.Ry., 

12.23 

758 

62 

44 

6.29 

10.71 

57 

America,  Nile,         ,, 

1  1.  10 

736 

66.3 

70 

8.8 

9-31 

58 

Pallas, 

16.04 

818 

51 

38 

6 

10.47 

59 

Brindley,                   ,, 

9-15 

802 

87.65 

54 

7.2 

60 

Orion,  G.  &  S.-W.  Ry., 

9.24 

495 

53.6 

84 

9-4 

8  28 

61 

Queen, 

10.5 

65-5 

87 

10 

8^7 

PORTABLE   STEAM-ENGINE  BOILERS. 


80 1 


EVAPORATIVE   PERFORMANCE   OF   PORTABLE   STEAM- 
ENGINE   BOILERS  WITH   COAL.     1872. 

The  results  of  the  excellently  conducted  trials  of  portable  steam-engines 
exhibited  at  the  show  of  the  Royal  Agricultural  Society,  at  Cardiff,  in  1872, 
were  fully  reported  by  the  judges,  Mr.  F.  J.  Bramwell  and  Mr.  W.  Mene- 
laus.1  To  this  report,  with  the  valuable  tables  appended  to  it,  prepared  by 
the  consulting  engineers,  Messrs.  Eastons  &  Anderson,  the  author  is 
indebted  for  the  data  with  which  he  has  formed  the  table  No.  281.  The 
fuel  used  was  Llangennech  (Welsh)  coal;  an  analysis  of  it,  by  Mr.  G.  J. 
Snelus  is  given  at  page  415,  ante.  The  quantity  of  ash  and  clinker  averaged, 
so  far  as  it  was  observed,  about  6  per  cent,  of  the  fuel.  The  boiler  was  of 
the  ordinary  pattern,  having  a  firebox  and  multitubular  flues ;  but  Messrs. 
Davey,  Paxman,  &  Co.'s  boiler  contained,  in  addition,  ten  circulating  wrought- 
iron  bent  water-tubes,  2^  inches  in  diameter,  in  the  firebox,  rising  from  the 
sides  to  the  top. 

Table  No.  281. — PORTABLE  STEAM-ENGINE  BOILERS. — PROPORTIONS 
AND  RESULTS  OF  EVAPORATIVE  PERFORMANCE.     1872. 

(Compiled  and  reduced  from  the  Report  of  the  Judges,  Royal  Agricultural 
Society's  Show,  Cardiff.) 

Fuel :— Llangennech  (Welsh)  Coal. 


Area  of  Fire- 

Coal 

No. 

Constructors. 

grate. 

Heating 
Surface 
(Tubes 
measured 

Ratio  of 
Heating 
Surface 
to  Trial 

Con- 
sumed 
per  Sq. 
Foot  of 

Equivalent 
Water 
Evaporated 
from  and  at 

Equiva- 
lent 
Water 
Evapor- 

As Re- 

Nor- 
mal. 

duced 
for 
Trial. 

on  the 
Outside). 

Fire- 
grate. 

Trial- 
grate 

H^ur. 

212°  F.  per 
Square  Foot  of 
Grate,  per  Hour. 

ated  per 
Pound 
of  Coal. 

sq.  ft. 

sq.  ft. 

sq.  ft. 

ratio. 

Ibs. 

Ibs. 

cu.  ft. 

Ibs. 

I 

J  Marshall,        ) 
(  Sons,  &  Co.   \ 

4-4 

3-o 

283.5 

94-5 

15-7 

161 

2.58 

10.23 

2 

\  Clayton   &     \ 
\  Shuttleworth 

5-3 

3-2 

220.0 

69 

12.8 

151 

2.42 

11.83 

„ 

1  Clayton   & 
\  Shuttleworth  j 

it 

,, 

,, 

J» 

12-5 

148 

2.36 

ii.  81 

3 

Hayes  

5-i 

5-i 

170.6 

33 

14.8 

66.5 

1.06 

4-59 

4 

(  Davey,  Par-     ) 
\    man,  &  Co.   \ 

3-75 

3-75 

168.4 

45 

10.3 

114 

1.83 

11.02 

5 

Tuxford  &  Sons 

6.13 

— 

193.0 

— 

— 

— 

— 

— 

6 

Brown  &  May... 

3-2 

3-2 

I59-I 

5° 

9-53 

104 

1.66 

10.89 

7 

Tasker  &  Sons... 

4-7 

4-7 

158.0 

34 

13.0 

119 

1.91 

9-33 

8 

j  Reading  Iron-  ) 
|     Works  } 

7.2 

2-37 

211.  0 

89 

20.4 

214 

3-43 

10.49 

9 

Lewin  

4-3 

1.6 

I5I.6 

— 

— 

— 

— 

— 

10 

j  E.  R.  &  F. 
(    Turner  

3-5 

3-5 

187.8 

54 

20.7 

204 

3-26 

9-93 

ii 

1  Barrows    & 
j    Stewart  

5-0 

5-o 

129.8 

26 

13-6 

1  20 

1-93 

8-97 

12 

j  Ashbey,  Jef- 
\  fery,  &  Luke 

5-5 

2.0 

204.5 

1  02 

3i-i 

319 

5.10 

9.27 

1  The  Trials  of  Portable  Steam- Engines  at  Cardiff;  Report  by  the  Judges.      1872. 

51 


8O2  EVAPORATIVE   PERFORMANCE  OF  STEAM-BOILERS. 

RELATIONS   OF   GRATE-AREA  AND   HEATING  SURFACE 
TO   EVAPORATIVE   PERFORMANCE. 

SPECIAL   EXPERIMENTS  ON   THE  RELATIVE  VALUE  OF  THE   DIFFERENT 

PARTS  OF  HEATING  SURFACE. 

Mr.    Graham's  Experiments  •,  1858.  —  Mr.    John   Graham   published,  in 
,1  an  account  of  his  experiments  on  the  proportional  evaporative  value 


of  the  different  parts  of  the  heating  surfaces  of  boilers. 

ist  Series.  Four  open  tin  pans,  12  inches  square,  in  a  row,  set  in  brick- 
work. A  grate  1  2  inches  square  was  set  directly  under  the  first  pan,  29  inches 
below  it,  from  which  a  flash-flue  3  inches  deep  conducted  the  .  gaseous 
products  under  the  other  pans  towards  a  chimney.  The  first  pan  showed 
"  the  direct  heating  effect  of  fire;"  the  second,  the  effect  of  an  "equal  surface 
of  blaze,"  the  third  and  fourth,  the  effect  of  heated  air  only.  With  a 
"  moderately  strong  draught  "  the  quantities  of  water  evaporated  per  hour 
were  proportionally  as  follows  :  — 

Percentage  of 
Evaporative  Duty. 

ist  pan,  ........  .  ...............  as  100     .........     67.6 

2d     „     ........................    „     27     .........     18.2 

3<*     „    ........................    „     13     .........       8.8 

4th   „     ........................    „       8     .........       5.4 


IOO.O 

Showing  that  two-thirds  of  the  whole  evaporation  was  effected  from  the  first 
pan,  and  only  a  twentieth  from  the  last  pan. 

2d  Series.  Three  cylinders  of  ^-inch  plate,  3  feet  in  diameter,  and  3  feet 
long,  open  to  the  atmosphere,  in  a  row  end  to  end,  were  set  in  brickwork. 
A  grate  was  placed  under  the  first  cylinder,  3  feet  long  and  2  feet  wide, 
and  9^  inches  below  the  cylinder;  with  a  flash-flue  under  the  second  and 
third  cylinders,  concentric  with  them,  of  4  inches  radial  width,  and  carried 
up  on  each  side  to  the  level  of  the  centre  of  the  cylinders.     The  average 
results  of  eleven  trials  for  evaporation,  with  the  calculated  heating  surfaces, 
were  as  follows  :  — 

Area  of  grate,  ........................................  6  square  feet. 

Heating  surface  of  ist  cylinder,  ............    !°-53          „ 

Do.  2d      do  .............    14.13         ,, 

Do.  3d      do  .............    14.13         „ 


38.79 

Worsley  coal  consumed,  7  2  Ibs.  per  hour,  or  1 2  Ibs.  per  square  foot  of 
grate  per  hour.  Water  evaporated  from  60°  F.,  4.55  Ibs.  per  pound  of  coal ; 
the  duty  was  proportionally  as  follows : — 

Percentage  of  Duty. 
For  Whole  Surface.  Per  Square  Foot. 

i st  boiler, as  100     66.4,       or       73     percent. 

2d     „     „     34.7  23.0,        „        18.5      „ 

3d     „     „     16     ^0.6,        „          8.5      „ 


IOO.O  IOO.O 

1  Transactions  of  the  Literary  and  Philosophical  Society  of  Mana/iester,  vol.  xv.,  1858. 


COMPARATIVE  EFFICIENCY   OF   HEATING  SURFACE.          803 

Showing  that  about  three-fourths  of  the  evaporative  work  per  square  foot 
of  surface  was  done  by  the  first  cylinder,  and  only  a  twelfth  by  the  third 
cylinder. 

Experiments  of  Messrs.  Woods  and  Dewrance,  I842.1 — Mr.  Edward  Woods 
and  Mr.  John  Dewrance,  in  1842,  tested  the  evaporative  duty  of  successive 
portions  of  the  flue-tubes  of  a  locomotive  boiler,  5  feet  6  inches  long, 
divided  into  six  compartments  by  vertical  diaphragms.  The  first  compart- 
ment was  6  inches  long,  and  each  of  the  others  12  inches.  It  was  found 
that  the  evaporative  duty  of  the  first  compartment  was  about  the  same  per 
square  foot  as  that  of  the  fire-box;  that  of  the  second  compartment  about 
a  third  of  that  value;  that  of  the  remaining  compartments  very  small;  and 
that  the  first  6  inches  did  more  work  than  the  remaining  60  inches  of 
tube. 

Experimental  Dedttdions  of  M.  Paul  Havrez,  1874. — The  important 
deduction,  that  the  evaporative  performance  of  similar  boilers  per  unit  of 
grate-area,  increases  with  the  square  of  the  surface-ratio,  is  confirmed  by  the 
deduction  made  by  M.  Paul  Havrez  of  the  following  law,  from  the  perfor- 
mances of  locomotive  boilers:2 — That  the  quantities  of  water  evaporated 
by  consecutive  equal  lengths  of  flue-tubes  decrease  in  geometrical  progres- 
sion, whilst  the  distances  from  the  commencement  of  the  series  increase  in 
arithmetical  progression.  The  point,  he  adds,  at  which  the  law  begins  to 
prevail,  is  that  at  which  the  radiation  of  heat  from  the  fuel  ceases,  and  heat 
is  communicated  by  conduction  alone.  One  of  the  experiments  of  which 
the  results  were  investigated  by  M.  Havrez,  was  made  by  M.  Petiet,  of  the 
Northern  Railway  of  France,  who  repeated  the  experiment  of  Mr.  Woods 
and  Mr.  Dewrance,  and  tested  the  evaporative  value  of  the  different  parts 
of  a  locomotive  boiler  having  tubes  of  a  length  of  1 2  feet  3  inches  divided 
into  five  compartments.  The  first  compartment  consisted  of  the  fire-box, 
with  3  inches  of  length  of  the  tubes;  and  the  four  tube-sections  were  3.02 
feet  long.  Using  coke  and  briquettes  as  fuel,  the  average  results  were  as 
follows : — 

Fire-box         ist  Tube         ad  Tube        sd  Tube        4th  Tube 
Section.          Section.          Section.          Section.          Section. 

{60.28  box 
16.15  tubes. 
76.43         179  179          179          1 79  sq.ft. 

Water  evaporated  per  \ 
square  foot  per  hour,  >       24.5  8.72         4.42         2.52         1.68  Ibs. 

with  coke  ) 

Water  evaporated  per  \ 

square  foot  per  hour,  >       36.9         11.44         5.72         3.52         2.31  Ibs. 
with  briquettes j 

M.  Havrez's  law  of  progression  is  traceable  here,  and  whether  it  be  exact, 
or  only  approximately  true,  the  rapidly  diminishing  evaporations  are  corro- 
borative of  the  results  of  previous  experiments.  If  the  successive  evapora- 

1  The  Engineer,  March,  1858. 

2  "  Evaporation  in  Steam-boilers  decreasing  in  Geometrical  Progression,"  by  M.  Paul 
Havrez,  Annales  du  Genie  Civil,  August  and  September,  1874;  abstracted  in  the  Proceed- 
ings  of  the  Institution  of  Civil  Engineers,  vol.  xxxix.,  page  398,  1874-75. 


804          EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 

tions  be  set  off  as  ordinates  to  a  base  line  representing  the  advance  of  the 
heating  surface,  and  contoured,  the  area  of  the  figure  is  a  measure  of  the 
total  evaporation.  The  area  would  bulk  largely  at  the  first  part,  and  taper 
down  quickly  towards  the  end;  and  it  is  easily  comprehended  that  such 
areas  of  evaporation  for  boilers  of  different  total  lengths  or  quantities  of 
surface,  may  increase  practically  as  the  squares  of  the  total  surfaces, — 
supposing  that  the  final  temperatures  of  the  gases  in  leaving  the  boilers 
were  the  same. 


FORMULAS  FOR  THE  RELATIONS  OF  GRATE-AREA,  HEATING  SURFACE, 
WATER,  AND  FUEL. 

It  is  well  known  that,  in  a  given  boiler,  in  which  the  grate  and  the  heat- 
ing surface  are  constant  —  and,  of  course,  also  the  ratio  of  the  surface  to  the 
grate-area,  —  the  greater  the  quantity  of  fuel  consumed  per  hour  the  greater 
also  is  the  quantity  of  water  evaporated;  but  that  the  production  of  steam 
increases  at  a  less  rate  than  the  combustion,  in  other  words,  that  the  quan- 
tity of  water  evaporated  per  pound  of  fuel  is  diminished.  But  it  has  remained 
a  question:  —  At  what  rate  does  this  diminution  of  efficiency  take  place? 
The  answer  is  supplied  by  the  fact,  generalized  from  the  experimental  obser- 
vations on  stationary,  portable,  marine,  and  locomotive  boilers,  detailed  or 
noticed  in  preceding  pages,  that  the  total  quantity  of  water  evaporated  per 
square  foot  of  grate  is  expressed  by  a  constant  quantity,  A,  plus  a  constant 
multiple,  B  c,  of  the  fuel  consumed  per  square  foot  of  grate,  or  by  the  general 
formula 


The  sense  of  this  equation  is,  that  though  the  water  evaporated  per 
square  foot  of  grate  does  not  keep  pace  with  the  fuel  consumed,  yet  that  the 
quantity  of  water  increases  by  equal  increments  for  equal  increments  of  fuel 
per  square  foot  of  grate. 

Again,  on  the  inverse  supposition,  that  the  efficiency  of  the  fuel  remains 
constant,  how  is  the  performance  of  a  boiler  affected  by  the  proportions  of 
the  grate-area  and  the  heating  surface?  The  author,  in  1852,  investigated 
this  question  by  the  aid  of  the  observations  already  noticed,  of  the  evapora- 
tive performance  of  locomotive-boilers,  using  coke;  and  he  deduced  from 
them,  that,  assuming  throughout  a  constant  efficiency  of  the  fuel,  or  pro- 
portion of  water  evaporated  to  the  fuel,  the  evaporative  performance  of  a 
locomotive  boiler,  or  the  quantity  of  water  which  it  was  capable  of  evapo- 
rating per  hour,  decreases  directly  as  the  grate-area  is  increased;  that  is  to 
say,  the  larger  the  grate  the  smaller  is  the  evaporation  of  water,  at  the 
same  rate  of  efficiency  of  fuel,  even  with  the  same  heating  surface. 
2d.  That  the  evaporative  performance  increases  directly  as  the  square  of 
the  heating  surface,  with  the  same  area  of  grate  and  efficiency  of  fuel. 
3d.  The  necessary  heating  surface  increases  directly  as  the  square  root  of 
the  performance  ;  that  is  to  say,  for  example,  for  four  times  the  performance, 
with  the  same  efficiency,  twice  the  heating  surface  only  is  required.  4th. 
The  necessary  heating  surface  increases  directly  as  the  square  root  of  the 
grate,  with  the  same  efficiency;  that  is  to  say,  for  instance,  if  the  grate  be 
enlarged  to  four  times  its  first  area,  twice  the  heating  surface  would  be 


COMPARATIVE   EFFICIENCY  OF   HEATING  SURFACE. 


805 


required,  and  would  be  sufficient,  to  evaporate  the  same  quantity  of  water 
per  hour  with  the  same  efficiency  of  fuel. 

Let  W  be  the  quantity  of  water  evaporated  per  hour,  and  C  the  weight 
of  coke  consumed  per  hour,  W  and  C  varying  so  as  to  preserve  a  constant 
ratio  to  each  other;  let  h=  the  heating  surface,  and^=  the  area  of  grate,  in 
square  feet;  then 


in  which  m  is  a  constant.  When  the  water,  W,  is  expressed  in  cubic  feet, 
and  9  Ibs.  of  water  is  evaporated  per  pound  of  fuel,  the  value  of  m, 
deduced  from  the  results  of  forty  experiments,  was  found  to  be  .00222,  and 


.OO222  — 
g 


(3) 


Reduced  to  the  standard  of  one  square  foot  of  grate,  let  w  and  c  be  the 
weights  of  water  and  fuel  respectively,  per  square  foot  of  grate,  in  constant 
ratio  to  each  other;  then,  dividing  the  above  formulas  respectively 


(4) 


and  w  (cubic  feet)  =  .00222  (  —  )2 

o 


(  5  ) 


Showing  that,  when  the  ratio  of  water  to  fuel  is  constant,  the  performance 
of  the  boiler,  per  square  foot  of  grate,  increases  as  the  square  of  the  ratio 
of  the  heating  surface  to  the  grate-area.  The  following  table  of  examples, 
extracted  from  Railway  Machinery?-  shows  how  closely  the  evaporation 
proceeded  according  to  the  square  of  the  surface-ratio,  when  9  Ibs.  of  water, 
at  the  ordinary  temperatures  and  pressures,  was  evaporated  per  pound  of 
coke. 

Table  No.  282.  —  OF  RELATIVE  HEATING  SURFACES  AND  RATES  OF 
CONSUMPTION  OF  WATER  IN  LOCOMOTIVE  BOILERS. 

(Railway  Machinery.) 


Consump- 

Classified Groups  of  Locomotives. 

Surface- 
ratio. 

tion  of 
Water  per 
Hour  per 
Sq.  Foot  of 

Water  per 
Pound  of 
Coke. 

Number 
of  Ex- 
periments. 

Grate. 

ratio. 

cubic  feet. 

Ibs. 

Orion,  Sinus,  Pallas,  E.  &  G.  Ry,  

52 

6.15 

9 

13 

C.  R.  Passenger  Engines,  

66 

8    " 

Q.I 

17 

Snake  L  &  S  W  Ry. 

72 

12 

8q 

2 

Sphynx  A  Hercules,  

QO 

18 

**y 

8.Q2 

8 

The  quantities  of  water  are  thrown  into  the  parabolic  curve,  AC, 
Fig.  328,  next  page,  being  ordinates  to  the  base-line,  AB,  on  which  the 
relative  surface-ratios  are  measured. 

It  was  thus  found,  that,  practically,  there  can  never  be  too  much  heating 
surface,  as  regards  economical  evaporation,  but  there  maybe  too  little;  and 

1  Railway  Machinery,  page  158. 


8o6 


EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 


that,  on  the  contrary,  there  may  be  too  much  grate-area  for  economical 
evaporation,  but  there  cannot  be  too  little,  so  long  as  the  required  rate  of 
combustion  per  square  foot,  does  not  exceed  the  limits  imposed  by  physical 
conditions.  ~ 


A  o 


Heating  Surface-ratios. 


Fig.  328.—  Diagram  to  show  Rate  of  Economical  Consumption  of  Water  per  hour  per  foot 
of  Grate,  for  given  Surface-ratios. 

To  co-relate  the  formula  (  i  ),  in  which  the  surface-ratio  is  constant, 
with  the  formula  (  4  ),  in  which  the  evaporative  efficiency  of  the  fuel  is 
constant,  it  may  suffice  for  the  present  to  observe  that  the  quantity  B  c  is 
constant  for  all  surface-ratios,  and  that  the  quantity  A  varies  as  the  square 

of  the  surface-ratio.     Let  the  surface-ratio  —  =  r,  then  A  =  ar2,  in  which  a  is. 

<5 

a  constant  which  is  specific  for  each  kind  of  boiler;  and 

(6) 


w  =  the  water  evaporated  in  pounds  per  square  foot  of  grate  per  hour. 
c  —  the  fuel  consumed  in  pounds  per  foot  per  hour. 

E  =  —  =  the  efficiency  of  the  fuel,  or  the  weight  of  water  evaporated 

c  per  pound  of  fuel. 

A  =  ar  2  =  a  constant,  which  is  specific  for  each  kind  of  boiler. 
B  =  a  constant  multiplier,  specific  for  each  kind  of  boiler. 

r  -  —  =  the  ratio  of  the  heating  surface  to  the  grate-area. 

o 

a  =  a  constant,  specific  for  each  kind  of  boiler. 

When  the  water  and  fuel  per  foot  of  grate  per  hour  are  given,  the  value  of 
the  required  surface-ratio  is  found  from  the  above  formula,  for  ar2  =  w  —  ~Bf, 
and 

I  -  =FT- 

..........................  (7) 


When  the  water  per  foot  of  grate  per  hour,  and  the  surface-ratio,  are 
given,  to  find  the  fuel  per  foot  of  grate  per  hour  required  to  evaporate  the 
water:  ^c-w  —  ar2,  and 

w  -  ar* 


B 


(8) 


IV 


When  the  efficiency  E  =  — ,  of  the  fuel  is  given,  that  is,  the  weight  of  water 
evaporated  per  pound  of  fuel;  also,  the  surface-ratio;  to  find  the  fuel  that 


COMPARATIVE   EFFICIENCY  OF  HEATING  SURFACE.          807 
may  be  consumed  per  square  foot  of  grate  per  hour  corresponding  to  that 


efficiency.   As  -  =E  = 
c 


f  =B  +  —  ;  then  ar*  =  c  (E  -  B)  ;  and 
c  c 


c- 


ar* 


(9) 


When  the  efficiency  E  or  —  ,  and  the  fuel  consumed  per  foot  of  grate  per 
hour,  are  given,  to  find  the  surface-ratio  required  to  effect  that  evaporation. 
As  already  found,  ar*  =  c  (E  -  B),  and  r*  = 


c       ~ 


;  whence, 


Newcastle  Marine  Boiler,  page  785. 

Select  for  comparison,  from  tables  Nos.  274  and  275,  pages  786  and  788, 
the  performance  of  this  boiler  with  a  grate-area  of  22  square  feet,  and  749 
square  feet  of  heating  surface,  34.05  times  the  grate,  with  increasing  rates  of 
combustion  of  coal  per  square  foot  per  hour.  Find  the  corresponding 
weights  of  water  evaporated  per  square  foot,  and  plot  them  to  a  vertical  scale, 


•ace 

-ISC 

.too 

•so 


ccal- 


Fig.  329. — Newcastle  Marine  Boiler. — Diagram  to  show  Relation  of  Water  and  Coal  per  square  foot  of 
Grate-area,  22  square  feet.     Surface-ratio,  34.05. 

upon  a  base-line  AB,  Fig.  329,  measuring  the  weights  of  coal  consumed. 
They  are  found  to  lie  in,  or  close  to,  a  straight  line,  D  C,  drawn  obliquely 
upwards  from  a  point,  D,  in  the  ordinate  of  zero,  at  a  level  which  is  25  Ibs. 
above  the  base-line,  and  the  general  formula  (  6  )  becomes 

w  =25  +  971^; (  IJ  ) 

in  which  ar*  =  25,  and  6  =  9.71.  The  annexed  table,  No.  283,  shows  the 
correspondence  of  the  actual  quantities  of  water  evaporated,  with  those 
which  are  calculated  from  the  coal  consumed,  by  this  formula  (  1 1  ).1 

1  The  diagonal  line  C  D,  in  Fig.  329,  does  not  exactly  strike  the  average  of  the  results 
for  the  grate  of  22  square  feet  alone ;  but  it  is  the  average  for  the  results  obtained  from  the 
various  sizes  of  grate  taken  together.  For  reference  to  the  line  A  E,  see  page  817. 


8o8 


EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 


Table  No.  283. — NEWCASTLE  MARINE  BOILER — RELATIONS  OF  COAL 

AND  WATER. 

Grate  22  square  feet.     Surface-ratio  34.05. 


Nos.  of 
Experi- 
ments in 
Tables 
No.  274 
and  275. 

Coal  per 
Foot  of 
Grate 
per  Hour. 

Water  per 
Pound  of 
Coal,  from 
and  at 

212°  F. 

Total  Water  per  Foot  of  Grate 
per  Hour. 

Water 
per  Pound 
of  Coal, 
according 
to 
Formula. 

Observed. 

By 
Formula 
(n). 

Difference 
by 
Formula. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

per  cent. 

Ibs. 

5 

17.27 

11.70 

202.0 

192.7 

-4.6 

ii.  16 

H 

22.08 

11.41 

251.9 

2394 

-5.0 

10.84 

15 

23.04 

10.62 

244.7 

248.7 

+  1.6 

10.79 

16 

25.97 

11.17 

290.1 

277.2 

-4.4 

10.67 

17 

26.05 

10-33 

269.1 

277.9 

+  3-3 

10.67 

6 

26.98 

10.80 

291.4 

287.0 

-i-5 

10.64 

18 

28.51 

10.58 

301.6 

301.8 

0.0 

10.58 

Since  A  =  ar*  =  25,  in  the  present  instance;  a  —  -|  =  —  5i_i  =  .02156,  and 

ar*  =  .02  1  56  ?-2.     By  substitution,  the  following  formula  is  obtained,  which 
applies  to  all  surface-ratios  in  the  Newcastle  boiler:  — 


12 


Table  No.  284.  —  NEWCASTLE  MARINE  BOILER  —  RELATIONS  OF  COAL 

AND  WATER. 

Varying  grate-area  and  surface-ratio.      Calculations  for  normal  surface-ratio  34.05 

by  formula  (  1  1  ). 


Coal  per  Square  Foot  of 
Grate  per  Hour. 

Water  per  Sq.  Foot  of  Grate  per  Hour, 
for  Normal  Surface-ratio,  34.05. 

No.  of 
Experi- 
ment. 

Grate-area. 

Surface- 
ratio. 

Actual. 

Reduced  in 
the  Ratio  of 
the  Squares 
of  the  Sur- 
face-ratios, 
for  Normal 

Reduced  in 
the  same 
Ratio  as 
for  the  Coal. 

Calculated 
from 
Column  5 
by 
Formula 

Difference 
by 
Formula. 

Ratio,  34.05. 

(  "  )• 

(i) 

Ujj 

sq.  feet. 

(3) 
ratio. 

Ibs. 

Ibj 

(6) 
Ibs. 

(7) 
Ibs. 

(8) 
per  cent. 

10 

42 

17.83 

1  6.0 

58.35 

563.1 

591.6 

+    5.1 

II 

„ 

„ 

17.6 

63.82 

583.3 

644.7 

+  10.5 

12 
13 

33 

22.7 

18.13 
20.36 

66.12 
45.8i 

5924 
4237 

667.0 
469.8 

+  12.6 
+  10.9 

2 

28.5 

26.28 

19.0 

31.90 

355-0 

334-8 

-   5-7 

5 

22 

34.05 

17.27 

17.27 

202.0 

192.7 

-  4.6 

14 

„ 

22.08 

22.08 

251.9 

239-4 

-  5-0 

15 

„ 

„ 

23.04 

23.04 

244.7 

248.7 

+   1.6 

16 

„ 

}) 

25.97 

25.97 

290.1 

277.2 

-  4-4 

17 

?J 

}) 

26.05 

26.05 

269.1 

277.9 

+  3-3 

6 

„ 

26.98 

26.98 

291.4 

287.0 

18 

„ 

„ 

28.51 

28.51 

301.6 

301.8 

o.o 

4 

19.25 

38.91 

17.25 

13-21 

165.5 

153.3 

-  74 

21 

18 

4I.6l 

18.67 

12.50 

139.7 

146.4 

+  4.8 

22 

„ 

„ 

24.89 

16.67 

182.7 

186.9 

+   2.3 

7 

„ 

„ 

27.36 

18.32 

208.3 

202.9 

-    2.6 

8 

15-5 

48.32 

3740 

18.57 

1974 

205.3 

+  4.0 

COMPARATIVE   EFFICIENCY   OF   HEATING  SURFACE. 


809 


The  results  of  the  other  experiments  with  the  Newcastle  boiler,  made 
with  different  areas  of  grate,  may  be  reduced  for  direct  comparison  with 
those  made  with  the  22-feet  grate,  by  reducing  both  the  coal  and  the  water 
per  square  foot  per  hour,  in  the  ratio  of  the  squares  of  the  respective  surface- 
ratios,  whilst  the  ratio  of  the  coal  and  water,  or  the  efficiency,  remains 
constant.  The  table  No.  284  shows  the  reduced  water  (column  6) 
corresponding  to  the  reduced  coal  (column  5),  for  the  normal  surface-ratio 
34.05.  In  column  7,  the  reduced  waters  are  given  as  calculated  by  the 
formula  (  1 1  ) ;  and  the  differences  by  the  formula,  which  are,  upon  the 
whole,  inconsiderable,  are  given  in  the  last  column. 

To  show  the  suitability  of  the  formula  (  1 2  )  for  the  calculation  of  water 
evaporated,  from  the  given  surface-ratios,  as  they  are,  the  annexed  table, 
No.  285,  shows,  by  comparison  (columns  5  and  6),  the  actual  and  calculated 
quantities  of  water  evaporated  by  the  coals  (column  4),  with  the  ratios  in 
column  3.  The  percentages  of  differences  are  identical  with  those  already 
exhibited  in  the  previous  table. 

Table  No.  285. — NEWCASTLE  MARINE  BOILER — RELATIONS  OF  COAL 

AND  WATER. 

Varying  grate-areas  and  surface-ratios.     Calculations  for  the  actual 
ratios,  by  formula  (  12  ). 


Number 
of 
Experi- 
ment. 

Grate- 
area. 

Surface- 
ratio. 

Coal 
per  Square 
Foot  of 
Grate  per 
Hour. 

Water  per  Square  Foot  of  Grate  per 
Hour,  for  the  given  Surface-ratios. 

Actual, 
as  from 
and  at 
212°  Fahr. 

Calculated 
by 
Formula 

(12) 

Difference 
by 
Formula. 

W 

(2) 

(3.) 

(4) 

(5} 

(6) 

(7) 

square  feet. 

ratio. 

Ibs. 

Ibs. 

Ibs. 

per  cent. 

10 

42 

17.83 

16.0 

1544 

162.2 

+    5-1 

II 

„ 

17.6 

160.9 

177-7 

+  10.5 

12 

w 

JL 

18.13 

162.5 

182.8 

+  12.6 

13 

33 

22.7 

20.36 

188.3 

231.5 

+  IO.9 

2 

28.5 

26.28 

19.0 

2II.5 

1994 

-   5-7 

5 

22 

34.05 

17.27 

202.0 

192.7 

-  4.6 

15 

|| 

22.08 
23.04 

251.9 

244.7 

2394 
248.7 

+    1.6 

16 

„ 

„ 

25.97 

290.1 

277.2 

-  4-4 

17 

w 

B 

26.05 

269.1 

277.9 

+   3-3 

6 

„ 

„ 

26.98 

291.4 

287.0 

18 

M 

28.51 

301.6 

301.8 

0.0 

4 

19.25 

38.91 

17.25 

216.1 

200.1 

-  74 

21 

1  8 

4I.6l 

18.67 

208.5 

218.6 

+  4.8 

22 

„ 

„ 

24.89 

272.8 

279.0 

+   2.3 

8 

15-5 

4^32 

27.36 
37-40 

311.1 
397.6 

303.0 
413.5 

+  4.0 

The  consistency  of  the  results  of  the  application  of  the  formula  under 
widely  varying  proportions  of  boiler,  and  varying  rates  of  combustion, 
affords  evidence  of  the  correctness  of  the  principles  on  which  it  is  based. 

Wigan  Marine  B oiler ,  page  781. 

The  trials  of  this  boiler  were  made  with  a  constant  grate  of  10.3  square 
feet  area,  and  a  constant  surface  of  508  square  feet,  giving  a  surface-ratio 


8io 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


of  50.  The  average  results  of  the  trials  selected  for  the  present  purpose, 
are  placed  in  the  following  table,  together  with  the  quantities  of  water 
evaporated,  as  calculated  by  the  following  formula  deduced  from  the 
plotting  of  the  results: — 

10-75'  (13) 


Showing  a  smaller  constant  and  a  greater  multiple  than  the  formula  of  the 
Newcastle  boiler.  Substituting  for  25  the  general  expression  a  r*,  and 
reducing  for  the  value  of  w,  the  general  formula  is, 


10.75 


which  may  be  employed  for  different  surface-ratios. 

Table  No.  286. — WIGAN  MARINE  BOILER — RELATIONS  OF  COAL 
AND  WATER. 

Grate  10.3  square  feet;  surface-ratio  50. 


DESCRIPTION  OF  COALS. 

Coal 
per  Foot 
of  Grate 

Hour. 

Water 
per  Pound 
of  Coal, 
from  and 
at  212° 
Fahr. 

Total  Water  per  Square  Foot 
of  Grate  per  Hour. 

Observed. 

By 

Formula 
(IS]- 

Difference 
by 
Formula. 

South   Lancashire   and   Cheshire  ) 
coals  —  Mr  Fletcher's  trials          \ 

Ibs. 
27.63 

27.50 
41.25 
28.83 
26.2O 

Ibs. 
11.54 

11.92 
11.36 
11.95 
12.44 

Ibs. 
318.8 

327.8 
468.6 
344-5 
325-9 

Ibs. 
322.1 

320.6 
468.6 

334-9 
306.6 

per  cent. 
+  1.0 

-2.2 
O.O 
-2.8 
-6.0 

South    Lancashire   and    Cheshire  ( 
coals  —  Messrs.  Nicol  &  Lynn...  \ 
Hartley's  (Newcastle)  coals  

Welsh  coals  

It  appears  from  this  table  that  the  South  Lancashire  and  Cheshire  coals, 
and  the  Newcastle  coals,  were  equally  efficient;  and  that  the  Welsh  coals 
had  a  slightly  greater  evaporative  action  than  the  others. 


Experimental  Marine  Boiler,  Navy  Yard,  New  York,  U.S.,  page  795. 

This  boiler  affords  examples  of  very  low  surface-ratios.  With  its  normal 
proportions,  10.8  square  feet  of  grate  and  150.3  square  feet  of  surface,  the 
surface-ratio  is  14.  When  the  flue-tubes  were  stopped  off,  the  surface-ratio 
was  only  4.21.  By  the  plotting  of  the  experimental  results,  reduced  for  a 
uniform  surface-ratio  of  14,  the  following  formula  was  derived: — 

w  =  . 0204  r*+  7.624  c  (  15  ) 

It  is  seen  in  the  following  table,  that  the  calculated  evaporation  is  con- 
siderably in  excess  of  the  actual  reduced  evaporation,  in  the  extreme 
instances  of  the  flash-flue  and  the  small  surface-ratio,  4.21.  It  is  obvious 
that  such  dissimilar  cases  as  those  of  a  flash-flue  and  a  multitubular  boiler, 
are  not  directly  comparable. 


COMPARATIVE   EFFICIENCY   OF   HEATING   SURFACE.          8ll 


Table  No.  287. — EXPERIMENTAL  MARINE  BOILER,  NAVY  YARD,  NEW 
YORK — RELATIONS  OF  COAL  AND  WATER. 

Varying  grate-area  and  surface-ratio.     Calculations  for  normal 
surface- ratio  14. 


Coal  per  Square  Foot 
of  Grate  per  Hour. 

Water  per  Square  Foot  of  Grate  per 
Hour,  for  Normal  Ratio  14. 

Index 
to 

Grate- 

Surface- 

Reduced  in 
the  Ratio  of 

Calculated 

r_.^__ 

Experi- 
ment. 

area. 

ratio. 

Actual. 

the  Squares 
of  the  Sur- 

Reduced 
in  the 

irom 
Column  5 

Difference 
by 

face-ratios, 
for  Normal 

Same  Ratio. 

by 
Formula 

Formula. 

Ratio  14. 

(15)- 

square  feet. 

ratio. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

per  cent. 

X 

10.8 

4.21 

11.77 

130.2 

865 

996.1 

+  15 

V 

?? 

j) 

16.57 

183.2 

1082 

1400 

+  29.4 

w 

?? 

)) 

16.58 

183.3 

III9 

1401 

+  25.2 

A 

?) 

14 

5-57 

5-57 

51.63 

46.4 

-  IO.I 

B 

jj 

)) 

10.99 

10.99 

98.39 

87.7 

-  10.9 

C 

j? 

» 

16.57 

16.57 

I3L7 

130.3 

-     I.O 

D 

>> 

D 

22.10 

22.10 

172.5 

172.4 

0.0 

E 

j) 

j) 

27.76 

27.76 

205.4 

215.5 

+  4.9 

I 

8.64 

17.24 

15 

9.88 

84.80 

79-3 

-  6.5 

L 

JJ 

jj 

20.73 

13-66 

109.60 

I08.I 

-   14 

J 

6.48 

22.84 

15 

5.64 

46.67 

47.0 

+  0.7 

M 

JJ 

>j 

22.84 

10.30 

76.54 

82.5 

+  7-1 

K 

4.32 

34-03 

15 

2.54 

22.67 

23-3 

+    2.8 

N 

» 

•>•> 

27.58 

4.67 

33-81 

39-6 

+  17 

Wigan  Stationary  Boilers,  page  771. 

The  data  afforded  by  these  typical  boilers  are  specially  useful,  as  they 
represent  classes  of  boilers  in  general  use  in  England.  The  several 
experimental  results,  required  for  the  present  purpose,  are  collected  in  the 
annexed  table.  The  first  two  are  the  results  for  flash-draughts,  for  which 
the  side  and  bottom  flues  were  cut  off,  and  the  gases  were  conducted  direct 
to  the  chimney  after  having  passed  through  the  tubes.  By  plotting  the 
coal  and  water  reduced  according  to  the  squares  of  the  surface-ratios,  for 
a  uniform  ratio  of  30,  this  formula  was  obtained,  — 


w  =  20  +  9.56*: 
And  in  the  general  form,  for  various  ratios,  — 

w  =  .0222  r2  +  9.56^ 


(  16  ) 


( 


By  the  formula  (  16  ),  the  quantities  of  water  in  column  6  of  the  table  No. 
288  were  calculated  from  the  reduced  coals  in  column  5. 

The  agreement  of  the  reduced  and  the  calculated  quantities  of  water 
(columns  6  and  7)  is  very  close,  excepting  for  the  flash-draught. 


812 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


Table  No.  288. — WIGAN  STATIONARY  BOILERS — RELATION 
OF  COAL  AND  WATER. 

Varying  grate-area  and  surface-ratio.      Calculations  for  ratio  30. 


BOILERS 
(Without  Economizer). 

Grate- 
area. 

Surface- 
ratio. 

Coal  per  Square  Foot 
of  Grate  per  Hour. 

Water  per  Square  Foot  of 
Grate  per  Hour  for  Ratio  30. 

Actual. 

Reduced  in 
the  Ratio 
of  the 
Squares  of 
the  Surface- 
ratios,  for 
Ratio  30. 

Re- 
duced 
in  the 
same 
Ratio. 

Calcu- 
lated 
from 
Column 
5  by 
Formula 
(16). 

Difference 
by  For- 
mula. 

(') 

Galloway,  flue-tubes  ) 
only  ( 

(2) 
sq.  feet. 

31.5 

» 
)) 
)) 

)) 
)) 
)) 

)) 
21 

)) 

» 
)) 

(3) 
ratio. 
13.70 

14.74 
22.8 

23-5 
24.4 

5) 
)) 

254 
34-3 

35-5 

36.5 

j> 

(4) 
Ibs. 
18.58 

19.91 

I8.3 
I4.O 
17.26 

1  8.6 
19.1 

16.71 

21.8 

23.0 

21.5 

22.7 

(5) 
Ibs. 
89.10 

82.47 
31.68 
22.82 

26.03 
28.12 
28.87 

23.31 

16.68 
16.43 

14.52 
15-33 

(6) 
Ibs. 
757-3 

678.8 
322.9 
230.4 

271.5 
290.2 
293-7 
251.0 
179.6 
179.2 

158.0 
165.1 

(7) 
Ibs. 
871.8 

808.4 
322.9 
238.2 

268.8 
288.8 
296.0 

242.8 

179.5 
I77.I 

158.8 
166.6 

(8) 
per  cent. 
+  15.0 

+  ig.O 
0.0 

+   3-4 

-     I.O 

-  0-5 
+   0.8 

-   3-3 
o.o 

-     1.2 

+    0.5 
+    0.9 

Lancashire,     flue-     \ 
tubes  only  ...            C 

Galloway,  complete... 
Lancashire  and  Gal-  ) 
lowav                         C 

Lancashire  

Do  

Do 

Do.     with  water  ) 
tubes  ] 

Galloway 

Lancashire  and  Gal-  } 
lowav  ...                 .  .  ( 

T      -y  •  ' 

Lancashire  

Do 

Stationary  Boilers  in  France,  page  796. 

The  proportions  and  the  results  of  performance-  are  treated  in  the  follow- 
ing table.  The  following  special  formulas  have  been  deduced  for  the  three 
boilers  respectively,  and  for  the  three  collectively : — 


"Fairbairn" 

Lancashire 

French 

All  the  boilers 


w  =  .oii26r* 
w  =  .omr2 


+  7.7  c  ............  (  18  ) 

+  8.0  c  ............  (  19) 

+  8.0  c  ............  (  20  ) 

+  7.82  c  ...........  (  21  ) 


It  is  seen  that  the  same  formula  applies  to  the  Lancashire  and  the  French 
boilers;  and  that,  therefore,  the  reporters  of  the  trials  were  justified  in 
asserting  that  these  boilers  were  equally  efficient.  The  comparatively  inferior 
quantity  evaporated  in  the  first  trial  in  the  table,  resulted  probably  from 
an  excessively  large  surplus  of  air  admitted  into  the  furnace:  the  total 
quantity  of  air  in  that  instance,  amounted  to  261  cubic  feet  per  pound  of 
coal. 


COMPARATIVE   EFFICIENCY   OF   HEATING  SURFACE. 


Table  No.  289. — STATIONARY  BOILERS  IN  FRANCE — RELATIONS 
OF  COAL  AND  WATER. 

Calculations  of  evaporative  performance  for  surface-ratio  30.     Ronchamp  coal. 


BOILERS. 

Grate- 
area. 

Surface- 
ratio. 

Coal  per  Square  Foot 
of  Grate  per  Hour. 

Water  per  Square  Foot  of  Grate     . 
per  Hour,  for  Surface-ratio  30. 

Actual. 

Reduced  in 
the  Ratio 
of  the 
Squares  of 
the  Surface- 
ratios  for 
Ratio  30. 

Reduced 
in  the 
same 
Ratio. 

Calculated 
from  Col- 
umn 5,  by 
Formulas 

(  18  ),  (  19  ), 
(20). 

I 

Difference 
by  For- 
mulas. 

"Fairbairn"... 

jj 
Lancashire  

jj 

55 

French          . 

sq.  feet. 
20.5 
55 
55 
55 
55 
20.1 

55 
55 

ratio. 

49-5 
55 
29.8 

55 
jj 
30.3 

55 
5) 

Ibs. 
10.70 
18.53 
10.41 
19.15 
19.50 
11.36 
19.87 
20.57 

Ibs. 

3-93 
6.8  1 
10.55 
19.41 
19.76 
11.14 
19.48 
20.16 

Ibs. 

34.8 

62.7 
94.1 
165.0 
166.8 

95-5 
165.4 
1  66.6 

Ibs. 
40.7 
63.3 
92.5 

161.8 
164.5 
97.1 
162.3 
167.6 

per  cent. 
+  17 
+    0.9 

—         *7 

-     -9 
-     4 
+     -7 
-     -9 
+  0.6 

55 
5J 

Locomotive-Boilers,  page  798. 

The  experimental  trials  from  which  the  evaporative  performances  of 
locomotives  have  been  tabulated,  have,  of  course,  been  conducted  under 
various  conditions.  There  is,  nevertheless,  a  remarkable  degree  of  har- 
mony amongst  them,  for,  when  plotted,  they  are  seen,  with  a  very  few 
exceptions  of  early  date,  to  follow  the  laws  of  evaporative  performance 
already  enunciated.  Even  the  performance  of  the  boiler  of  the  primitive 
Killingworth  engine,  when  the  evaporative  efficiency  is  increased  by  one- 
half  to  represent  the  value  of  coke  compared  with  coal  as  imperfectly 
burned  in  that  boiler, — range  as  well  as  should  have  been  expected,  with 
those  of  other  locomotives.  In  fact,  the  improved  Killingworth  boiler 
exhibits  a  performance  above  the  general  average. 

Using  good  coke  as  fuel,  the  evaporative  performance  of  locomotive- 
boilers  in  which  the  flue-tubes  are  spaced  sufficiently  apart  to  admit  of  a 
free  circulation  of  water  around  them,  is  substantially  embraced  by  the 
following  formula  when  the  surface-ratio  is  75,  which  is  a  good  practical 
ratio : — 

w=  100  +  7. 94<r (coke) (  22  ) 

For  any  given  surface-ratio,  the  general  formula  is, — 

#/  =  .oi78r2  +  7. 94 c  (coke)  (  23  ) 

Using  good  coal  as  fuel,  the  formulas  for  the  coal-burning  locomotive 
boilers  in  table  No.  280,  page  799;  namely,  Nos.  31-34,  and  Nos.  39-41, 
are: — 

Nos.  31-34,  S.  E.  R.  Nos.  39-41,  L.  &  S.  W.  R. 

For  surface-ratio  75,    w  =  $o  +  9.6^ w=$o  +  9.82^ (24) 

For  any  surface-ratio,   w  =  .ooc)r*  +  g.6c     w  =  .009^-1-9.82^..  (  25  ) 


EVAPORATIVE   PERFORMANCE  OF   STEAM-BOILERS. 


Portable-Engine  Boiler  s,  page  80  1. 

These  boilers  are  arranged  in  the  table  No.  290,  in  the  order  of  the  surface- 
ratios.  The  coal  and  the  water  per  square  foot  of  grate  are  reduced  for  the 
ratio  50  (columns  5,  6),  from  which  has  been  deduced,  by  plotting,  the  for- 
mula, — 

w=2o+  8.6<r  ..............................  (24) 

For  any  given  surface-ratio,  the  general  formula  is,  — 


25 


The  calculated  quantities  of  water  (column  7),  by  formula  (  24  ),  follow 
closely  the  reduced  quantities  (column  6),  except  in  the  first  three  instances, 
Nos.  12,  i,  and  8,  where  they  are  much  in  excess.  In  these  instances,  the 
•excessive  reduction  of  the  grate  has  involved  a  material  departure  from  the 
normal  disposition  of  a  firebox,  especially  for  No.  8,  in  which  the  grate  was 
reduced  to  a  third  of  its  normal  area.  The  surface-ratios  were  driven 
up  to  102,  94.5,  and  89.  The  first  two  boilers,  Nos.  12  and  i,  have  the 
greatest  numbers  and  the  smallest  diameters  of  tubes.  The  drift  of  the 
evidence  goes  to  show  that  fewer  tubes,  of  larger  diameter,  do  better  for 
the  combustion  of  coal,  the  circulation  of  water,  and  the  absorption  of  heat. 
There  is  another  exception,  No.  3,  with  a  surface-ratio  33,  in  which  the 
calculated  quantity  of  water  is  twice  as  much  as  the  reduced  actual 
quantity.  The  excess  in  this  case  is  satisfactorily  accounted  for  by  causes 
which  were  pointed  out  by  the  judges  in  their  report.1 

Table  No.  290.  —  PORTABLE-ENGINE  BOILERS:  —  RELATIONS 
OF  COAL  AND  WATER. 

Calculations  of  evaporative  performance  for  surface-ratio  50. 


Coal  per  Square  Foot 
of  Grate  per  Hour. 

Water  per  Square  Foot  of  Grate 
per  Hour  for  Surface-ratio  50. 

Grate- 

Reduced  in 

No.  of 

area  as 

Surface- 

the  Ratio 

Reduced 

Calculated 

Boiler. 

Reduced 

ratios. 

of  the 

in  the 

from  Col- 

Difference 

for 
Trial. 

Actual. 

Squares  of 
the  Surface- 

same 
Ratio. 

umn  5,  by 
Formula 

by  For- 
mula. 

ratios  for 

(  24  )• 

Ratio  50. 

00 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

sq.  feet. 

ratio. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

per  cent. 

12 

2.0 

102 

3I-I 

7473 

69.28 

84.27 

+    21.6 

I 

3-o 

94-5 

15-7 

4-395 

44.96 

57.80 

+     28.5 

8 

2-37 

89 

20.4 

5-73 

60.  1  1 

69.28 

+     15.2 

2 

3-2 

69 

12.8 

6.721 

79-51 

77.80 

—       2.1 

» 

» 

» 

12.5 

6.564 

77.52 

76.45 

-        1.4 

10 

3-5 

54 

20.7 

17-75 

176.2 

172.6 

-       2.0 

6 

3-2 

50 

9-53 

9-53 

103.8 

102.0 

-      1-7 

4 

3-75 

45 

10.32 

12.72 

I40.I 

129.4 

-     7.6 

7 

4-7 

34 

13.0 

28.11 

262.3 

261.7 

-       0.2 

3 
ii 

5-i 

5.0 

11 

14.8 
13-6 

33.97 
50.30 

155.9 
45I-I 

312.10 
452.6 

-f  1  00.0 
+       0-3 

x"The  engine  was  indifferently  managed."  .  .  .  "It  would  appear  that  the 
boiler  did  about  one-half,  or  rather  less  than  one-half,  its  duty  in  making  steam." — Report 
of  the  Judges,  page  17. 


COMPARATIVE  EFFICIENCY  OF  HEATING  SURFACE.          815 

Looking  to  the  evaporating  capabilities  of  the  portable-engine  boilers  in 
their  ordinary  condition,  with  unrestricted  grates,  it  may  be  useful  to  show 
at  what  rates  they  are  capable  of  evaporating  water  from  and  at  212°, 
in  the  ratio  of  10  Ibs.  of  water  per  pound  of  coal  consumed.  For  the 
calculation  of  these  rates,  the  formula  (  9  ),  page  807,  may  be  employed. 
It  is, 


The  value  of  E  is  10,  of  B  is  8.6,  and  of  a  is  .008;  and,  by  substitution, 

.ooSr2 

c  =  -  :  or 
10-8.6' 


By  this  formula,  the  value  of  c,  the  quantity  of  coal  consumed  per  square 
foot  of  grate  per  hour,  is  found,  when  the  surface-ratio  r  is  given,  for  each 
boiler.  Thence,  multiplying  by  the  grate-area,  is  found  the  total  quantity 
of  coal  per  hour;  and  ten  times  the  coal  is  the  quantity  of  water.  In  this 
way  the  following  table,  No.  291,  is  calculated;  in  which  the  boilers  are 
placed  in  the  order  of  their  surface-ratios.  It  is  seen  that  No.  2  boiler  is 
capable  of  evaporating  at  the  given  rate  of  efficiency,  8.15,  say  8,  cubic  feet 
of  water  per  hour.  This  is  just  i  cubic  foot  per  nominal  horse-power:  — 
all  the  boilers  having  been  designated  of  8  horse-power.  No.  n  boiler 
would  only  evaporate  3  cubic  feet  per  hour,  at  the  given  rate  of  efficiency, 
whilst  No.  i  is  capable,  by  calculation,  of  evaporating  16^4  cubic  feet 
per  hour.  It  has  already  been  seen  that  there  is  reason,  in  the  design 
of  its  tube-surface,  for  doubting  whether  No.  i  is  capable  of  so  good  a 
performance. 

Standard  Average  Practice  for  Portable-Engine  Boilers.  —  The  last  line  01 
the  table  No.  291  may  be  assumed  as  a  standard  result  of  average  practice 
for  portable-engine  boilers  of  8  nominal  horse-power.  The  following  data 
may  be  taken,  in  round  numbers  :  — 

Nominal  horse-power,  ....................................       8  H.P. 

Area  of  fire-grate,  ......................  .  .................       5.5  square  feet, 

Area  of  heating-surface,  .................................   220.0  „ 

Ratio  of  heating-surface  to  grate-area,  or  surface-  )  . 

ratio,  ...............................................  / 

Coal  of  good  quality  consumed  per  hour,  ...........     50  Ibs. 

Do.  per  horse-power  per  hour,  ......       6.25  Ibs. 

Do.  per  square  foot  of  grate,  .  .  .say       9  Ibs. 

Water  evaporated  from  and  at  212°  F.  \ 

per  hour,  at  the  rate  of  10  Ibs.  per  >    500  Ibs.,  or  8  cubic  feet. 
pound  of  coal,  ........................  j 

Do.  per  horse-power,  ......    62.4  „     or  i          „ 

Do.  per  square  foot  of) 

grate,  9.  Ibs.,..  .....    ..........  say  \    9°      »     °"-45     „ 


8 16 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


Table  No.  291. — PORTABLE-ENGINE  BOILERS: — CALCULATED 
EVAPORATIVE  PERFORMANCE. 

From  and  at  212°  F.,  at  the  rate  of  10  Ibs.  of  water  per  pound  of  coal. 


Coal  Consumed  per 

Hour. 

No.  of  Boiler. 

Surface- 
ratio. 

Grate-area. 

Per  Square 

Total  Water  Evaporated 
per  Hour. 

Foot  of 

Total. 

Grate. 

ratio. 

square  feet. 

Ibs. 

Ibs. 

Ibs. 

cubic  feet. 

I 

64 

4-4 

2340 

102.96 

1029.6 

16.50 

10 

54 

3-5 

16.66 

58.31 

583.1 

9-34 

6 

5o 

3-2 

14.30 

45.76 

457.6 

7.33 

4 

45 

3-75 

11.57 

43-24 

4324 

6-93 

2 

4i 

5-3 

9.60 

50.88 

508.8 

8.15 

12 

37 

5-5 

7.82 

43.01 

430.1 

6.89 

9 

35 

4-3 

7.00 

30.10 

301.0 

4.82 

7 

34 

4-7 

6.60 

31.02 

310.2 

4-97 

3 

33 

5-i 

6.22 

31.72 

317.2 

5.08 

5 

3i 

6.13 

5.50 

33-71 

337-1 

5.40 

8 

29 

7.2 

4-23 

30.45 

304.5 

4.88 

ii 

26 

5.0 

3-86 

19.30 

193.0 

3-09 

Averages, 

40 

4.84 

9.14 

44.24 

442.4 

7.09 

To   evaporO 

ate  8  cubic 

feet       ofl 

40 

5.46 

9.14 

49.92 

499-2 

8.0 

water  per 

hour,  J 

GENERAL  FORMULAS  FOR  PRACTICAL  USE. 

By  the  French  experiments  with  stationary  boilers,  the  Lancashire  and 
French  boilers  were,  by  the  formulas,  page  812,  identical  in  performance; 
and  the  so-called  "Fairbairn"  boiler  was  nearly  as  effective  as  these, — 
within  3}^  per  cent.  The  three  forms  of  boiler  may,  therefore,  be  accepted 
as  equally  efficient;  and  they  may  be  classed  with  the  Wigan  boiler,  as  of 
equal  efficiency,  with  the  same  coal,  and  with  the  same  management. 

The  performance  of  the  Howard  boiler,  likewise,  is  conformable  to  the 
formula  for  the  Wigan  boiler;  and  the  Howard  boiler  is  a  type  of  the 
"  sectional "  kind  of  boilers. 

The  formula  for  the  Wigan  boiler  is,  therefore,  applicable  to  all  stationary 
boilers,  other  than  multitubular,  with  good  coal  and  good  management. 

The  performances  of  the  Newcastle  and  the  Wigan  marine  boilers,  are  nearly 
alike.  Thus,  for  a  surface-ratio  30,  the  corresponding  quantities  of  water, 
w,  for  different  rates  of  coal,  c,  per  square  foot  of  grate  per  hour,  are  as 
follows : — 


Coal, c=        10  20  30 

Newcastle  boiler, w=  116.5  2I3-6  310.7 

Wigan  boiler, w=  116.5  224.0  331.5 

Differences, w=         o.o  10.4  20.8 

Less  than  Wigan, ...                    o.o         4.6  6.3 


40     Ibs. 
407.8    „ 
439-0    » 

31-2    » 
7.1  per  cent. 


GENERAL  FORMULAS   FOR   PRACTICAL  USE.  817 

Halve  the  difference,  and  take  a  mean  of  the  formulas  ;  the  mean  will 
be  a  satisfactory  general  formula  for  marine  boilers  :  — 

Newcastle,  ......  w  =  .  02156  T^  +  Q.yi  c 

Wigan,  ...........  w  =  .  01  r*+  10.75  c 

Mean,  .........  o/  =  .oi6  r*  +  10.25  c  ...............  (28) 

For  coal-burning  locomotive  boilers  a  mean  of  the  two  formulas  adduced, 
page  813,  which  are  nearly  identical,  will  be  a  satisfactory  formula:  — 


S.  E.  Railway,  ........  w  =  .oog  f2-f  9.6  c 

L.  &  S.  W.  Railway,  w  =  .009  r*  +  9.82  c 

Mean,  ...............  w  =  .oog  r2  +  9.y  c   ..............  (  29  ) 


The  general  formulas  which  have  been  deduced  are  here  collected  to- 
gether :  — 

Formulas  for  the  Relation  of  Coal  and  Water  consumed  in  Steam-boilers  per 
square  foot  of  grate-area  per  hour,  and  the  ratio  of  the  heating-surface  to 
the  area  of  the  fire-grate. 

Water  taken  as  evaporated  from  and  at  2  12°  F. 

Stationary  Boilers,  .........  w  =  .o222  r2+  9.56^:  ............  (3°) 

Marine  Boilers,  ...  ..........  w=    .016  r2+  10.25  c  ............  (  31  ) 

Portable-engine  Boilers,..  w=    .008  r2  +  8.6  c  ............  (  32  ) 

Locomotive  Boilers  )  /  x 

(coal-burning),  /  ""  w=   •°°9r+  *"l  <  ........  (33) 

Locomotive  Boilers  )  /  x 

(coke-burning),}-"  ^=-°'78>"+  7-94^-            .  (  34  ) 

Limits  to  the  Application  of  the  Formulas  (  30  )  to  (  34  ). 

There  are  minimum  rates  of  consumption  of  fuel  below  which  these 
formulas  are  not  applicable.  The  limit  varies  for  each  kind  of  boiler,  and 
it  varies  with  the  surface-ratio.  It  is  imposed  by  the  fact  that  the  maximum 
evaporative  power  of  fuel  is  a  fixed  quantity,  and  is  naturally  at  that  point 
of  the  scale,  say  E  in  Fig.  329,  page  807,  where  the  reduction  of  the  rate 
of  combustion  for  a  given  ratio,  procures  the  absorption  into  the  boiler  of 
the  whole  of  the  proportion  of  the  heat  which  is  available  for  evaporation. 
In  the  combustion  of  good  coal  the  limit  of  evaporative  efficiency  may  be 
taken  as  measured  by  1  2  ^  Ibs.  of  water  from  and  at  212°  F.  ;  and  in  that 
of  good  coke  by  12  Ibs.  of  water  from  and  at  212°  F.  The  dotted  line 
EA,  Fig.  329,  represents  the  correct  course  of  the  diagram  towards  the 
zero  point,  indicating  a  constant  proportion  of  w—  12.5  c,  for  coal;  or  w  = 
1  2  c,  for  coke. 

To  ascertain  the  minimum  rates  of  combustion  of  coal  for  stationary 
boilers,  to  which  the  formula  (  30  )  applies  :  —  The  limit  is  reached  when  w 
becomes  equal  to  12.5^;  or  when  I2.5^ 


9.  5  6)  c  =  2.  94.*:.    By  reduction,  c  =  '°222  r*  =•  .  oo  7  5  5  r2.    For  a  given  surface- 

2.94 

52 


8i8 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


ratio  r,  the  limiting  value  of  c  is  found  by  multiplying  the  square  of  the 
ratio  by  .00755. 

For  the  other  kinds  of  boiler,  the  limiting  values  of  c  are  found  in  the  same 
way.  They  are  here  placed  all  together : — 

Stationary  boilers,  limiting  value  of  c  -  .007 5 5  r*. 

Marine  boilers, „  „  =.oojr2. 

Portable-engine  boilers, „          .    „  =.oo2r*. 

Locomotive  boilers  (coal-burning),         „  „  =. 00325  r2. 

„      (coke-burning),        „  „  =.0044^. 

For  lower  values  of  c,  or  consumptions  of  fuel  per  square  foot  of  grate  per 
hour,  the  values  of  w,  the  corresponding  quantities  of  water,  are  simply 
12.5  c  for  coal,  and  12  c  for  coke. 

The  annexed  table,  No.  292,  contains  the  limiting  values  of  c  for  given 
surface-ratios  r. 


Table  No.  292. —  MINIMUM  VALUES  OF  c,  OR  MINIMUM  QUANTITIES  OF 
FUEL  CONSUMED  PER  SQUARE  FOOT  OF  GRATE  PER  HOUR,  FOR 
GIVEN  SURFACE-RATIOS,  TO  WHICH  THE  FORMULAS  (  30 )  TO  ( 34 ) 
ARE  APPLICABLE. 


5 

10 

SUR 

15 

FACE-RAl 

20 

IDS. 

30 

40 

50 

Minimum  Consumption  of  Fuel  per  Square  Foot  of 
Grate  per  Hour. 

Stationary   . 

Ib. 
.2 

•17 
.05 
.1 
.1 

lb. 

•7 
•7 

.2 

•3 

•4 

lb. 

i-7 
1.6 

•4 
•7 

I.O 

Ibs. 

3-0 

2.8 

.8 

1:1 

Ibs. 

6.8 
6-3 

2.9 
4.0 

Ibs. 
I2.I 
II.  2 
3-2 
5.2 
7.0 

Ibs. 

18.9 
17.5 

K 

II.O 

Marine,  

Portable 

Locomotive  (coal-burning),. 
Do.          (coke-burning), 

Locomotive  (coal-burning),. 
Do.          (coke-burning), 

60 

Surface- 

70    ;    75 

ratios  (con 
80 

tinued}. 
90 

100 

II.7 

16 

15.9 

21 

18.3 

25 

20.8 

28 

26.3 
36 

32.5 

44 

The  only  limit  to  the  application  of  the  formulas  (  30  )  to  (  34 ),  to  ascend- 
ing values  of  c,  or  quantities  of  fuel  per  square  foot  per  hour,  is  the  limit  of 
endurance  of  the  fuel  itself  under  the  action  of  the  draught: — from  100  Ibs. 
to  1 20  Ibs.  per  square  foot  per  hour,  for  ordinary  hard  coal  or  coke.  Beyond 
this  limit,  the  fuel  is  liable  to  be  shaken  and  partly  dispersed,  unconsumed, 
by  the  force  of  the  draught;  although  coke  has  been  known  to  withstand 
the  draught  of  a  locomotive  when  consumed  at  the  rate  of  130  Ibs.  per 
square  foot  per  hour. 


EVAPORATIVE   EFFICIENCY   OF   STEAM-BOILERS. 


819 


Table  No.  293. — EVAPORATIVE  PERFORMANCE  OF  STEAM-BOILERS,   FOR 
INCREASING  RATES  OF  COMBUSTION  AND  DIFFERENT  SURFACE-RATIOS. 

For  best  coal  and  best  coke ;  surface-ratio  30. 


Kind  of  Boiler,          Water  from  and  at 
and  Fuel.                212°  F.  per  Hour. 

Fue 
5 

per  Squ 
IO 

are  Foot 
15 

of  Grate 
2O 

per  Hou 
30 

r,  in  pouj 
40 

ids. 
50 

Stationary,  coal, 
formula  (  30  ). 

Per  square  foot 
Per  Ib.  of  coal 

Ibs. 
62.5* 
12-5 

Ibs. 

116 
11.56 

Ibs. 
I63 
10.89 

Ibs. 
211 
10.56 

Ibs. 
307 
IO.23 

Ibs. 
402 
1  0.06 

Ibs. 
498 
9.96 

Marine,  coal,  for- 
mula (31). 

Per  square  foot 
Per  Ib.  of  coal 

62.5* 
12.5 

"7 

11.69 

168 
11.25 

219 
JO-PS 

322 
10.69 

424 
I0.6I 

527 
10.54 

Portable,     coal, 
formula  (  32  ). 

Per  square  foot 
Per  Ib.  of  coal 

50 
IO 

93 
9-3 

136 
9.01 

179 

8.95 

265 

8.83 

351 

8.77 

437 
8.74 

Locomotive  (coal- 
burning), 
formula  (  33  ). 

Per  square  foot 
Per  Ib.  of  coal 

57 
11.4 

i°5 
10.5 

'54 
10.26 

202 
IO.  IO 

299 

9-97 

396 
9.90 

493    ! 
9-86  j 

Locomotive  (coke- 
burning), 
formula  (34). 

Per  square  foot 
Per  Ib.  of  coke 

56 
11.14 

95 
9-54 

135 
9.02 

175 
8-75 

254 
8-47 

334 
8-35 

413 
8.03 

Surface-ratio  50. 

Kind  of  Boiler, 
and  Fuel. 

Water  from  and  at 
212°  per  Hour. 

Fue 

5 

i  per  Sqt 
10 

are  Foot 

'5 

of  Grate 
2O 

per  Hoi 
30 

ir,  in  pou 
40 

nds. 
50 

Stationary,  coal, 
formula  (30). 

Per  square  foot 
Per  Ib.  of  coal 

Ibs. 
62.5* 
12-5 

Ibs. 
125* 
12.5 

Ibs. 
187.5* 
12.5 

Ibs. 
247 
12-33 

Ibs. 
342 
II.4I 

Ibs. 
438 
10.95 

Ibs. 

534 
10.67 

Marine,  coal,  for- 
mula (31). 

Per  square  foot 
Per  Ib.  of  coal 

62.5* 
12-5 

125* 
12-5 

187.5* 
12.5 

245 
12.25 

348 
11.58 

450 
11.25 

552 
11.05 

Portable,    coal, 
formula  (  32  ). 

Per  square  foot 
Per  Ib.  of  coal 

62.5* 
12-5 

106 
10.6 

149 

9-93 

I92 
9.6 

278 
9.27 

364 
9.10 

450 
9.00 

Locomotive  (coal- 
burning), 
formula  (  33  ). 

Per  square  foot 
Per  Ib.  of  coal 

62.5* 
12.5 

1  20 
ii-95 

168 
ii.  20 

217 
10.85 

3H 
10.45 

4" 

10.26 

508 
10.15 

Locomotive  (coke- 
burning), 
formula  (34). 

Per  square  foot 
Per  Ib.  of  coke 

60* 

I2.O 

120* 
I2.O 

164 
10.91 

203 

io.  16 

283 
9.42 

362 
9-05 

8*83 

Surface-ratios  75. 

Kind  of  Boiler, 
and  Fuel. 

Water  from  and  at 
212°  per  Hour. 

Fue 
30 

1  per  Sqi 
40 

iare  Foot 

5° 

of  Grate 
60 

per  Hoi 

75 

ir,  in  pou 
90 

nds. 
100 

Locomotive  (coal- 
burning), 
formula  (33). 

Per  square  foot 
Per  Ib.  of  coal 

Ibs. 
342 

"•39 

Ibs. 

439 
10.97 

Ibs. 

536 

10.71 

Ibs. 

63I 
10.65 

Ibs. 

778 
10.37 

Ibs. 

927 
10.26 

Ibs. 

IO2O 
10.20 

Locomotive  (coke- 
burning), 
formula  (34). 

Per  square  foot 
Per  Ib.  of  coke 

338 
11.27 

418 
10.44 

497 
9-94 

576 
9.61 

695 
9.26 

8l5 
9-05 

894 
8-94 

1  These  quantities  fall  below  the  scope  of  the  formulas  for  the  water,  as  explained  in 
the  text. 


820 


EVAPORATIVE   PERFORMANCE   OF   STEAM-BOILERS. 


APPLICATIONS  OF  THE  GENERAL  FORMULAS  FOR  THE  EVAPORATIVE 
PERFORMANCE  OF  STEAM-BOILERS. 

The  table  No.  293,  preceding  page,  contains  the  relative  quantities  of  fuel 
consumed  and  water  evaporated,  for  surface-ratios  and  rates  of  combustion 
per  square  foot  of  grate  per  hour,  within  the  range  of  ordinary  practice.  It 
is  seen  that,  with  the  surface-ratios  30  and  50,  the  boilers  are  in  the  order 
of  evaporative  efficiency  as  follows : — 


SURFACE-RATIO  30. 

Marine. 

Stationary. 

Locomotive  (coal-burning). 

Portable. 

Locomotive  (coke-burning). 


SURFACE-RATIO  50. 

Marine. 
Stationary. 

Locomotive  (coal-burning). 
Do.         (coke-burning). 
Portable. 


Table  No.  294. — EQUIVALENT  WEIGHTS  OF  BEST  COAL  AND  INFERIOR  FUELS. 

To  be  used  with  formulas  (  30  )  to  (  34  ),  page  817. 


Relative 
Heating 
Power. 

Equivalenl 
Weight 
of  Best 
Coal. 

Equivalen 
Weight  of 
Inferior 
Fuel. 

Relative 
Heating 
Power. 

Equivalent 
Weight 
of  Best 
Coal. 

Equivalen 
Weight  of 
Inferior 
Fuel. 

Relative 
Heating 
Power. 

Equivalen 
Weight 
of  Best 
Coal. 

Equivalen 
Weight  of 
Inferior 
Fuel. 

best  coal=i 

best  coal=i 

best  coal=i. 

best  coal=;i 

best  coal  =  i 

best  coal=i. 

100 

I 

I. 

70 

.70 

143 

40 

.40 

2.50 

99 

•99 

I.OI 

-    -69 

1.45 

39 

•39 

2.56 

98 

.98 

1.02 

68 

.68 

1.47 

38 

.38 

2.63 

97 

•97 

1.03 

67 

.67 

1.49 

37 

•37 

2.70 

96 

.96 

1.04 

66 

.66 

1.52 

36 

.36 

2.78 

95 

•95 

1.05 

65 

.65 

1.54 

35 

•35 

2.86 

94 

•94 

1.06 

64 

.64 

1.56 

34 

•34 

2-94 

93 

•93 

i.  08 

63 

•63 

1.59 

33 

•33 

3-03 

92 

.92 

1.09 

62 

.62 

1.61 

32 

•32 

3-i3 

9i 

.91 

1.  10 

61 

.61 

1.64 

3i 

•3i 

3-23 

90 

.90 

i.  ii 

60 

.60 

1.67 

30 

•30 

3-33 

89 
.     88 

!88 

1.  12 
I.I4 

59 
58 

•59 
•58 

1.69 
1.72 

3 

.29 
.28 

3-45 
3-57 

87 

.87 

I.I5 

57 

•57 

i-75 

27 

•27 

3-7o 

86 

.86 

1.16 

56 

•56 

1.79 

26 

.26 

3-85 

85 

.85 

1.18 

55 

•55 

1.82 

25 

•25 

4.00 

84 

•84 

1.19 

54 

•54 

1.85 

24 

.24 

4.17 

83 

-83 

1.20 

53 

•53 

1.89 

23 

•23 

4-35 

82 

.82 

1.22 

52 

.52 

1.92 

22 

.22 

4-55 

81 

.81 

1.23 

5i 

•5i 

1.96 

21 

.21 

476 

80 

.80 

1.25 

5o 

.50 

2.OO 

20 

.20 

5.00 

79 

•79 

1.27 

49 

•49 

2.04 

JQ 

.19 

5.27 

78 

•78 

1.28 

48 

•48 

2.08 

18 

.18 

'5.56 

77 

•77 

1.30 

47 

•47 

2.13 

17 

•17 

5.88 

76 

.76 

1.32 

46 

.46 

2.17 

16 

.16 

6.25 

75 

•75 

i-33 

45 

•45 

2.22 

15 

•15 

6.67 

74 

•74 

1-35 

44 

•44 

2.27 

14 

.14 

7.14 

73 

•73 

1-37 

43 

•43 

2-33 

13 

•13 

7.69 

72 

.72 

i-39 

42 

.42 

2.38 

12 

.12 

8-33 

7i 

•7i 

1.41 

4i 

.41 

2-44 

II 

.11 

9.09 

10 

.10 

IO.O 

EMPLOYMENT   OF   FORMULAS   FOR   INFERIOR   FUELS.        821 

Portable-engine  boilers  are  clearly  inferior  in  efficiency  to  coal-burning 
locomotive  boilers,  and  they  may  be  constructed  like  these  with  sensible 
advantage. 

Employment  of  the  Formulas  (30)  #7(34)  for  Fuels  of  Inferior  Heating 
Power. — i  st.  To  find  the  evaporative  performance  of  a  given  weight  of 
inferior  fuel,  per  square  foot  of  grate  per  hour.  Substitute,  for  the  given  weight 
of  inferior  fuel,  the  equivalent  weight  of  best  coal,  and  find  by  the  formula 
the  water  evaporated. 

The  equivalent  weight  of  best  coal  is  found  by  multiplying  the  weight 
of  inferior  fuel  by  the  number  in  column  2  of  the  table  No.  294,  opposite 
the  relative  heating  power  of  the  inferior  fuel. 

2d.  To  find  the  weight  of  an  inferior  fuel  required  for  a  given  evaporative 
performance.  Find,  by  the  formula,  in  its  inverted  form,  on  the  model  of 
the  equation  (  9  ),  page  807,  the  weight  of  best  coal  required,  and  substitute 
for  this  weight  the  equivalent  weight  of  the  inferior  fuel. 

The  equivalent  weight  of  inferior  fuel  is  found  by  multiplying  the  weight 
of  best  coal  by  the  number  in  column  3  of  the  table,  opposite  the  relative 
heating  power  of  the  inferior  fuel. 

A  table  of  relative  heating  powers  of  fuels  is  given  at  page  769. 


STEAM    ENGINE. 


ACTION  OF  STEAM  IN  A  SINGLE  CYLINDER. 

PRESSURE  OF  STEAM  DURING  EXPANSION  IN  A  CYLINDER. 

When  steam  is  admitted  to  a  cylinder  during  a  portion  of  the  stroke,  then 
cut  off,  and  expanded  in  the  cylinder,  upon  the  piston,  for  the  remainder  of 
the  stroke,  the  pressure  on  the  piston,  during  the  period  of  admission,  is  or 
ought  to  be  uniform,  whilst  the  pressure  during  the  period  of  expansion 
falls  as  the  piston  advances  and  the  steam  expands.  In  engines  in  good 
working  order,  the  expansion  follows  substantially  the  law  of  Boyle,  or 
Mariotte,  according  to  which  the  pressure  falls  in  the  inverse  ratio  of  the 
expansion.  Substantially,  it  is  said,  for  the  actual  changes  of  pressure  seldom 
follow  the  law  exactly.  The  pressure  usually  falls  more  rapidly  in  the  first 
portion  of  the  expansion,  and  less  rapidly  in  the  last  portion,  than  is  indi- 
cated by  the  law  of  the  inverse  ratio;  and  thus,  the  final  pressure  may  be, 
and  it  usually  is,  greater  than  that  which  would  be  deduced  from  the  ratio 
of  expansion.  But  the  fulness  of  the  expansion-curve  depicted  on  the 
indicator-diagram,  near  the  end,  compensates  for  the  hollowness  near  the 
beginning;  and,  sinking  details,  it  is  found  that,  practically,  the  area  bounded 
by  the  curve  is  equal  to  that  which  would  be  bounded  by  a  hyperbolic 
curve  formed  according  to  Mariotte's  law.1 

It  is,  therefore,  assumed,  for  purposes  of  illustration  and  the  calculation  of 
power,  that  the  expansion  of  steam  in  the  cylinder  takes  place  according  to 
Mariotte's  law:  the  curve  representing  the  diminishing  pressures  due  to 
the  increasing  volume  being  a  portion  of  a  hyperbola. 

To  formulate  the  method  of  describing  a  hyperbolic  curve  of  expansion 
over  a  given  base-line : — 

Let  L  =  the  length  of  the  stroke,  in  feet,  supposing  that  there  is  not  any 

clearance. 

/=the  period  of  admission,  or  the  cut-off,  in  feet. 
s  =  any  greater  part  of  the  stroke  measured  from  the  commencement,  in 

feet. 

P  =  the  total  initial  pressure,  in  pounds  per  square  inch. 
P'  =  the  total  pressure  in  pounds  per  square  inch,  at  the  end  of  a  given 

part  of  the  stroke  s. 

P"  =  the  total  final  pressure,  or  the  pressure  at  the  end  of  the  stroke,  in 
pounds  per  square  inch. 

1  Mr.  David  Thomson  arrived  at  the  same  conclusion  in  his  excellent  paper  "On 
Compound  Engines,"  in  the  Proceedings  of  the  London  Association  of  Foremen  Engineersy. 
September,  1873. 


PRESSURE  OF   STEAM   DURING   EXPANSION.  823 

ThenF.±i£;  ..  (  z  ) 


expressed  by  the  following  rule : — 

RULE  i.  To  find  the  pressure  at  any  point  of  the  period  of  expansion  when 
the  initial  pressure  is  given. — Multiply  the  initial  pressure  in  pounds  per 
square  inch  by  the  period  of  admission  in  feet,  and  divide  the  product  by 
the  distance  of  the  given  point  from  the  beginning  of  the  stroke.  The 
quotient  is  the  pressure  in  pounds  per  square  inch. 

The  pressure  P'  may  also  be  found  from  the  final  pressure  P"  by  the 
formula 


giving  the  rule : — 

RULE  2.  To  find  the  pressure  at  any  point  of  the  period  of  expansion  when 
the  final  pressure  is  given. — Multiply  the  final  pressure  in  pounds  per  square 
inch  by  the  length  of  the  stroke,  and  divide  the  product  by  the  distance  of 
the  given  point  from  the  beginning  of  the  stroke.  The  quotient  is  the 
pressure  in  pounds  per  square  inch. 

Note. — When  there  is  clearance,  it  is  to  be  reckoned  in  parts  of  the  stroke, 
and  added  to  the  values  of  L,  /,  and  s,  before  using  these  for  calculation. 

Let  the  base-line  mn,  be  the  length  of  the  stroke,  say  6  feet;  me  the 
initial  pressure,  say  63  Ibs. ;  cd  the  period  of  admission,  say  one-third  of 
the  stroke.  Suppose,  for  simpli- 
city, that  there  is  not  any  clear- 
ance. Draw  the  perpendicular^', 
from  the  point  of  cut-off,  and 
divide  the  period  of  expansion  d'n 
into  any  suitable  number  of  parts, 
say  10  parts,  at  the  points  i,  2, 
3,  &c.  Calculate  by  the  rule  the 

Several     pressures     at     the     points  Fig.  330.— Construction  of  a  Hyberbolic  Curve. 

i,  2,  3,  &c.,  and  set  them  off  by 

the  scale  of  pressure  on  vertical  ordinates  from  the  points ;  the  curve  dg 
traced  through  the  ends  of  the  ordinates  is  the  hyperbolic  curve  of  expansion. 
At  the  successive  points  of  the  base  of  the  expansion-line,  which  are, — 

d'>     J>       2>      3>       4,       5>        6,        7>        8>         9>      .«» 
the  values  of  the  ordinates,  or  pressures,  are — 

63>  52-5>  45.  39-4,  35.  S^S*  28-6>  26.1,  24-2,  22.5,  2 1  Ibs.  per  sq.  in. ; 
or,  putting  the  initial  pressure  =  i,  they  are  relatively  as 

i,  .833,  .714,  .625,  .555,  .500,  .455,  .417,   .385,  .357,  .333. 

The  extreme  ordinate  ng  is  thus  found  to  be  a  third  of  dd',  or  21  Ibs., 
and  the  ordinate  No.  5  is  a  half,  or  31.5  Ibs.  As  an  example  of  the  cal- 
culations for  an  ordinate,  take  No.  2.  The  period  of  admission  is  2  feet,  the 

divisions  of  the  base  of  expansion  are  — -  foot,  and  the  length  m  2  is  2.8  feet; 

10 

then,  by  the  rule,  the  pressure  measured  by  No.  2  ordinate,  is, — 


824  STEAM   ENGINE— SINGLE-CYLINDER. 

The  calculation  may  generally  be  simplified  by  taking,  as  a  datum,  the 
length  of  stroke  =  i.  In  this  instance,  the  period  of  admission  would 
be  =  .333,  the  period  of  expansion  =  .666,  and  each  tenth  division  =  .0666. 
By  the  rule, 

63lbs.  x.333     _    21    _ 
•333  +  (.0666x2)  -.467" 

as  before. 

To  illustrate  generally  the  application  of  the  hyperbolic  law  of  expansion, 
showing  that  the  product  of  the  pressure  and  the  volume  at  any  point  of  the 
expansion-curve  is  constant,  let  the  base-line  A  B  represent  the  course  of  a 
piston  in  a  cylinder,  and  the  volume  described  by  it.  Sup- 
posing that  there  is  no  clearance,  let  steam  of  10  Ibs.  total 
pressure  A  c,  be  admitted  for  a  space  i  foot  in  length,  A  D. 
The  rectangle  A  E  is  the  product  of  the  pressure  and  volume 
of  the  steam  admitted.  If  expanded  to  the  double  volume 
A  d,  and  to  half  the  pressure  de,  the  area  of  the  elongated 
rectangle  AC  is  equal  to  that  of  the  initial  rectangle  A  E. 
Expanding  further,  to  four  volumes  A  d',  and  to  the  fourth 
part  of  the  initial 
pressure,  d'  e,  the 
new  rectangle  Ae' 
is  equal  to  each  of 
the  others  Ae  and 

A  E.       Similarly,    the        Fig.  331— To  Illustrate  the  Hyperbolic  Law  of  the  Expansion  of  Steam. 

rectangles  Ae"  and 
Ae'",  for  a  fifth  and  a  sixth  of  the  initial  pressure,  and  five  times  and 
six  times  the  initial  volume,  are  each  equal  to  the  initial  rectangle  AE. 
The  hyperbolic  curve  containing  these  rectangles  may  be  indefinitely 
extended  at  either  end,  to  embrace,  on  the  one  part,  intense  pres- 
sures and  small  volumes,  and,  on  the  other  part,  very  low  pressures 
and  large  volumes. 

THE  WORK  OF  STEAM  BY  EXPANSION. 

Proceeding,  now,  to  a  consideration  of  the  area  of  the  diagram,  Fig.  331; 
— as  the  area  of  the  rectangle  A  E,  is  the  product  of  the  pressure  and  volume, 
and  expresses  the  work  done  upon  the  piston  by  the  steam  in  entering  and 
occupying  the  cylinder,  so,  likewise,  the  hyperbolic  area,  D  E  d'"  e'",  expresses 
the  work  done  by  the  steam  by  expansion  within  the  cylinder  after  it  is 
shut  in.  'This  area,  and  consequently  the  quantity  of  work  done,  may  be 
computed  by  means  of  the  known  relations  of  hyperbolic  superficies  with 
their  base-lines: — according  to  which,  if  the  base-lines  AD,  Ad,  Ad',  &c., 
extend  in  a  geometrical  ratio,  or  as  i,  2,  4,  8,  16,  &c.,  the  successive  areas 
De,  ne',  &c.,  increase  in  an  arithmetical  ratio,  or  as  i,  2,  3,  4,  &c.  On  the 
principles  of  logarithms,  which  represent,  in  arithmetical  ratio,  natural 
numbers  in  geometrical  ratio,  special  tables  of  so-called  hyperbolic  logarithms 
are  compiled,  to  facilitate  the  calculation  of  the  areas  of  work  due  to 
various  degrees  of  expansion.  The  hyperbolic  numbers  consist,  in  fact, 
of  the  multiples  of  common  logarithms  by  2.302585,  which,  thus  modified, 
become  direct  expressions  of  the  proportions  borne  by  the  work  by  expansion 


THE  WORK  OF  STEAM   BY   EXPANSION.  825 

pertaining  to  different  degrees  of  expansion,  to  the  initial  work  done  by  the 
steam  during  its  admission  into  the  cylinder;  but  they  are  not  employed 
as  logarithms.  For  example,  the  initial  volume  being  expressed  by  i,  and 
the  total  volumes  by  expansion,  by  the  following  numbers  in  geometrical 
ratio, 

i,  2,  4,  8,  16, 

the  hyperbolic  logarithms  of  these  numbers  are,  in  arithmetical  ratio, 

.000,        .693,       1.386,       2.079,       2-772, 
being  as  o,  i,  2,  3,  4, 

and  these  logarithms  express  the  actual  ratio  of  the  whole  work  by  expansion, 
for  different  degrees  of  expansion,  to  the  initial  work  of  the  steam,  expressed 
as  i.  The  total  work  done  by  a  quantity  of  steam  expanded  successively 
from  the  initial  volume, 

i       to       2,  4,  8,  1 6  volumes, 

will  therefore  be  in  the  proportions  of 

i,     i +.693,     1  +  1.386,     1  +  2.079,     1  +  2.772, 
or       i,         1.693,  2.386,  3.079,  3.772, 

showing  that,  for  an  expansion  of  16  times,  the  initial  work  done  by  the 
steam  during  its  admission  is  nearly  quadrupled. 

But  it  is  necessary  to  make  a  deduction  for  the 
back  pressure  from  the  condenser,  to  find  the  effec- 
tive work  of  the  steam.  Suppose  a  cylinder  of  5  feet 
stroke,  represented  by  A  B,  Fig.  331,  with  a  piston 
having  an  area  of  i  square  inch,  into  which  steam 
of  10  Ibs.  pressure  per  square  inch  is  admitted  for 
i  foot  of  the  stroke,  A  D,  against  a  uniform  back 
pressure  of,  say,  2  Ibs.  per  square  inch,  for  the  whole 
stroke.  Let  the  steam  be  expanded  through  the  re- 
maining  four-fifths  of  the  stroke,  and  construct  the 
diagram  of  work,  Fig.  332,  in  which  the  2-lb.  zone  of  resistance  or  back 
pressure  is  shaded.  Then, 

At  the  end  of  the ist,      2d,          3d,  4th,  5th  foot  of  stroke, 

The  total  pressures  are....  10          5  3  */3  2)4  2  Ibs.  per  sq.  inch; 

The  back  pressures  are...  222  2  2  Ibs.     do.    do.; 

The  effective  pressures....  8          3  i  i/3  yz  o  Ibs.     do.    do. 

The  total  work  done  by  expansion  up  to  the  end  of  each  foot  of  stroke, 
is  represented  by  the  hyperbolic  logarithm  of  the  ratio  of  expansion,  the 
initial  work  being  =  i.  Thus, — 

At  the  end  of  the ist,      2d,          3d,          4th,          $th  foot  of  stroke, 

The  steam  is  expanded  into  —         234  5  volumes, 

Of  which  the  hyperbolic   )  *l:-c-',  ,  <T 

logarithms  are  [     ~  ^         *'10        *'&  l£l 

The  initial  duty  being  as...     i          i  i  i  i  (unity), 

And  the  total  duty  as i  1.69        2.10        2.39  2.61 


826 


STEAM   ENGINE— SINGLE-CYLINDER. 


As  the  initial  work,  represented  by  i,  is  10  foot-pounds,  being  10  Ibs.  exerted 
through  i  foot,  and  the  resistance  is  2  foot-pounds  for  each  foot  of  the 
stroke, — 

At  the  end  of  the 

ist,  2d,  3d,  4th,  5th  foot  of  the  stroke, 

The  work  by  expansion  is 

o.o  6.9  n.o  13.9          16.1  foot-pounds; 

The  total  work  done  is 

10  16.9  21. o 

The  total  resistance  is 

246 
The  total  effective  work  is 
8  12.9 

And  the  gain  by  expansion  is 
o  61 


15.0 


23.9 


8 


15.9 


26.1 


10 


16.1 


do. 


do. 


do. 


87 


99 


101  per  cent. 


From  the  foregoing  particulars,  it  appears  that  the  total  work  of  the 
steam,  by  expanding  it  to  five  times  the  initial  volume,  is  fully  2^  times 
the  initial  work  done  without  expansion.  When  the  back  pressure  is  allowed 
for,  the  effective  work,  16.1  foot-pounds,  is  only  twice  the  initial  work,  8  foot- 
pounds; making  a  gain  of  101  per  cent,  when  the  expansion  is  extended 
to  the  extreme  limit,  where  the  positive  pressure  becomes  equal  to  the 
back  pressure. 

It  further  appears  that  the  effective  work  of  the  steam  expanded  down  to 
the  back  pressure  from  the  condenser,  is  just  equal  to  the  work  developed 
by  expansion  alone.  The  initial  work  is  balanced  in  amount  by  the  resist- 
ance, each  of  them  being  10  foot-pounds. 

The  same  conclusions  apply  to  a  non-condensing  cylinder  discharging  the 
steam  into  the  atmosphere.  Let  the  total  initial  pressure,  A  C,  Fig.  332,  be 
75  Ibs.  per  square  inch,  and  suppose  the  steam  to  be  expanded  five  times, 
as  before,  down  to  a  pressure  of  1 5  Ibs.  per  square  inch,  and  then  exhausted 
into  the  atmosphere,  maintaining  a  back  pressure  of  15  Ibs.  per  square 
inch  throughout  the  stroke,  represented  by  the  shaded  zone.  On  a  piston 
of  i  square  inch  area,  the  proportions  of  work  will  be  as  follows : — 

At  the  end  of  the ist, 

The  total  work  done  is  as 
The  total  work  done  is  ) 

actually ]      75 

The  total  resistance  is 

The  total  effective  work  is 
The  gain  by  expansion  is 

In  this  case,  where  steam  of  five  atmospheres  is  expanded  five  times,  and 
exhausted  into  the  atmosphere  at  a  pressure  of  one  atmosphere,  the  pro- 
portions of  work  done  are  the  same  as  when  steam  of  10  Ibs.  pressure  per 
square  inch  is  expanded  five  times  and  exhausted  at  a  pressure  of  one-fifth, 
or  2  Ibs.  per  inch;  and  they  indicate  equal  degrees  of  efficiency  of  the 
steam  in  the  way  it  is  applied. 


ist, 

2d, 

3d, 

4th, 

5th  foot  of  stroke, 

I 

1.69 

2.10 

2-39 

2.6  1 

75 

126.7 

157.5 

179.2 

195.7  foot-pounds. 

15 

30 

45 

60 

75            do. 

60 

96.7 

112.5 

119.2 

120.7         do. 

o 

61 

87 

99 

101      per  cent. 

CLEARANCE  IN   STEAM-CYLINDERS.  827 

It  may  be  concluded,  generally,  that  when  the  steam  is  expanded  down 
to  the  back  pressure  in  the  cylinder,  whether  from  the  condenser  or  from 
the  atmosphere,  the  effective  work  done  in  the  cylinder  is  just  equal  to  the 
total  work  done  by  expansion,  the  total  initial  work  being  just  balanced  and 
neutralized  in  amount  by  the  resistance  of  back  pressure. 

And  the  utmost  useful  ratio  of  expansion,  looking  to  the  operations 
within  the  cylinder,  is  measured  by  the  number  of  times  which  the  total 
back  pressure  is  contained  in  the  total  initial  pressure  of  the  steam  in  the 
cylinder.  Indeed,  it  may  be  affirmed  that  four-fifths  of  this  measure  of 
expansion  is  sufficient  as  a  limit,  for  it  has  been  shown  that  whilst  the  gain 
by  expansion  to  four  times  is  99  per  cent.,  that  of  a  fivefold  expansion  is 
101  per  cent,  which  is  only  2  per  cent.  more. 

Another  reason  usually  advanced  for  arresting  the  fall  of  pressure,  in 
expanding,  at  a  higher  limit  than  the  back  pressure,  is  based  on  the 
frictional  or  passive  resistance  of  the  engine.  This  resistance  is  to  be 
opposed  by  the  steam  in  the  cylinder;  and  the  total  pressure,  it  is  said, 
should  not  fall  below  that  which  is  equivalent  to  the  back  pressure,  plus  the 
frictional  resistance,  since,  it,  is  argued,  if  the  pressure  at  any  part  of  the 
stroke  do  fall  below  the  sum  of  these  resistances,  the  excess  of  these  above 
the  positive  pressure  is  so  much  dead  resistance,  and  is  so  much  in  reduc- 
tion of  the  useful  efficiency  of  the  steam.  This  argument  is  plausible,  but 
fallacious ;  and  it  would  be  valid  only  on  the  supposition  that  the  engine 
could  move  without,  at  the  same  time,  doing  its  proper  duty  in  driving 
shafting  and  machinery.  The  supposition  is,  of  course,  impossible.  But, 
why  draw  the  line  of  so-called  useless  resistance  at  the  fly-wheel  shaft? 
The  shafting  for  driving  the  machinery  also  opposes  dead  resistance,  and 
before  the  engine  can  move  at  all,  the  resistance  of  the  shafting  must  be 
overcome.  The  resistance  of  all  the  machinery  must  likewise  be  overcome. 
The  useful  work  to  be  done  must  likewise  be  overcome;  in  fact,  the  whole 
of  the  work,  dead  and  alive,  must  be  overcome.  So  the  argument  leads  to 
the  absurd  conclusion  that  the  pressure  in  the  cylinder  should  not  fall 
below  the  total  mean  pressure  exerted;  and  as  it  is  not  to  fall  below, 
neither  can  it  reach  above  the  mean  pressure,  for  that  would  imply  an 
additional  initial  force,  which  would  render  a  greater  mean  pressure,  which 
is  absurd.  If  the  argument  had  any  truth  in  it,  it  would  lead  necessarily 
to  the  abandonment  of  all  expansive  working,  and  to  the  employment  of  a 
uniform  pressure,  with  the  admission  of  steam  throughout  the  whole  of  the 
stroke. 

CLEARANCE  IN  STEAM-CYLINDERS. 

The  clearance,  or  free  space,  between  the  piston  when  at  the  beginning 
of  a  stroke,  and  the  slide-valve,  is  filled  with  steam  of  the  initial  pressure  at 
the  commencement  of  each  stroke ;  and  this  padding,  as  it  may  be  called, 
does  no  work  directly,  and  is  entirely  non-effective  in  non-expansive  engines. 
But  in  expansive-working  cylinders,  the  clearance-steam  does  its  proper 
quota  of  work,  in  conjunction  with  the  other  steam,  during  the  period  of 
expansion. 

The  volume  of  the  clearance  may  be  measured  in  parts  of  the  stroke 
supposed  to  be  multiplied  into  the  area  of  the  piston ;  and  it  is  here  taken, 
for  purposes  of  discussion,  at  7  per  cent,  of  the  stroke. 


828  STEAM   ENGINE—  SINGLE-CYLINDER. 

FORMULAS  FOR  THE  WORK  OF  STEAM  IN  THE  CYLINDER. 

Now,  let  L  =  the  length  of  stroke,  in  feet, 

/  =  the  period  of  admission,  or  the  cut-off,  in  feet,  excluding 

clearance, 
c  =  the  total  clearance  at  one  end  of  the  cylinder,  the  volume 

being  measured  in  feet  of  the  stroke, 
L'  =  the  length  of  the  stroke,  plus  the  clearance,  or  L  +  c, 
I'  =  the  period  of  admission,  plus  the  clearance,  or  /+  c, 
R  =  the  nominal  ratio  of  expansion,  or  L  -r-  /, 
R'  =  the  actual  ratio  of  expansion,  or  L'  +  /', 
a  =  the  area  of  the  piston  in  square  inches, 
P  =  the  total  initial  pressure  in  Ibs.  per  square  inch,  supposed 

to  be  uniform  during  admission, 
p  =  the  average  total  pressure,  in  Ibs.  per  square  inch,  for  the 

whole  stroke, 
p'  =  the  average  back  pressure,  in  Ibs.  per  square  inch,  for  the 

whole  stroke, 

w  =  the  whole  work  done  in  one  stroke,  in  foot-pounds, 
w'  =  the  work  of  back  pressure  for  one  stroke,  in  foot-pounds, 
W  =  the  net  work  done  in  foot-pounds. 

The  actual  ratio  of  expansion  is 


. 

l+c      I' 

The  work  done  during  admission  is  equal  to  the  total  pressure  on  the 
piston,  a  x  P,  multiplied  by  the  period  of  admission,  or  a  P  /,  which  is  the 
work  in  foot-pounds,  and  this  work  is  done  by  a  volume  of  steam  measured 
by  the  period  of  admission,  plus  the  clearance,  or  by  /  +  c  -  l'\  and  as 
1=1'  -c,  then 

whole  work  done  during  admission  =  a  P  /=  a  P  (/'  -  c)  .....  (  3  ) 

To  find  the  work  done  by  expansion  to  the  end  of  the  stroke,  the  total 
pressure  on  the  piston,  a  P,  is  to  be  multiplied  by  /',  the  period  of  admis- 
sion plus  the  clearance,  and  by  the  hyperbolic  logarithm  of  R',  the  actual 
ratio  of  expansion,  or 

whole  work  done  during  expansion  =  a  P  /'  x  hyp  log  R',  ...  (  4  ) 

which  is  the  work  done  by  expansion,  in  foot-pounds.  Add  together  these 
two  quantities  of  work,  (  3  )  and  (  4  ),  and  reduce;  then,  for  the  total  work, 
w,  done  by  the  steam  in  one  stroke  of  the  piston, 

a/  =  0P[/'(i+hyplogR')-f]  ......................  (  5  ) 

The  work  of  back  pressure  for  one  stroke  is 

w'  =  ap'*L;   ...............................   (  6  ) 

and  the  net  work,  such  as  may  be  measured  by  an  indicator-diagram,  is 
w  —  w'\  or, 

W  =  *[P(/'(i+hyplogR')-')-/L]  ............   (   7  ) 


INITIAL   PRESSURE  IN   THE  CYLINDER.  829 

RULE  3.  To  find  the  net  work  done  by  steam  in  the  cylinder  for  one  stroke  of 
the  piston,  with  a  given  cut-off.  —  i.  To  the  hyperbolic  logarithm  of  the  actual 
ratio  of  expansion,  allowing  for  clearance,  add  i;  multiply  the  sum  by  the 
period  of  admission,  plus  the  clearance,  in  feet;  from  the  product  subtract 
the  clearance,  and  multiply  the  remainder  by  the  total  initial  pressure  in 
Ibs.  per  square  inch.  The  product  is  the  total  work  done  in  foot-pounds 
per  square  inch  on  the  piston.  2.  Multiply  the  average  back  pressure  in 
Ibs.  per  square  inch  by  the  length  of  the  stroke  ;  the  product  is  the  negative 
work  of  back  pressure  in  foot-pounds  per  square  inch.  3.  Subtract  the 
second  product  from  the  first  product;  the  remainder  is  the  net  work  in 
foot-pounds  per  square  inch  on  the  piston.  4.  Multiply  the  area  of  the 
piston  by  the  net  work  per  square  inch;  the  product  is  the  net  work  in 
foot-pounds  done  in  the  cylinder  for  one  stroke. 

Note.  —  When  the  period  of  admission  and  the  clearance  are  expressed  as 
percentages  of  the  stroke,  the  percentages  are  to  be  converted  into  feet  of 
the  stroke.  The  actual  ratio  of  expansion  is  found  by  dividing  TOO  plus 
the  percentage  of  clearance,  by  the  sum  of  the  percentages  of  admission 
and  clearance. 

To  exemplify  the  application  of  the  rule,  take  a  non-condensing  steam- 
cylinder  3  feet  in  diameter  with  a  stroke  of  5  feet,  and  initial  steam  of  a 
total  pressure  of  70  Ibs.  per  square  inch  on  the  piston,  cut  off  at  one-fourth 
of  the  stroke,  and  expanded  during  the  remaining  three-fourths.  The  aver- 
age back  pressure  is  17  Ibs.  per  square  inch,  and  the  clearance  is  5  per 
cent,  of  the  stroke.  What  is  the  whole  work  done  in  one  stroke?  The 
steam  is  cut  off  at  15  inches,  to  which  the  clearance,  which  is  5  per  cent,  of 
the  stroke,  or  3  inches,  is  to  be  added.  The  sum  is  18  inches,  or  1.5  feet, 

and  the  actual  ratio  of  expansion  is  5  —  L_3  =  3.5,  of  which  the  hyperbolic 

logarithm  is  1.204;  to  this  add  i,  making  2.204,  to  be  multiplied  by  1.5, 
making  3.306.  From  this  product  subtract  the  clearance  .25  feet,  leaving 
3.056.  Then  3.056  x  70  Ibs.  =  213.92  foot-pounds  of  total  work  per  square 
inch  of  piston;  and  213.92x1017.87  square  inches  area  of  piston  = 
217,750  foot-pounds,  the  total  work  done  in  one  stroke.  The  back  pressure 
1  7  Ibs.  per  square  inch  x  5  =  85  foot-pounds  per  square  inch  for  the  whole 
stroke;  and  85  x  1017.87  =  8653  foot-pounds,  the  negative  work  of  back- 
pressure. Finally  — 

foot-pounds. 

Total  work  done  on  the  piston,  for  one  stroke,  .......  2  1  7,750 

Negative  work  of  back  pressure,  for  one  stroke,  ......      8,653 

Difference,  or  net  work  for  one  stroke,  ........  209,097 

INITIAL  PRESSURE  IN  THE  CYLINDER. 

Inverting  formula  (  7  ),  the  required  initial  pressure  for  a  given  net  quan- 
tity of  work  in  one  stroke,  is  as  follows  :  — 


P- 


fl[/;(H.hyplogR')-']  ' 


fThe  initial  pressure  required   to  produce  a  given  average  total  pressure 
per  square  inch  for  a  given  actual  ratio  of  expansion,  is  found  by  sub- 


830  STEAM   ENGINE—  SINGLE-CYLINDER. 

stituting,  for  W,  its  equivalent  a  L  (p-p')>  in  formula  (  8  );  and  reducing. 
Then 

p-  ^L  _  .  ..(9) 

/'(i+hyplogR')-' 

AVERAGE  TOTAL  PRESSURE  IN  THE  CYLINDER. 

The  average  total  pressure,  /,  in  the  cylinder,  in  terms  of  the  initial  pres- 
sure^ for  a  given  actual  ratio  of  expansion,  is  found  by  dividing  the  second 
member  of  the  equation  (  5  ),  by  the  area  of  the  piston  and  by  the  length 
of  the  stroke;  or  by  a  simple  inversion  of  equation  (  9  )  :  — 


The  average  total  pressure,  /,  in  terms  of  the  total  work  done  for  one 
stroke,  is  also, 


AVERAGE  EFFECTIVE  PRESSURE  IN  THE  CYLINDER. 

The  average  effective  pressure  is  found  by  subtracting  the  average  back 
pressure  from  either  of  the  above  values  of  p,  formula  (  10  )  or  (  1  1  ),  or  it 
is  found  by  dividing  the  second  member  of  equation  (  7  )  by  the  area  of 
the  piston  and  by  the  length  of  the  stroke  :  giving,  by  reduction, 

(p  -/)  =  P  [/'(i+  hyp  log  R')-j  _f  _  (I2} 


THE  PERIOD  OF  ADMISSION  AND  THE  ACTUAL  RATIO  OF  EXPANSION. 

The  actual  rate  of  expansion  required  for  the  production  of  a  given 
average  total  pressure  from  a  given  initial  total  pressure  may  be  found  ten- 
tatively by  inverting  the  formula  (  10  ),  for  initial  pressure,  and  reducing, 
by  which  the  following  formula  is  obtained  :  — 


gR'^-f-^-  -i  .......................  (13) 

Here,  there  are  two  unknown  quantities,  namely,  hyp  log  R'  and  /'. 

RULE.  —  Multiply  the  length  of  stroke  by  the  mean  pressure,  and  divide  by 
the  initial  pressure;  and  to  the  quotient  add  the  clearance,  making  a  sum  A. 
Assume  a  period  of  admission,  and  add  to  it  the  clearance,  to  make  a  value 
for  the  divisor  /',  and  find  the  corresponding  value  for  hyp  log  R',  the  hyper- 
bolic logarithm  of  a  ratio  of  expansion.  Find  the  ratio  in  a  table  of  hyper- 
bolic logarithms,  and  by  it  divide  the  sum  of  the  stroke  and  the  clearance. 
If  the  quotient  be  equal  to  the  assumed  period  of  admission  plus  the  clear- 
ance, it  follows  that  the  assumed  period  is  the  required  period  of  admission, 
and  the  ratio  of  expansion  is  the  required  actual  ratio.  But  if  the  quotient 
be  greater  than  the  sum  of  the  assumed  period  and  the  clearance,  then  the 
.  assumed  period  of  admission  is  too  long.  If  the  quotient,  on  the  contrary, 


ACTUAL   RATIO  OF   EXPANSION.  831 

be  less,  the  assumed  period  is  too  short.  Try  again,  and  assume  a  shorter 
or  a  longer  period  of  admission,  as  the  case  may  require,  until  the  required 
period  of  admission  and  ratio  of  expansion  have  been  arrived  at. 

This  is  a  long  rule,  but  the  operation  of  it  is  less  tedious  than  may  be 
imagined.  For  example,  reverting  to  previous  data,  take  the  stroke  =  5  feet; 
clearance  .25  feet,  total  initial  pressure  =  70  Ibs.,  and  average  total  pressure  = 
42.78  Ibs.  per  square  inch;  to  find  the  required  period  of  admission.  Then 

42.78  x  q 

— +•  25  =  3. 306 (Sum  A) 

70 

Assume  a  period  of  admission,  1.75  feet;  then 

i.75+-25  =  2.oo  (Sum  B) 

And,  3.306-^2  =  1.653,  from  which  deduct  i;  the  remainder  .653  is  the 
hyperbolic  logarithm  of  the  ratio  of  expansion,  1.92.  Now,  the  stroke  plus 
the  clearance  is  5.25,  and 

=  2.73  feet,  as  a  period  of  admission  plus  clearance; 


and  2. 73-. 25  =  2. 48  feet.  But  this  is  greater  than  the  assumed  period 
namely,  1.75  feet.  Try,  therefore,  a  smaller  period  to  begin  with,  say  i  foot 
then 

i+. 25  =  1. 25   (Sum  B) 

3.306^1.25  =  2.61;  and  2.61-1  =  1.61,  which  is  the  hyperbolic  logarithm 
of  the  ratio  5 ;  then 

^-^  =  1.05  feet;  and  1.05 -.25  =  . 80  foot. 

But  .80  foot  is  less  than  the  assumed  period,  namely,  i  foot;  and  i  foot 
is  too  short.  The  required  period  must  be  less  than  1.75,  and  more  than 
i  foot;  and  nearer  to  i  foot  than  to  1.75  feet.  Try  1.25  feet,  then 

1.25 +  .25  =  1.50  (Sum  B) 

3.306^-1.50=2.2040;  and  2.2040-1  =  1.2040,  which  is  the  hyperbolic 
logarithm  of  the  ratio  3.5;  then 

^^  =  1.5  feet;  and  1.5 -.25  =  1.25  feet, 

which  is  equal  to  the  period  last  assumed.  The  required  period  of  admis- 
sion is,  therefore,  1.25  feet;  and  the  ratio  of  expansion  is  3.5. 

Note. — Calculation  for  this  rule  may  be  shortened  by  using  the  following 
table  (No.  295),  page  836,  particularly  when  the  clearance  is  7  per  cent, 
of  the  stroke,  as  is  assumed  in  the  composition  of  that  table.  When 
the  clearance  deviates  by  i  or  2  per  cent,  from  the  standard  of  the  table, 
suitable  allowances  may  be  made  on  the  results  drawn  from  the  table,  by 
which  near  approximations  may  be  made.  Take  the  last  example,  in 
which  the  clearance  is  5  per  cent,  of  the  stroke.  Reduce  the  given  mean 
pressure  to  the  expression  .611,  which  is  its  relative  value  when  the  initial 


832  STEAM   ENGINE  —  SINGLE-CYLINDER. 

pressure  is  taken  as  i,  thus  42.78-^-7o  =  .6n.  Looking  down  the  fourth 
column  of  the  table,  the  nearest  values  are  .619  and  .608,  corresponding 
to  the  ratios  of  expansion  3.5  and  3.6,  the  exact  ratio  being  3.5.  The 
corresponding  periods  of  admission  in  column  3  are  23.6  and  22.7  per 
cent,  of  the  stroke,  and  adding  to  these  2  per  cent.,  to  compensate  for 
the  difference  of  clearance  —  5  per  cent,  in  the  example,  as  against  7  per 
cent,  in  the  table  —  the  sums  average  about  25  per  cent.,  which  is  the  correct 
admission. 

RULE  4.  To  find  the  Period  of  Admission  required  for  a  given  Actual 
Ratio  of  Expansion.  —  Divide  the  length  of  stroke  plus  the  clearance  by  the 
actual  ratio  of  expansion;  and  deduct  the  clearance  from  the  quotient. 
The  remainder  is  the  period  of  admission. 

2.  When  the  Quantities  are  given  as  Percentages  of  the  Stroke.  —  Add  the 
percentage  of  clearance  to  100,  and  divide  the  sum  by  the  actual  ratio  of 
expansion;  and  deduct  the  percentage  of  clearance  from  the  quotient.  The 
remainder  is  the  period  of  admission  as  a  percentage  of  the  stroke. 

The  Period  of  Admission  required  for  a  given  Actual  Ratio  of  Expansion  is 


RULE  5.  The  Pressure  of  Steam  expanded  in  the  Cylinder,  at  the  end  of  the 
Stroke,  or  at  any  other  point  of  the  Expansion,  is  found  by  dividing  the  initial 
pressure  by  the  ratio  of  actual  expansion  calculated  to  the  given  point 
of  the  stroke.  The  quotient  is  the  pressure  at  that  point. 

Or,  multiply  the  initial  pressure  by  the  period  of  admission  plus  the  clear- 
ance, and  divide  the  product  by  the  length  of  the  part  of  the  stroke  described 
up  to  the  given  point,  plus  the  clearance.  The  quotient  is  the  pressure  at 
that  point. 

THE  RELATIVE  PERFORMANCE  OF  EQUAL  WEIGHTS  OF  STEAM 
WORKED  EXPANSIVELY. 

The  steam  may  be  said  to  be  measured  off  for  each  stroke  of  the  piston, 
a  cylinder-full  at  a  time,  of  expanded  steam;  whilst  the  final  pressure  is  a 
measure  of  the  density,  and  therefore  of  the  weight,  of  this  steam.  The 
mean  pressures,  again,  are  measures  of  the  total  performance  of  the  same 
body  of  steam.  It  follows,  that  the  relative  total  performance  is  directly  as 
the  mean  pressure,  and  inversely  as  the  weight  of  steam  condensed  or  as 
the  final  pressure,  and  that,  if  the  former  be  divided  by  the  latter,  the 
quotients  will  show  the  relative  total  performance  of  a  given  weight  of  the 
steam,  as  admitted  and  cut  off  at  different  points,  and  expanded  to  the  end 
of  the  stroke,  with  a  clearance  of  7  per  cent,  of  the  stroke,  as  follows  :  — 

When  the  steam  is  cut  off  at 

i,          #,        fa        Y^        Vs.      fa       Vio,       VisOfstroke> 
the  relative  total  performance  per  unit  of  steam  is  directly  as  the  average 
pressures, 

i.ooo,     .969,     .860,     .637,     .567,     .457,     .413,     .348,...  (A) 
and  is  inversely  as  the  final  pressures, 

i.ooo,     .769,     .532,     .298,     .250,     .182,     .159,     .128. 


WORK   DONE   BY   ADMISSION   AND    EXPANSION.  833 

The  relative,  or  proportional,  total  performance  of  given  equal  weights  of 
steam  are  therefore  in  the  ratio  of  the  second  last  row  of  figures  divided  by 
the  last  row  of  figures;  the  total  performance  for  steam  admitted  for  the 
whole  stroke,  without  any  expansion,  being  taken  as  i.  Thus, 

_i         .969       .860       .637       .567       .457       -413       -348 
i'        -769'      .532'      .298'      .250'      .182'      .159'      .128 

or  the  quotients, 

i.oo,      1.26,      1.62,      2.13,      2.27,      2.51,      2.60,      2.72....  (B) 

These  quotients  may  be  found,  otherwise,  from  the  actual  ratios  of  expansion, 
which  are  inversely  as  the  final  pressures,  by  multiplying  the  average  pressures 
by  the  respective  ratios.  For  example,  when  the  steam  is  cut  off  at  ^,  the 
actual  ratio  of  expansion  is  1.3,  and  the  mean  pressure  .969  x  1.3  =  1.26, 
which  is  the  relative  efficiency,  as  already  found  above. 

It  is  seen  that  the  total  work  or  performance  of  a  given  weight  of  steam 
is  fully  doubled  by  cutting  off  and  expanding  at  a  fourth  of  the  stroke,  as 
compared  with  the  admission  of  steam  for  the  whole  of  the  stroke. 

In  these  comparisons  of  the  relative  performance  of  steam  worked  expan^ 
sively,  the  opposition  of  back  pressure  has,  for  simplicity,  been  omitted 
from  the  calculations.  Taking  the  back  pressure  as  constant  with  all  ratios 
of  expansion,  it  would  constitute  a  uniform  quantity  to  be  deducted  from 
each  of  the  total  mean  pressures,  of  which  the  ratios  are  given  in  line  A  ; 
and  as  the  remainders  would  thus  decrease  more  rapidly  than  the  total 
pressures,  it  would  follow  that  the  quotients,  line  B,  would  increase  less 
rapidly  than  as  they  are  there  shown  to  increase. 

PROPORTIONAL  WORK  DONE  BY  ADMISSION  AND  BY  EXPANSION. 

To  ascertain  in  what  proportions  the  whole  work  for  the  stroke  is  done 
by  admission  and  by  expansion,  leaving  unconsidered  the  back  pressure  : 
the  work  by  admission  is  in  proportion  to  the  period  of  admission,  and  if 
this  be  subtracted  from  the  proportional  mean  pressure,  the  remainder  is 
the  proportional  work  by  expansion.  Thus,  when  the  steam  is  cut  off  at 


xAo, 


these  fractions  are  the  periods  of  admission,  and  are  proportional  to  the 
work  by  admission,  and  are  decimally  as  follows  :  — 

i.ooo,     .750,     .500,     .333,     .250,     .200,     .125,     .100,     .066, 

which  being  subtracted  from  the  relative  total  average  pressures,  the  re- 
mainders are  the  relative  work  by  expansion  :  — 

.000,     .219,     .360,     .393,     .387,     .367,     .332,     .313,     .282; 
the  sum  of  the  last  two  rows,  or  the  total  average  pressures,  being  as 
i.ooo,     .969,     .860,     .726,     .637,     .567,     .457,     .413,     .348; 

which  are  the  same  as  the  values  in  line  A,  page  832. 

Here  it  appears  that  the  quantity  of  work  done  by  expansion,  arrives  at 
a  maximum  when  the  period  of  admission  is  about  one-third  of  the  stroke. 

53 


^34 


STEAM   ENGINE— SINGLE-CYLINDER. 


With  a  greater  or  a  less  admission  it  is  reduced.  But  the  proportion  of 
work  by  expansion,  relative  to  the  work  by  admission,  increases  regularly  as 
the  admission  is  reduced.  Thus,  taking  the  work  for  the  periods  of  admis- 
sion successively,  as 

i,         i,          i,  i,  i,  i,  i,  i,  i, 

the  corresponding  proportions  of  work  done  by  expansion,  are  successively 
as  o,  .29,  .72,  1.31,  1.55,  1.83,  2.66,  3.13  4.27. 

The  loss  by  clearance-space  neutralizes  a  considerable  proportion  of  the 
gain  by  expansion,  as  appears  from  the  following  examples. 


THE  INFLUENCE  OF  CLEARANCE  IN  REDUCING  THE  PERFORMANCE  OF 
STEAM  IN  THE  CYLINDER. 

To  note  the  effect  of  clearance  in  reducing  the  efficiency  of  steam  in  the 
cylinder,  let  the  steam  be  admitted  for  one-fourth  of  the  stroke,  and  let 
cdgnm  be  the  indicator-diagram  described,  with  a  perfect  vacuum,  of  which 

the  base  m  n  is  the  length  of  the  stroke 
=  100,  and  the  extension  of  the  base, 
m  m',  is  the  length  of  the  clearance  =  7. 
The  average  pressure,  //,  is,  by  the  for- 
mula (  10  ),  .637,  when  the  initial  pres- 
sure is  i.  The  loss  of  pressure  by 
clearance,  is  represented  by  the  initial 
area  mm'c'c,  the  pressure  being  =  i,  and 
the  volume  =  7  per  cent,  of  that  of  the 
stroke.  Averaged  for  the  whole  stroke, 
that  is,  multiplying  i  by  7  and  dividing 
by  100,  the  average  loss  of  pressure  for 

the  whole  stroke  is  i  x  _?—  —  .070;  and 


100 


Fig. 


-Dmgmmht^how  influence  of  Clear-  jf  thjs  average  loss  be  added  to  the  aver- 

age pressure,  the  sum,  .637  +  .070  =  .707, 

expresses  the  relative  efficiency  with  which  a  given  weight  of  steam  would 
be  worked  if  there  were  no  loss  by  clearance.  It  shows  that  there  would 
be  a  gain  of  1  1  per  cent.  This  greater  relative  efficiency  is  represented  on 
the  diagram  by  the  upper  line/'/'. 

The  relative  efficiency  may  be  otherwise  found  by  means  of  formula 
(  10),  for  the  average  total  pressure,  the  item  of  clearance  being  eliminated 
from  it.  Suppose  the  clearance  in  the  diagram,  Fig.  333,  to  be  included  as 
part  of  the  stroke,  then  the  period  of  admission  becomes  32  per  cent,  and 
the  length  of  stroke  107  per  cent.;  and,  when  the  initial  pressure  is  i, 


32  (i+hyp  log  ^  or  3.35) 
107 


107 


=  .66i, 


the  average  pressure,  as  against  .637  the  average  pressure  with  clearance. 
But,  as  the  strokes  are  different,  the  average  pressures  are  to  be  multiplied 


INFLUENCE  OF  CLEARANCE.  835 

by  their  respective  strokes,  to  give  the  proportion  of  the  efficiencies;  thus, 

.637  x  100  =  63.7  relative  efficiency,  for  25  %  admission,  with  7  %  clearance; 
.661x107  =  70.7  do.  32        do.  without  clearance; 

being  in  the  same  ratio  to  each  other,  as  the  values  .637  and  .707  already 
found. 

The  comparison  is  extended  for  other  periods  of  admission  by  simply 
adding  the  average  loss  .070,  to  the  corresponding  average  pressures  in  the 
4th  column  of  the  table,  No.  295.  Thus, 

When  the  steam  is  cut  off  at 

full  stroke,  #,         ^,          */3  >        #i         Y*>          'Ao,          Vis  of  stroke, 

the  average  pressures  representing  the  relative  work,  when  the   pressure 
during  admission  =  i,  are 

i.  ooo,     .969,    .860,        .726,     .637,       .457,       .413,        .348, 

and,  adding  the  loss  by  7  per  cent,  of  clearance,  .070,  the  increased  relative 
work  done  by  a  given  weight  of  steam,  if  there  were  no  clearance,  would  be 


1.070,  1.039,    -93°>        -796,     .707,       -527>       -483, 
showing  that  the  gain  would  be 

7,        7.2,        8.1,          9.6,     ii.o,       15.3,  17,          20  percent, 

which  is  lost  by  clearance. 

Table  No.  295.  —  RATIOS  OF  EXPANSION  OF  STEAM,  WITH  RELATIVE 
PERIODS  OF  ADMISSION,  PRESSURES,  AND  TOTAL  PERFORMANCE. 

To  facilitate  calculations  about  steam  expanded  in  cylinders,  the  table 
No.  295  has  been  composed.  The  actual  ratios  of  expansion,  column  i, 
range  from  i.o  to  8.0,  for  which  the  hyperbolic  logarithms  are  given,  for 
ready  reference,  in  column  2.  The  3d  column  contains  the  periods  of 
admission  relative  to  the  actual  ratios  of  expansion,  as  percentages  of  the 
stroke,  calculated  by  Rule  4.  The  4th  column  gives  the  values  of  the 
mean  pressures  relative  to  the  initial  pressures,  the  latter  being  taken  as  i, 
calculated  by  formula  (  10  ).  The  5th  column  gives  the  values  of  the  initial 
pressures  relative  to  the  mean  pressures,  when  the  latter  are  taken  as  i. 
These  values  are  the  reciprocals  of  those  of  the  4th  column;  at  the  same 
time  they  may  be  calculated  by  formula  (  9  ).  In  the  calculation  of  these 
last  three  columns,  3,  4,  and  5,  clearance  is  taken  into  account,  and  its 
amount  is  assumed  at  7  per  cent,  of  the  stroke.  In  the  6th  column,  of 
final  pressures,  they  are  such  as  would  be  arrived  at  by  the  continued 
expansion  of  the  whole  of  the  steam  to  the  end  of  the  stroke,  the  initial 
pressure  being  equal  to  i.  They  are  the  reciprocals  of  the  ratios  of 
expansion,  column  i,  as  indicated  by  Rule  5. 

The  7th  column  contains  the  relative  total  performance  of  equal  weights 
of  steam  worked  with  the  various  actual  ratios  of  expansion:  the  total 
performance  when  steam  is  admitted  for  the  whole  of  the  stroke  without 
expansion  being  equal  to  i.  They  are  calculated  on  the  principle  exem- 
plified at  page  832. 


836 


STEAM   ENGINE — SINGLE-CYLINDER. 


Table  No.  295. — EXPANSIVE  WORKING  OF  STEAM: — ACTUAL  RATIOS  OF 
EXPANSION;  WITH  THE  RELATIVE  PERIODS  OF  ADMISSION,  PRES- 
SURES. AND  PERFORMANCE. 

Clearance  at  each  end  of  the  Cylinder,  7  per  cent,  of  the  stroke. 
(SINGLE  CYLINDER.) 


I 
ACTUAL 
RATIO  OF 
EXPANSION  ; 
Or  Number 
of  Volumes 
to  which  the 
Initial  Volume 
is  Expanded. 

2 

HYPERBOLIC 
LOGARITHM 
of  Actual 
Ratio  of 
Expansion. 

3 

CORRESPONDING 
PERIOD  OF 
ADMISSION,  or 
CUT-OFF. 
Clearance,  7  per 
cent,  of  the  Stroke. 

4 

AVERAGE 
TOTAL 
PRESSURE. 

5 

TOTAL 
INITIAL 
PRESSURE. 

6 

TOTAL 
FINAL 
PRESSURE. 

7 
RATIO  OF 
TOTAL  PER- 
FORMANCE 
of  Equal 
Weights  of 
Steam.  (Col. 
4-=-Col.  6.) 

initial  volume 

stroke  =  100. 

initial  pres- 

mean  pres- 

initial pres- 

with  100  % 
of  admission 

SSI. 

sure=i. 

sure;^. 

sure=i. 

=  1.000. 

1.0 

.0000 

100 

1.  000 

.000 

1.  000 

1.  000 

1.05 

.0488 

95.0 

•9997 

.003 

.952 

1.050 

.1 

•0953 

90.3  or  9/10 

.996 

.004 

.909 

1.096 

.15 

.1398 

86.0 

.990 

.010 

.870 

1.138 

.18 

.1698 

83.3  or  s/6 

.986 

.014 

.847 

1.164 

.2 

.1823 

82.1 

•983 

.017 

.833 

1.180 

•23 

.2070 

80.0  or  4/5 

.980 

.020 

.813 

1.206 

.25 

.2231 

78.6 

•977 

.024 

.800 

1.  221 

•3 

.2624 

75-3  or  # 

.969 

.032 

.769 

I.26l 

•35 

.3000 

72-3 

.961 

.041 

.741 

1.297 

•39 

•3293 

70.0  or  7/xo 

•953 

.049 

.719 

I-325 

•4 

•3365 

69.4 

.951 

.052 

.714 

1-332 

45 

.3716 

66.8  or  2/3 

•942 

.062 

.690 

1.365 

•5 

•4055 

64-3 

•932 

•073 

.666 

1-399 

•54 

•4317 

62.5  or  % 

•925 

.O8l 

.649 

1.425 

•55 

.4382 

62.0 

.922 

.085 

.645 

1.429 

.6 

.4700 

59.9  or  3/s 

•913 

.095 

.625 

1.461 

.65 

.5008 

57-9 

.107 

.606 

1.490 

•7 

.5306 

56.0 

.894 

.II9 

.588 

1.520 

•75 

•559* 

54.1 

•883 

.132 

•571 

1.546 

.8 

.5878 

52.4 

•873 

.145 

•555 

1-573 

.85 

•6i53 

50.8 

.864 

•157 

.541 

1-597 

.88 

.6314 

50.0  or  yz 

.860 

.163 

•532 

1.616 

•9 

.6419 

49-3 

.854 

.171 

.526 

1.624 

•95 

.6678 

47-9 

.846 

.182 

•5J3 

1.649 

2.0 

.6931 

46.5 

.836 

.196 

.500 

1.672 

2.1 

.7419 

44.0 

.818 

.222 

.476 

1.718 

2.2 

•7885 

41.6 

•799 

.251 

•455 

1.756 

2.28 

.8241 

40.0  or  2/5 

.787 

.271 

•439 

1-793 

2-3 

.8329 

39-5 

.782 

.279 

•435 

1.798 

2.4 

•8755 

37.6  or  y* 

.766 

•305 

.417 

1-837 

2.5 

.9163 

35-8 

.750 

•333 

.400 

1.875 

2.6 

•9555 

34-2 

.736 

•359 

.385 

1.912 

2.65 

•9745 

33-3  or  i/3 

.726 

•377 

•377 

1.925 

2.7 

•9933 

32.6 

.719 

•391 

•370 

1-943 

2.8 

1.030 

31.2 

.706 

.416 

•357 

1.978 

2.9 

1.065 

29.9  or  3/10 

.692 

•445 

•345 

2.006 

3-o 

1.099 

28.7 

.679 

•473 

•333 

2.039 

3-i 

1.131 

27.5 

.665 

.504 

•323 

2.059 

3-2 

1.163 

26.4 

.652 

•534 

•313 

2.083 

3-3 

1.194 

25.4 

.641 

.560 

•303 

2.115 

EXPANSIVE   WORKING  AND   PERFORMANCE  OF   STEAM.        837 


Table  No.  295  (continued}. 


I 
ACTUAL 
RATIO  OF 
EXPANSION  ; 
Or  Number 
of  Volumes 
to  which  the 
Initial  Volume 
is  expanded. 

2 

IYPERBOLIC 
LOGARITHM 
of  Actual 
Ratio  of 
Expansion. 

3 

CORRESPONDING 
PERIOD  OF 
ADMISSION,  or 
CUT-OFF. 
Clearance,  7  per 
cent,  of  the  stroke. 

4 

AVERAGE 
TOTAL 
PRESSURE. 

5 

TOTAL 
INITIAL 
PRESSURE. 

6 

TOTAL 
FINAL 
PRESSURE. 

7 
RATIO  OF 
TOTAL  PER- 
FORMANCE 
of  equal 
Weights  of 
Steam.  (CoL 
4-5-CoL  6.) 

initial  volume 

stroke  =  100. 

initial  pres- 

mean  pres- 

initial pres- 

with  100  % 
of  admission 

SKI. 

sure=i. 

sures^. 

sure=i. 

=  1.000. 

3-35 

1.209 

25.0  or  ]i 

.637 

1.570 

.298 

2.129 

3-4 

1.224 

24-5 

.631 

1.585 

.294 

2.146 

3-5 

1.253 

23.6 

.619 

1.615 

.286 

2.104 

3-6 

I.28l 

22.7 

.608 

1.645 

.278 

2.187 

3-7 

1.308 

21.9 

•597 

1.675 

.270 

2.2II 

3-8 

1-335 

21.2 

.589 

1.698 

.263 

2.240 

3-9 

1.361 

20.4 

•579 

1.727 

.256 

2.202 

4.0 

1.386 

19.7  or  Vs 

.567 

1.764 

.250 

2.278 

4.1 

1.411 

19.1 

•559 

1.789 

.244 

2.291 

4.2 

1-435 

18.5 

•551 

I.8I5 

.238 

2.315 

4-3 

1.459 

17.9 

.542 

1.845 

•233 

2.326 

44 

1.482 

17-3 

•533 

1.876 

.227 

2.348 

4-5 

1.504 

1  6.8  or  i/6 

.526 

I.90I 

.222 

2.370 

4.6 

1.526 

16.3 

.518 

1.930 

.217 

2.387 

4-7 

1.548 

15.8 

.511 

1-957 

.213 

2.399 

4.8 

1.569 

15-3 

.503 

1.988 

.208 

2.4l8 

4.9 

1.589 

14.8 

494 

2.024 

.204 

2.422 

5.0 

1.609 

14.4  or  */7 

.488 

2.049 

.200 

2.440 

5.2 

1.649 

13.6 

.476 

2.IOI 

•193 

2.466 

5-4 

1.686 

12.8 

.462 

2.164 

.185 

2.497 

5-5 

1.705 

12.5  or  x/8 

•457 

2.188 

.182 

2.5II 

5.6 

1.723 

I2.I 

.450 

2.222 

.178 

2.528 

5.8 

1.758 

11.4 

.438 

2.283 

.172 

2-547 

5-9 

1-775 

1  1.  1  or  i/g 

432 

2.315 

.I69 

2.556 

6.0 

1.792 

10.8 

.427 

2.342 

.167 

2.567 

6.2 

1.825 

10.3 

.419 

2.387 

.161 

2.585 

6.3 

1.841 

10.0  or  i/zq 

413 

2.421 

.159 

2.597 

6.4 

1.856 

9-7 

.407 

2.457 

.156 

2.609 

6.6 

1.887 

9.2  or  i/n 

.398 

2.513 

.152 

2.6l9 

6.8 

1.917 

8.7 

.388 

2-577 

.147 

2.639 

7.0 

1.946 

8.3  or  1/12 

.381 

2.625 

•143 

2.664 

7.2 

1.974 

7-9 

•373 

2.681 

•139 

2.683 

7-3 

1.988 

7-7  or  1/13 

.369 

2.710 

•137 

2.693 

7-4 

2.001 

7-5 

•365 

2.740 

•135 

2.703 

7.6 

2.028 

7.1  or  i/I4 

•357 

2.801 

.132 

2.7II 

7.8 

2.054 

6.7  or  '/is 

.348 

2.874 

.128 

2.719 

8.0 

2.079 

6.4  or  1/16 

•342 

2.924 

.125 

2.736 

The  pressures  have  been  calculated  on  the  supposition  that  the  pressure 
of  steam,  during  its  admission  into  the  cylinder,  is  uniform  up  to  the  point 
of  cutting  off,  and  that  the  expansion  is  continued  regularly  to  the  end  of 
the  stroke.  In  practice,  of  course,  there  are  deviations  from  these  ideal 
conditions.  Wiredrawing  action  occasionally  causes  a  fall  of  pressure  during 
admission,  and  the  opening  of  the  exhaust  before  the  piston  arrives  at  the 
•end  of  the  stroke  causes  the  expansion-line  to  fall  away  towards  the  end. 


838  STEAM   ENGINE— SINGLE-CYLINDER. 

The  allowances  necessary  to  be  made  for  these  deviations,  as  well  as  for 
the  back  pressure  of  the  air  in  non-condensing  engines,  and  that  from  the 
condenser  in  condensing  engines,  and  for  compression  of  exhaust  steam 
towards  the  end  of  the  return  stroke,  will  be  considered  at  a  subsequent 
stage.  The  calculations  have  been  made  for  periods  of  admission  ranging 
from  100  per  cent,  or  the  whole  of  the  stroke,  to  6.4  per  cent,  or  x/i6th  of 
the  stroke.  And  though,  nominally,  the  expansion  is  16  times  in  the  last 
instance,  it  is  actually  only  8  times,  as  given  in  the  first  column.  The  great 
difference  between  the  nominal  and  the  actual  ratios  of  expansion  is 
caused  by  the  clearance,  which  is  equal  to  7  per  cent,  of  the  stroke,  and 
causes  the  nominal  volume  of  steam  admitted,  namely,  6.4  per  cent.,  to  be 
augmented  to  6.4+7  =  13.4  per  cent,  of  the  stroke,  or  more  than  double, 
for  expansion.  When  the  steam  is  cut  off  at  I/g  th,  the  actual  expansion  is 
only  6  times ;  when  cut  off  at  J/5  th,  the  expansion  is  4  times ;  when  cut  off 
at  y§d,  the  expansion  is  2^3  times;  and  to  effect  an  actual  expansion  to 
twice  the  initial  volume,  the  steam  is  cut  off  at  46  *4  per  cent,  of  the  stroke, 
not  at  half  stroke. 

Though  a  uniform  clearance  of  7  per  cent,  at  each  end  of  the  stroke  has 
been  assumed  as  a  fair  average  proportion  for  the  purpose  of  compiling  the 
table,  the  clearance  of  cylinders  with  ordinary  slides  varies  considerably — 
say,  from  5  to  8  or  9  per  cent.  With  the  mean  clearance,  7  per  cent.,  that 
has  been  assumed,  the  table  gives  approximate  results  sufficient  for  most 
practical  purposes;  they  will  economize  calculation,  and  they  are  certainly 
more  trustworthy  than  such  as  can  be  deduced  by  calculations  based  on 
simple  tables  of  hyperbolic  logarithms,  where  clearance  is  neglected. 

It  has  already  been  exemplified  at  page  831,  how  the  table  may  serve  in 
making  approximate  calculations  when  the  clearance  is  other  than  7  per  cent. 

TOTAL  WORK  DONE  BY  ONE  POUND  OF  STEAM  EXPANDED  IN 
A  CYLINDER. 

If  i  Ib.  of  water  be  converted  into  steam  of  atmospheric  pressure — 
14.7  Ibs.  per  square  inch,  or  2116.8  Ibs.  per  square  foot — it  gradually 
occupies  a  volume  equal  to  26.36  cubic  feet;  and  the  work  done  in  acquir- 
ing this  volume  under  one  atmosphere  is  equal  to  2116.8  Ibs.  x  26.36  feet 
=  55,799  foot-pounds.  The  equivalent  quantity  of  heat  expended  is 
i  unit  per  772  foot-pounds,  or,  altogether,  55,799-^-772  =  72.3  units.  This 
is  precisely  the  work  of  i  Ib.  of  steam  of  one  atmosphere,  acting  on  a 
piston  without  expansion. 

The  gross  work  thus  done  on  a  piston  by  i  Ib.  of  steam,  generated  at 
total  pressures  varying  from  15  Ibs.  to  100  Ibs.  per  square  inch,  varies,  in 
round  numbers,  from  56,000  to  62,000  foot-pounds,  equivalent  to  from 
72  to  80  units  of  heat. 

The  simple  work  of  a  pound  of  steam,  without  expansion,  thus  exempli- 
fied, is  reduced  by  clearance  according  to  the  proportion  it  bears  to  the 
net  capacity  of  the  cylinder.  If  the  clearance  be  7  per  cent,  of  the  stroke, 
then  107  parts  of  steam  are  consumed  in  doing  the  work  of  a  stroke, 
which  is  represented  by  100  parts,  and  the  work  of  a  given  weight  of  steam 
without  expansion,  admitted  for  the  whole  of  the  stroke,  is  reduced  in  the 
ratio  of  107  to  100.  Having  determined,  by  this  ratio,  the  quantity  of  work 
by  i  Ib.  of  steam  without  expansion,  as  reduced  by  clearance,  the  work  for 
various  ratios  of  expansion  may  be  deduced  from  that,  in  terms  of  the 


WORK  DONE  BY  ONE   POUND   OF   STEAM.  839 

relative  performance  of  equal  weights  of  steam,  as  exemplified,  page  835, 
and  given  in  the  7th  column  of  table  No.  295. 

To  find  the  total  actual  work  of  i  Ib.  of  steam,  for  any  ratio  of  expan- 
sion, it  is  only  necessary  to  multiply  the  simple  work,  without  expansion,  as 
reduced  by  clearance,  by  the  ratio  or  relative  performance  just  referred  to. 
The  simple  work  of  a  pound  of  steam  does  not  greatly  vary  with  the  pres- 
sure; and,  for  present  purposes,  the  work  of  steam  of  a  total  pressure  of 
100  Ibs.  per  square  inch  will  be  calculated  and  tabulated.  This  pressure 
corresponds  to  a  net  pressure,  above  the  atmosphere,  of  85  Ibs.  per  square 
inch — a  convenient  average  standard  of  pressure.  The  volume  of  i  Ib.  of 
saturated  steam  of  100  Ibs.  per  square  inch  is  4.33  cubic  feet,  and  the 
pressure  per  square  foot  is  i44x  100=14,400  Ibs.;  then,  the  total  simple 
work — or  total  initial  work,  as  it  may  be  called — is, 

14,400x4.33  =  62,352  foot-pounds. 
This  amount  is  to  be  reduced  for  a  clearance  of,  say,  7  per  cent,  thus: — 

62,352  x  —  =  58,273  foot-pounds, 
107 

which  is  the  total  simple  work  of  i  Ib.  of  steam  of  100  Ibs.  total  pressure 
per  square  inch,  after  the  loss  by  clearance  is  deducted ;  and,  divided  by 
Joule's  equivalent,  772,  it  is  equal  to  75.5  units  of  heat.  Now,  the  total  or 
constituent  heat  of  i  Ib.  of  loo-lb.  steam,  reckoned  from  a  temperature  of 
212°  F.,  is  1001.4  units;  reckoned  from  102°  F.,  the  temperature  of  water 
from  the  condenser  under  a  pressure  of  i  Ib.  per  square  inch,  the  con- 
stituent heat  is  1111.4  units.  The  equivalent  of  the  net  simple  work, 
75.5  units,  is,  then,  7.5  per  cent,  of  the  total  heat  reckoned  from  2i2°F., 
or  6.7  per  cent,  if  reckoned  from  102°  F.  For  shorter  admissions,  with  com- 
plementary expansion,  the  work  is  increased  as  in  the  following  examples : — 

When  the  steam  is  cut  off  at 

i,         k,         *A,  %,          Vs,  */*,         Vio,         Vis  of  stroke, 

the  actual  ratios  of  expansion  are, 

i,        1.3,       1.88,        3.35,          4.0,          5.5,          6.3,          7.8  times; 
the  comparative  performances  of  i  Ib.  of  steam  are  as 

i,    1.261,    1.616,      2.129,      2.278,      2.511,      2.597,      2.719, 
and  the  total  actual  work  of  i  Ib.  of  xoo-lb.  steam  is  in  the  same  proportion, 
58,273,  73,513,  94,2oo,  124,066,  132,770,  146,325,  ISl,3?o,  158,414  foot-pounds. 
The  equivalents,  as  heat,  of  the  actual  work  done,  are 

75-5,      95-2,     122.0,       160.7,       171-9,       189.5,       196.1,       205.2  units, 
which  are,  in  parts  of  the  constituent  heat  reckoned  from  102°  F.,  equal  to 
6.7,        8.5,       1 1.0,         14.5,         15-5,         i7-o,         T7-6,         18.5  per  cent. 

From  these  examples,  it  appears  that  the  total  work  done  by  i  Ib.  of 
steam,  without  making  any  allowance  for  back  pressure  or  other  contin- 
gencies, varies  from  about  60,000  foot-pounds  when  applied  without  expan- 
sion, to  about  double  that,  or  about  120,000  foot-pounds,  when  expanded 
three  times,  cutting  off  at  about  27  per  cent,  of  the  stroke;  and  to  about 


840  STEAM   ENGINE— SINGLE-CYLINDER. 

150,000  foot-pounds,  or  2^  times  the  first  performance,  when  expanded 
about  six  times,  cutting  off  at  about  10  per  cent,  of  the  stroke. 

Also,  that,  of  the  heat  consumed  in  the  formation  of  steam,  not  7  per 
cent,  is  converted  into  total  work  when  there  is  no  expansive  action;  that 
substantially  with  an  expansion  of  six  times  there  is  only  17%  per  cent 
converted ;  and  that  even  with  an  expansion  of  eight  times,  when  the  steam 
is  cut  off  at  Vis**1?  IGSS  than  20  Per  cent.,  or  one-fifth  of  the  heat  consumed, 
is  converted  into  work.  The  remainder  of  the  heat  is  lost,  as  for  the  pur- 
pose of  the  steam-engine. 

CONSUMPTION  OF  STEAM  WORKED  EXPANSIVELY  PER  HORSE-POWER  OF 
TOTAL  WORK  PER  HOUR. 

The  measure  of  a  horse-power  is  the  performance  of  33,000  foot-pounds 
per  minute,  or  of  33,000  x  60  =  1,980,000  foot-pounds  per  hour.  This  work 
is  to  be  divided  by  the  work  of  i  Ib.  of  steam,  and  the  quotient  is  the 
weight  of  steam  or  water  required  per  horse-power  per  hour.  For  example, 
the  total  actual  work  done  in  the  cylinder  by  i  Ib.  of  zoo-lb.  steam,  without 
expansion,  and  with  7  per  cent,  of  clearance,  is  58,273  foot-pounds;  and 

1,9  0,000  _  _^  ^  Of  steam,  is  the  weight  of  steam  consumed  for  the  total 
58>273 

work  done  in  the  cylinder  per  horse-power  per  hour.  For  any  shorter  period 
of  admission,  with  expansion,  the  weight  of  steam  per  horse-power  is  less, 
as  the  total  work  by  i  Ib.  of  steam  is  more,  and  may  be  found  by  dividing 
1,980,000  foot-pounds  by  the  respective  total  work  done;  or  by  dividing 
34  Ibs.  by  the  ratio  of  performance,  column  7,  table  No.  295.  In  this  way 
it  is  found  that,  when  the  steam  is  cut  off  at 

i,         Y^         fa         fa         x/s»         fa        x/.o,        Vis  of  stroke, 
the  quantities  of  steam,  or  water  as  steam,  consumed  per  horse-power  of 
total  work  per  hour,  are 

34.0,      26.9,      21.0,       1 6.0,       14.9,       13.5,       13.1,       12. 5  Ibs. 

Further,  allowing  that  10  Ibs.  of  steam  are  generated  by  the  combustion  of 
i  Ib.  of  coal,  the  fuel  consumed  per  horse-power  of  total  work  per  hour  is, 

3.40,      2.69,      2.10,       i. 60,       1.49,       1.35,       1.31,       1.25  Ibs. 

TABLE  (No.  296)  OF  THE  TOTAL  WORK  DONE  BY  i  POUND  OF  STEAM 
OF  100  LBS.  TOTAL  PRESSURE  PER  SQUARE  INCH. 

The  table  No.  296,  which  follows,  is  compiled  on  the  basis  of  the  con- 
ditions above  laid  down,  which  are  repeated  under  the  heading  of  the 
table,  for  ready  reference.  The  ist,  2d,  and  3d  columns  are  repeated  from 
table  No.  295.  The  4th  column,  of  total  actual  work  done  by  i  Ib.  of 
steam  of  100  Ibs.  total  pressure,  is  calculated  by  multiplying  the  work 
without  expansion,  namely,  58,273  foot-pounds,  by  the  ratios  in  column  3, 
for  the  proportional  work  when  expanded.  The  5th  column  contains  the 
equivalent  of  heat  converted  into  work,  which  is  found  by  dividing  the 
work  in  foot-pounds  by  Joule's  equivalent,  772;  and  the  6th  and  7th 
columns  give  these  values  as  percentages  of  the  total  heat  of  steam  raised 
from  212°  and  102°  F.  respectively.  The  8th  column  contains  the  quantity 
of  steam  consumed  for  the  total  work  done  per  horse-power  per  hour. 


WORK   DONE   BY  ONE   POUND   OF   STEAM. 


841 


Table  No.  296. — TOTAL  WORK  DONE  BY  ONE  POUND  OF  STEAM  OF 

100    LBS.  TOTAL   PRESSURE   PER   SQUARE   INCH. 

ASSUMPTIONS. — That  the  initial  pressure  is  uniform;  that  the  expansion  is 
complete  to  the  end  of  the  stroke;  that  substantially  the  pressure  in 
expansion  varies  inversely  as  the  volume;  that  there  is  no  back  pressure; 
and  that  there  is  no  compression. 

Volume  of  I  Ib.  of  steam  of  100  Ibs.  pressure  per  square  inch,  or  14,400  Ibs. 
per  square  foot,  4.33  cubic  feet. 

Product  of  initial  pressure  and  volume, 62,352  foot-pounds. 

Constituent  heat  of  I  Ib.  of  this  steam — 

Reckoned  from  212°  F.,    1001.4  units. 

Reckoned  from  102°  F.,    1111.4  units. 

Clearance  at  each  end  of  the  cylinder,  7  per  cent,  of  the  stroke. 


ACTUAL 
RATIO 

OF 

EXPAN- 
SION. 

CORRESPONDING 
PERIOD  OF 
ADMISSION 
or  CUT-OFF,  in 
percentage  of 
Stroke. 

TOTAL  ACTUAL  WORK 
DONE  by  i  Ib.  of  loo-lb. 
Steam. 

EQUIVALENT  OF  HEAT 
converted  into  Work. 

Quantity  of 
Steam  con- 
sumed per 
Horse-power 
of  actual 
Work  done 
per  Hour. 

Ratio  of 

Work  done 
(col.  7,  table 
No.  295). 

Actual 
Work 
done. 

Heat 

con- 
verted. 

Percentage  of  Consti- 
tuent Heat  converted, 
as  calculated  from 
212°  F.  and  102°  F. 

(i) 

(2) 

(3) 

(4) 

(5) 

(6) 

(?) 

(8) 

initial 
vol.  =  i. 

per  cent. 

foot- 
pounds. 

units. 

%from 

2I2°;F. 

%from 

102°  F. 

Ibs. 

1.0 

100 

.000 

58,273 

75.5 

7-5 

6.7 

34-0 

1.05 

95 

.050 

6l,I93 

79-3 

7-9 

7-1 

32-4 

I.I 

90.3  or  9/IO 

.096 

63,850 

82.7 

8-3 

7-5 

31.0 

I.I5 

86.0 

.138 

66,310 

85.9 

8.6 

7.8 

29-9 

1.18 

83-3  or  s/6 

.164 

67,836 

87.9 

8.8 

7-9 

29.2 

1.2 

82.1 

.180 

68,766 

89.1 

8.9 

8.0 

28.8 

1.23 

80.0  or  4/5 

1.  206 

70,246 

91.0 

9.1 

8.2 

28.2 

1.25 

78.6 

1.  221 

71,151 

92.2 

9.2 

8-3 

27.8 

1-3 

75-3  or  # 

I.26l 

73,513 

95.2 

9-5 

8.5 

26.9 

i-35 

72.3 

1.297 

75,575 

97-9 

9-8 

8.8 

26.2 

1-39 

70.0  or  7/IO 

I-325 

77,242 

100.  1 

IO.O 

9.0 

25.6 

1.4 

69.4 

1.332 

77,6i6 

100.6 

10.  1 

9.1 

25.5 

1.45 

66.8  or  2/3 

1.365 

79,555 

102.9 

10.3 

9-3 

24.9 

i-5 

64-3 

1-399 

81,546 

105.6 

10.6 

9-5 

24-3 

1.54 

62.5  or  % 

1.425 

83,055 

107.6 

10.8 

9-7 

23.8 

i-55 

62.0 

1.429 

83,299 

107.9 

10.8 

9-7 

23-7 

1.6 

59.9  or  3/5 

1.461 

85,125 

110.3 

II.O 

9-9 

23-3 

1.65 

57-9 

1.490 

86,828 

112.5 

n-3 

10.2 

22.8 

i-7 

56.0 

1.520 

88,598 

114.8 

11.5 

10.4 

22.4 

1-75 

54.1 

1.546 

90,115 

116.7 

11.7 

10.5 

22.0 

1.8 

52.4 

1-573 

91,680 

118.7 

11.9 

10.7 

21.6 

1.85 

50.8 

1-597 

93,o65 

120.5 

12.0 

10.8 

21.3 

1.88 

50.0  or  yz 

1.616 

94,200 

122.0 

12.2 

II.O 

21.0 

1.9 

49-3 

1.624 

94,610 

122.5 

12.3 

ii.  i 

20.9 

1.95 

47-9 

1.649 

96,100 

124.5 

12.4 

1  1.2 

20.6 

2.0 

46.5 

1.672 

97,432 

126.2 

12.6 

1  1-3 

20.3 

2.1 

44.0 

1.718 

100,266 

129.7 

13.0 

ii  -7 

I9.8 

2.2 

41.6 

1.756 

102,366 

132.5 

13.2 

11.9 

194 

2.28 

40.0  or  2/5 

1-793 

104,466 

135-3 

13-5 

12.2 

19.0 

2-3 

39-5 

1.798 

104,855 

135-7 

13.6 

12.3 

18.9 

2.4 

37.6  or  3/s 

1.837 

107,050 

138.6 

13.9 

12.5 

18.5 

2.5 

35-8 

1.875 

109,266 

I4I.5 

I4.I 

12.7 

18.1 

2.6 

34-2 

1.912 

1  1  1,400 

144.3 

144 

13.0 

17.8 

842 


STEAM   ENGINE — SINGLE-CYLINDER. 


Table  No.  296  (continued}. 


ACTUAL 
RATIO 

OF 

EXPAN- 
SION. 

CORRESPONDING 
PERIOD  OF 
ADMISSION 
or  CUT-OFF,  in 
percentage  of 
Stroke. 

TOTAL  ACTUAL  WORK 
DONE  by  i  Ib.  of  loo-lb. 
Steam. 

EQUIVALENT  OF  HEAT 
converted  into  Work. 

Quantity  of 
Steam  con- 
sumed per 
Horse-power 
of  actual 
Work  done 
per  Hour. 

Ratio  of 
Work  done 
(col.  7,  table 
No.  295). 

Actual 
Work 
done. 

Heat 
con- 
verted. 

Percentage  of  Consti- 
tuent Heat  converted, 
as  calculated  from 
212°  F.  and  102°  F. 

(*) 

(2) 

(3) 

(4) 

(5} 

(6) 

(?) 

(8) 

initial 
vol.  =  i. 

per  cent. 

foot- 
pounds. 

units. 

%  from 

212°  F. 

%from 

102°  F. 

Ibs. 

2.65 

33-3  or  1/3 

1.925 

112,220 

1454 

14.5 

I3-1 

177 

2.7 

32.6 

1-943 

113,244 

146.7 

14.7 

13.2 

I7.6 

2.8 

30.2 

1.978 

115,244 

149.2 

14.9 

13.4 

17.2 

2.9 

29.9  or  3/IO 

2.006 

116,885 

I5I.4 

I5.I 

13.6 

16.9 

3-o 

28.7 

2.039 

118,820 

153-9 

154 

13-9 

I6.7 

3.1 

27.5 

2.059 

119,970 

155-4 

15-5 

13-9 

I6.5 

3-2 

26.4 

2.083 

121,386 

157.2 

15-7 

14.1 

I6.3 

3-3 

25.4 

2.115 

123,278 

159.6 

1  6.0 

14.4 

16.1 

3-35 

25.0  or  ]i 

2.129 

124,066 

160.7 

16.1 

14.5 

1  6.0 

3-4 

24.5 

2.146 

125,066 

162.0 

16.2 

14.6 

15.8 

3-5 

23-6 

2.164 

126,125 

163.4 

16.3 

14.7 

15-7 

3-6 
3-7 

22.7 
21.9 

2.187 

2.2II 

127,450 
128,860 

165.1 
166.9 

16.5 
16.7 

14.9 
15.0 

15-5 
15.4 

3-8 

21.2 

2.240 

130,533 

169.1 

16.9 

15.2 

15.2 

3-9 

20.4 

2.262 

131,800 

170.7 

17.1 

15.4 

15.0 

4.0 

19.7  or  1/5 

2.278 

132,770 

I7I.9 

17.2 

15-5 

14.9 

4.1 

l%1 

2.291 

133,500 

172.9 

17.3 

15.6 

14.8 

4.2 

18.5 

2.315 

134,900 

174.8 

17-5 

15.8 

14.7 

4.3 

17.9 

2.326 

135,555 

175.6 

17.6 

15.8 

14.6 

4.4 

17-3 

2.348 

136,825 

177.2 

17.7 

15.9 

14.5 

4-5 

1  6.8  or  Ve 

2.370 

138,130 

178.8 

17.9 

16.1 

14-34 

4.6 

16.3 

2.387 

139,100 

180.2 

1  8.0 

16.2 

14.23 

4-7 

15.8 

2-399 

139,800 

181.1 

18.1 

16.3 

14.16 

4-8 

15-3 

2.418 

140,920 

182.5 

1  8.2 

16.4 

14.05 

4-9 

14.8 

2.422 

141,210 

182.8 

18.3 

16.5 

14.03 

5.0 

14.4  or  1/7 

2.440 

I42,l8o 

184.2 

18.4 

16.6 

13.92 

5.2 

13-6 

2.466 

143,720 

186.2 

18.6 

.16.9 

I3-78 

5-4 

12.8 

2.497 

145,525 

188.5 

18.8 

16.9 

13.60 

5-5 

12.5  or  y8 

2.5II 

146,325 

189.5 

18.9 

17.0 

13-53 

5-6 

12.  1 

2.528 

147,320 

190.8 

19.1 

17.2 

13-44 

5.8 

1  1.4 

2.547 

148,390 

192.2 

19.2 

17.3 

13-34 

5-9 

1  1.  1  or  1/9 

2.556 

148,940 

192.9 

19-3 

17.4 

13.29 

6.0 

10.8 

2.567 

149,586 

193.7 

19.4 

17-5 

13-23 

6.2 

10.3 

2.585 

150,630 

19.5 

17.6 

I3-I4 

6-3 

i  o.o  or  VIQ 

2-597 

151,370 

196.1 

19.6 

17.6 

13.08 

6.4 

9-7 

2.609 

152,033 

196.9 

19.7 

17.7 

13.02 

6.6 

9.2  or  Vn 

2.619 

152,595 

197.7 

19.8 

17.8 

12.98 

6.8 

8-7 

2.629 

153,810 

199.2 

19.9 

17.9 

12.87 

7.0 

8.3  or  1/12 

2.664 

155,200 

201.  1 

20.1 

18.1 

12.75 

7-2 

7-9 

2.683 

156,330 

202.6 

20.3 

18.3 

12.66 

7-3 

7.7  or  i/,3 

2.693 

156,960 

203.3 

20.3 

18.3 

12.61 

7.4 

7-5 

2.703 

157,560 

204.1 

20-4 

18.4 

12.57 

7-6 

7-i  or  i/I4 

2.7II 

157,975 

204.6 

20.4 

18.4 

12.53 

7-8 

6.7  or  Vis 

2.719 

158,414 

205.2 

20.5 

18.5 

12.50 

8.0 

6.4  or  Vie 

2.736 

159,433 

2O6.5 

20.7 

1  8.6 

11.83 

CYLINDER-CAPACITY  AND  WORK  DONE. 


843 


APPENDIX  TO  TABLE  No.  296. 

TABLET  OF  MULTIPLIERS  for  the  total  Work  done  by  i  Ib.  of  Steam  of 
other  Pressures  than  100  Ibs.  per  Square  Inch,  to  be  applied  to  the 
total  actual  Work  as  given  in  the  table.  See  explanatory  notice  of  the 
table,  page  840. 


Total  Pressures  below  100  Ibs.  per 
Square  Inch. 

Total  Pressures  above  100  Ibs.  per 
Square  Inch. 

Ibs.  per  square  inch. 

multiplier. 

Ibs.  per  square  inch. 

multiplier. 

65 

•975 

IOO 

.000 

70 

.981 

no 

.009 

75 

.986 

120 

.Oil 

80 

.988 

130 

.015 

85 

.991 

140 

.022 

90 

95 

•995 
.998 

If? 

.025 
.031 

An  initial  total  pressure  of  100  Ibs.  per  square  inch  has  been  adopted  for 
the  table,  as  an  average  pressure  in  ordinary  good  practice,  and  the  contents 
of  the  table  are  good  as  approximate  values  for  other  pressures  considerably 
different  from  100  Ibs.,  more  or  less.  A  tablet  is,  however,  appended  to  the 
table  No.  296,  containing  multipliers  for  various  other  total  pressures,  which 
may  be  applied  to  the  total  actual  work  given  in  the  table  for  the  purpose 
of  determining  the  correct  total  quantities  of  work  for  steam  of  the  respec- 
tive pressures.  These  multipliers  are  arrived  at  by  multiplying  the  total 
pressure  of  any  other  given  steam  per  square  foot,  by  the  volume  in  cubic 
feet  of  i  Ib.  of  such  steam,  and  dividing  the  product  by  62,352,  which  is 
the  product  in  foot-pounds  for  steam  of  100  Ibs.  pressure.  The  quotient 
is  the  multiplier  for  the  given  pressure.  From  the  tablet  it  appears  that, 
between  the  extremes  of  65  Ibs.  and  160  Ibs.  per  square  inch,  the  deviation 
from  the  work  done  as  given  for  100  Ibs.  pressure  does  not  exceed  2^  and 
3  per  cent. 

NET  CYLINDER-CAPACITY  RELATIVE  TO  THE  STEAM  EXPENDED  AND 
WORK  DONE  IN  ONE  STROKE. 

The  quantity  of  cylinder-capacity  required  for  the  performance  of  a  given 
weight  of  steam  admitted  for  one  stroke  depends  on  the  volume  of  the 
steam,  and  on  the  ratio  of  expansion.  If  the  given  weight  admitted  be 
multiplied  by  the  volume  of  i  Ib.  of  the  steam  and  by  the  actual  ratio  of 
expansion,  the  product  is  the  gross  cylinder-capacity,  including  clearance. 
For  example,  if  i  Ib.  of  steam  of  100  Ibs.  pressure  per  square  inch  be 
admitted  for  the  whole  stroke,  without  expansion,  the  gross  capacity  is  the 
volume  of  i  Ib.  of  such  steam,  namely,  4.33  cubic  feet;  and  the  net 
capacity,  supposing  the  clearance  to  be  7  per  cent,  of  the  stroke,  is 


4-33 


IOO 

100  +  7 


=  4.047  cubic  feet. 


844  STEAM   ENGINE — SINGLE-CYLINDER. 

If,  again,  2  Ibs.  of  steam  of  100  Ibs.  pressure  be  admitted  and  expanded 
into  three  times  its  initial  volume,  the  gross  capacity  is, 

4.33  x  2  x  3  =  25.98  cubic  feet; 
and  the  net  capacity  is 

25.98  x  1^?  =  24.28  cubic  feet. 
107 

From  this  is  derived  following  rule  for  net  capacity : — 

RULE  6.  To  find  the  net  Capacity  of  Cylinder  for  a  given  Weight  of  Steam 
admitted  for  one  stroke,  and  a  given  actual  ratio  of  expansion. — Multiply  the 
volume  of  i  Ib.  of  the  steam  by  the  given  weight  in  pounds,  and  by  the 
actual  ratio  of  expansion.  Multiply  the  product  by  100,  and  divide  by 
100  plus  the  percentage  of  clearance.  The  quotient  is  the  net  capacity 
of  cylinder. 

Again,  the  quantity  of  cylinder-capacity  required  for  the  performance 
of  a  given  amount  of  total  actual  work,  in  one  stroke,  depends  on  the 
initial  pressure,  and  the  actual  ratio  of  expansion,  according  to  the  follow- 
ing rule : — 

RULE  7.  To  find  the  net  Capacity  of  Cylinder  for  the  performance  of  a  given 
amount  of  total  Actual  Work,  in  one  stroke,  with  a  given  initial  pressure,  and 
actual  ratio  of  expansion. — Divide  the  given  work  by  the  total  actual  work 
done  by  i  Ib.  of  steam  of  the  same  pressure,  and  with  the  same  actual  ratio 
of  expansion;  the  quotient  is  the  weight  of  steam  necessary  to  do  the  given 
work,  for  which  the  net  capacity  is  found  by  Rule  6,  preceding. 

Conversely,  the  weight  of  steam  admitted  per  cubic  foot  of  net  capacity, 
for  one  stroke,  is  the  reciprocal  of  the  cylinder-capacity  per  pound  of  steam, 
as  obtained  by  Rule  6. 

Likewise,  the  total  actual  work  done  per  cubic  foot  of  net  capacity,  for 
one  stroke,  is  the  reciprocal  of  the  cylinder-capacity  per  foot-pound  of  work 
done,  as  obtained  by  Rule  7. 

Finally,  the  total  actual  work  done  per  square  inch  of  piston,  per  foot  of 
the  stroke,  is  Vi44tn  Part  °f  tne  work  done  per  cubic  foot;  a  prism  i  inch 
square  and  i  foot  long  being  I/I44th  Part  °f  a  cubic  foot.  The  work,  in 
either  measure,  is  in  direct  proportion  to  the  mean  total  pressure  per 
square  inch. 

TABLE  OF  RELATIONS  OF  NET  CAPACITY  OF  CYLINDER  TO  STEAM 
ADMITTED  AND  WORK  DONE. 

The  table  No.  297  gives  the  net  capacity  of  cylinder  required  in  relation 
to  the  quantity  of  steam  of  100  Ibs.  total  pressure  per  square  inch  consumed, 
and  of  work  done,  in  one  stroke.  Columns  i,  2,  and  3,  are  the  ratios  of 
expansion,  periods  of  admission,  and  total  actual  work  done  by  i  Ib.  of 
steam  of  100  Ibs.  pressure,  repeated  from  columns  i,  2,  and  4,  in  the 
previous  table,  No.  296.  In  the  4th  column  are  the  net  capacities  of 
cylinder  required  for  each  pound  of  steam  admitted  for  one  stroke, 
found  by  Rule  6;  and  in  the  5th  column  are  the  net  capacities  of 
cylinder  for  100,000  foot-pounds  of  total  actual  work  done  in  one  stroke, 
found  by  Rule  7.  The  6th  column  contains  the  weights  of  steam  of 
100  Ibs.  pressure  admitted  to  the  cylinder  for  one  stroke,  per  cubic  foot  of 


CYLINDER-CAPACITY  AND   WORK   DONE.  845 

net  capacity.  These  values  are  simply  the  reciprocals  of  those  in  column  4. 
The  yth  column  contains  the  total  actual  work  done  by  steam  of  100  Ibs. 
initial  pressure,  for  one  stroke,  per  cubic  foot  of  net  capacity.  These 
values  are  the  products  of  the  reciprocals  of  those  in  column  5,  by  100,000, 
since  this  is  the  number  of  foot-pounds  for  which  the  values  in  column  5 
are  calculated.  The  8th  column  gives  the  total  actual  work  per  square 
inch  of  piston-area,  per  foot  of  stroke,  in  foot-pounds;  the  initial  pressure 
is  100  Ibs.,  and  the  following  pressures,  for  the  different  ratios  of  expansion, 
may  also  be  read  as  percentages  of  the  initial  pressure.  The  total  actual 
works  are  directly  proportional  to  the  mean  total  pressures  per  square  inch, 
as  given  in  column  4,  table  No.  295,  page  836,  where  the  initial  pressure 
is  taken  as  i;  and  the  former  have  been  found  by  multiplying  the  latter 
respectively  by  100. 

The  contents  of  the  table  No.  297  are  calculated  for  steam  of  100  Ibs. 
per  square  inch,  total  initial  pressure  ;  but  they  are  available,  with  the  aid 
of  a  set  of  multipliers,  for  other  pressures.  For  column  3,  the  total  actual 
work  done,  the  multipliers  have  already  been  given  in  the  tablet  appended 
to  the  table  at  page  848,  for  pressures  of  from  65  Ibs.  to  160  Ibs.  per 
square  inch.  For  column  4,  of  the  net  capacity  of  cylinder  per  pound  of 
steam  expended  in  one  stroke,  the  multipliers  are  simply  the  ratios  of  the 
volume  of  i  pound  of  zoo-lb.  steam  to  the  respective  volumes  of  i  pound  of 
steam  of  other  initial  pressures.  Thus,  for  steam  of  65  Ibs.  total  pressure,  of 
which  the  volume  of  j  pound  is  6.49  cubic  feet,  to  be  compared  with  4.33 
cubic  feet,  which  is  the  volume  of  a  pound  of  loo-lb.  steam,  the  multiplier  is 


and  the  net  capacity  of  cylinder  per  pound  of  65-00.   steam,  when  the 
steam  is  admitted  for  the  whole  of  the  stroke,  is 

4.05  cubic  feet  (for  loo-lb.  steam)  x  1.5  =  6.07  cubic  feet. 

For  column  5,  of  the  net  capacity  of  cylinder  per  100,000  foot-pounds 
of  total  actual  work  done  in  one  stroke,  the  capacity  for  other  pressures  is 
modified,  in  the  first  place,  in  the  inverse  ratio  of  the  multipliers,  as  found 
for  tablet,  page  848,  for  loo-lb.  steam,  and  steam  of  the  given  pressure; 
secondly,  in  the  ratio  of  the  volume  of  i  pound  of  loo-lb.  steam,  to  that  of  a 
pound  of  the  given  steam.  For  example,  for  steam  of  65  Ibs.  total  pressure 
per  square  inch,  the  weight  of  loo-lb.  steam  to  do  a  given  total  work,  as 
compared  with  the  weight  of  65-lb.  steam,  is  as  .975  to  i.ooo,  and  the 
volumes  of  a  pound  each  of  the  two  steams  are  respectively  4.33  and  6.49 
cubic  feet,  or  as  i  to  1.5  as  already  found.  The  values  in  column  5  are 
therefore  to  be  increased  in  the  compound  ratio  of 

.975  to  i.ooo 
and       i  to  1.5 


or,  combined,  as  .975  to  1.5, 
or  as       i  to  1.54. 

The  multiplier  for  65-lb.  steam  is  thus  1.54. 


846 


STEAM   ENGINE— SINGLE-CYLINDER. 


Table  No.  297. — NET   CYLINDER-CAPACITY,  WITH    RELATION  TO  STEAM 
ADMITTED,  AND  TOTAL  ACTUAL  WORK  DONE. 

For  Steam  of  100  Ibs.  total  pressure  per  square  inch. 
Clearance  at  each  end  of  the  Cylinder,  7  per  cent,  of  the  stroke. 


Net  Capacity  of 

Per  Cubic  Foot  of  Net 

Cylinder. 

Capacity  of  Cylinder. 

Perioojooo 

WEIGHT 

TOTAL 

TOTAL 
ACTUAL 

ACTUAL 
RATIO 

OF 

EXPAN- 
SION. 

PERIOD  OF 
ADMISSION,  or 
CUT-OFF,  as  a 
Percentage  of 
Stroke. 

TOTAL 
ACTUAL 
WORK  DONE 
by  i  Ib.  of 
loo-lb.  Steam. 

Per  pound 
of  loo-lb. 
Steam, 
admitted 
in  one 
Stroke. 

Foot- 
pounds of 
Total  Ac- 
tual Work 
done  by 
Steam  of 
100  Ibs. 

OF  STEAM 
of  100  Ibs. 
Total 
Pressure 
admitted 
for  one 
Stroke, 

ACTUAL 
WORK  DONE 
by  Steam  of 
100  Ibs., 
Total  Initia 
Pressure, 
in  one 

WORK 
DONE  per 
Sq.  Inch 
of  Piston, 
per  Foot  o 
Stroke, 
by  100  Ibs. 

pressure, 
in  one 

per  Cubic 

Stroke,  per 

Steam. 

Stroke. 

Foot. 

Cubic  Foot. 

(i) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

initial  vol- 
ume =  i. 

per  cent. 

foot-pounds. 

cubic  feet. 

cu.  feet. 

pound. 

foot-pounds. 

foot- 
pounds. 

1.0 

100 

58,273 

4.05 

6.94 

.247 

14,400 

100 

1.05 

95.0 

6l,I93 

4-25 

6.95 

•235 

14,388 

99-97 

I.I 

90.3  or  9/IO 

63,850 

445 

6.97 

.225 

14,347 

99.6 

1.1* 

86.0 

66,310 

4.65 

7.02 

.215 

14,245 

99.0 

1.18 

83.3  or  s/6 

67,836 

4.78 

7.04 

.209 

14,204 

98.6 

1.2 

82.1 

68,766 

4.86 

7.06 

.206 

14,164 

98.3 

1.23 

80.0  or  4/5 

70,246 

4.98 

7.09 

.201 

14,104 

98.0 

1.25 

78.6 

71,151 

5.06 

7.II 

.198 

14,065 

977 

i-3 

75-3  or  % 

73,513 

5.26 

7.l6 

.190 

13,966 

96.9 

1-35 

72.3 

75,575 

5.46 

7.23 

.183 

13,831 

96.1 

1-39 

70.0  or  7/IO 

77,242 

5.63 

7.28 

.178 

13,736 

95-3 

1.4 

69.4 

77,616 

5.67 

7.30 

.176 

13,699 

95.1 

145 

66.8  or  2/3 

79,555 

5.87 

7.38 

.170 

!3,55° 

94.2 

i-5 

64-3 

81,546 

6.07 

7-45 

.165 

13,423 

93-2 

1.54 

62.5  or  }i 

83,055 

6.23 

7.50 

.161 

13,333 

92.5 

i-55 

62.0 

83,299 

6.27 

7-53 

.159 

13,280 

92.2 

1.6 

59.9  or  3/5 

85,125 

6.47 

•J55 

I3,!4i 

9J-3 

1.65 

57-9 

86,828 

6.68 

7-69 

.150 

13,004 

i.7 

56.0 

88,598 

6.88 

7-77 

.145 

12,870 

89.4 

1-75 

54.1 

90,115 

7.08 

7.92 

.141 

12,626 

88.3 

1.8 

52,4 

91,680 

7.30 

7-95 

.137 

12,579 

87.3 

1.85 

50.8 

93,065 

7-49 

8.04 

•134 

12,438 

86.4 

1.88 

50.0  or  l/2 

94,200 

7.61 

8.08 

•!3i 

12,376 

86.0 

1.9 

49-3 

94,610 

7-69 

8.13 

.130 

12,300 

85.4 

1.95 

47-9 

96,100 

7.89 

8.22 

.127 

12,165 

84.6 

2.0 

46.5 

97,432 

8.09 

8.3I 

.124 

12,034 

83-6 

2.1 

44.0 

100,266 

8.50 

8.49 

.118 

11,778 

81.8 

2.2 

41.6 

102,366 

8.90 

8.70 

.112 

11,494 

79-9 

2.28 

40.0  or  2/5 

104,466 

9.23 

8.83 

.108 

n,325 

78.7 

2-3 

39-5 

104,855 

9-3i 

8.89 

.107 

11,249 

78.2 

2.4 

37-6  or  3/8 

107,050 

9.71 

9.07 

.103 

11,025 

76.6 

2.5 

35-8 

109,266 

10.12 

9.26 

.099 

10,799 

75.0 

2.6 

34-2 

111,400 

10.52 

945 

.095 

10,582 

73-6 

2.65 

33-3  or  1/3 

112,220 

10.72 

9.56 

•093 

10,460 

72.6 

2.7 

32.6 

113,244 

10.93 

9.65 

.091 

10,363 

71.9 

2.8 

30.2 

115,244 

11-33 

9.83 

.088 

10,173 

70.6 

2.9 

29.9  or  3/10 

116,855 

11.74 

10.04 

.085 

9,960 

69.2 

CYLINDER-CAPACITY  AND  WORK  DONE. 


847 


Table  No.  297  (continued). 


Net  Capacity  of 

Per  Cubic  Foot  of  Net 

Cylinder. 

Capacity  of  Cylinder. 

. 

Perioo.ooo 

TTt_  _. 

WEIGHT 

TOTAL 

TOTAL 
ACTUAL 

ACTUAL 
RATIO 

OF 

EXPAN- 
SION. 

PERIOD  OF 

ADMISSION,  or 
CUT-OFF,  as  a 
Percentage  of 
Stroke. 

TOTAL 
ACTUAL 
WORK  DONE 
by  i  Ib.  of 
loo-lb.  Steam 

Per  pounc 
of  loo-lb. 
Steam, 
admitted 
in  one 
Stroke. 

root- 
pounds  of 
Total  Ac- 
tual Work 
done  by 
Steam  of 
loo  Ibs. 

OF  STEAM 
of  100  Ibs. 
Total 
Pressure 
admitted 
for  one 
Stroke, 

ACTUAL 
WORK  DONE 
by  Steam  o 
100  Ibs., 
Total  Initia 
Pressure, 
in  one 

WORK 
DONE  per 
Sq.  Inch 
of  Piston, 
per  Foot  o 
Stroke, 
by  zoo  Ibs. 

pressure, 

per  Cubic 

Stroke,  per 

Steam. 

in  one 
Stroke. 

Foot. 

Cubic  Foot. 

(x) 

(2) 

(3) 

(4) 

(s) 

(6) 

(7) 

(8) 

ini  ticiJ  vol 
ume=i. 

per  cent. 

foot-pounds. 

cubic  feet 

cu.  feet. 

pound. 

foot-pounds. 

foot- 
pounds. 

3-0 

28.7 

118,820 

12.14 

10.22 

.082 

9,785 

67.9 

3-i 

27-5 

109,970 

12.55 

10.46 

.080 

9,560 

66.5 

3-2 

26.4 

121,386 

12.95 

10.67 

.077 

9,372 

65.2 

3-3 

25.4 

123,278 

13.35 

10.83 

.075 

9,234 

64.1 

3-35 

25.0  or  % 

124,066 

13.56 

10.93 

.074 

9,H9 

637 

34 

24-5 

125,066 

13.76 

11.00 

•073 

9,091 

63.I 

3-5 

23.6 

126,125 

14.16 

II.  21 

.071 

8,961 

61.9 

3-6 

22.7 

127,450 

14-57 

H.43 

.069 

8,749 

60.8 

37 

21.9 

128,860 

14.97 

1  1.  60 

.067 

8,621 

597 

3-8 

21.2 

1  30,533 

15.38 

11.78 

.065 

8,489 

58.9 

3-9 

20.4 

131,800 

15.78 

11.98 

.063 

8,347 

57-9 

4.0 

19.7  or  x/-5 

132,770 

16.19 

12.19 

.062 

8,203 

56.7 

4.1 

19.1 

133,500 

16.59 

12.39 

.060 

8,071 

55.9 

4.2 

18.5 

134,900 

17.00 

1  2.6o 

.059 

7,936 

55-i 

4-3 

17.9 

135,555 

17.40 

12.80 

.058 

7,8  1  2 

54-2 

4-4 

17.3 

136,825 

I7.8l 

13.01 

.056 

7,686 

53-3 

4-5 

1  6.8  or  x/6 

138,130 

18.21 

13.19 

.055 

7,58i 

52.6 

4-6 

10.3 

139,100 

18.62 

13.38 

.054 

7,474 

51.8 

4-7 

15.8 

139,800 

19.02 

13.58 

•053 

7,364 

51.1 

4-8 

15.3 

140,920 

19-43 

13.79 

.051 

7,252 

50.3 

4.9 

14.8 

141,210 

19.83 

14.01 

.050 

7,138 

49.4 

5.0 

14.4  or  i/7 

I42,l8o 

20.23 

14.23 

.049 

7,027 

48.8 

5.2 

13.6 

143,720 

21.04 

14.64 

.047 

6,831 

47-6 

5-4 

12.8 

H5,525 

21.85 

15.02 

.046 

6,658 

46.2 

5.5 

12.5  or  y8 

146,325 

22.25 

15.20 

.045 

6,579 

45-7 

5-6 

12.1 

147,320 

22.66 

15-38 

.044 

6,502 

45-0 

5.8 

1  1.4 

148,390 

23.47 

15.80 

•043 

6,329 

43.8 

5-9 

1  1.  1  or  I/9 

148,940 

23.87 

1  6.0  1 

.042 

6,246 

43-2 

6.0 

10.8 

149,586 

24.28 

16.23 

.041 

6,161 

42.7 

6.2 

10.3 

150,630 

25.09 

16.63 

.040 

6,013 

41.9 

6.3 

10.0  or  Vio 

151,370 

25.49 

16.83 

.039 

5,942 

4L3 

6.4 

9-7 

152,033 

25.90 

17.04 

•038 

5,868 

40.7 

6.6 

9.2  or  i/n 

152,595 

26.71 

1747 

•037 

5,724 

39-8 

6.8 

8-7 

153,810 

27.52 

17.89 

.036 

5,590 

38.8 

7.0 

8.3  or  x/12 

155,200 

28.33 

18.27 

•035 

5,473 

38.1 

7-2 

7-9 

156,330 

29.14 

18.64 

•0343 

5,365 

37-3 

7-3 

7-7  or  x/I3 

156,960 

29.54 

18.83 

•0339 

5,3H 

36.9 

7-4 

7-5 

157,560 

29.95 

19.01 

•0334 

5,260 

36.5 

7.6 

7.1  or  '/i4 

157,975 

30.76 

19.47 

.0325 

5,136 

35-7 

7-8 

6.7  or  Vis 

158,414 

31.57 

19-93 

.0317 

5,o  1  8 

34-8 

8.0 

6.4  or  Vie 

159,433 

32.38 

20.31 

•0309 

4,923 

34.2 

848 


STEAM   ENGINE— SINGLE-CYLINDER. 


APPENDIX  TO  TABLE  No.  297. 

TABLET  OF  MULTIPLIERS  FOR  NET  CYLINDER-CAPACITY,  STEAM 
ADMITTED,  AND  TOTAL  WORK  DONE. 

For  Steam  of  other  pressures  than  100  Ibs.  per  square  inch. 
(See  explanatory  notice  of  the  table,  page  844. ) 


MULTIPLIERS. 

Total  Pres- 

sures per 
Square  Inch. 

For  Column  3. 
Total  Work. 

For  Column  4. 
Capacity. 

For  Column  5. 
Capacity. 

For  Column  6. 
Weight  of 
Steam. 

For  Columns 
7  and  8. 
Work. 

Ibs. 

65 

•975      • 

1.50 

1.54 

.666 

.65 

70 

.981 

1.40 

1-43 

.714 

.70 

75 

.986 

•31 

i-33 

.763 

•75 

80 

.988 

.24 

1.25 

.806 

.80 

85 

.991 

•17 

1.18 

.855 

.85 

90 

•995 

.11 

i.  ii 

.901 

.90 

95 

.998 

.05 

1.05 

.952 

•95 

IOO 

1.  000 

.00 

1.  00 

.00 

.00 

no 

1.009 

.917 

.909 

.09 

.10 

120 

I.OII 

•843 

.833 

•17 

.20 

130 

1.015 

.781 

.769 

.28 

•30 

140 

1.022 

•730 

.714 

•37 

.40 

150 

1.025 

•683 

.667 

46 

•5° 

1  60 

I.03I 

.644 

.625 

•55 

.60 

For  column  6,  of  the  weights  of  steam  per  cubic  foot  of  net  capacity 
for  one  stroke,  the  weights  given  for  loo-lb.  steam  are  to  be  multiplied  by 
the  reciprocals  of  the  multipliers  for  column  4,  corresponding  to  other  pres- 
sures. Thus,  for  65-lb.  steam,  the  multiplier  is  the  reciprocal  of  1.5,  or 
.666.  For  column  7,  of  the  total  actual  work  done,  per  cubic  foot  of  net 
capacity,  for  one  stroke,  the  work  given  for  zoo-lb.  steam  is  to  be  multiplied 
by  the  reciprocal  of  the  multiplier  for  column  5,  as  determined  for  the  given 
other  pressure.  Thus,  for  65-^.  steam,  the  multiplier  is  the  reciprocal  of 
1.54,  or  .65. 

For  the  total  actual  work  done  per  square  inch  of  piston  per  foot  of 
stroke,  by  steam  of  any  other  pressure,  the  values  given  in  column  8,  for 
loo-lb.  steam,  are  to  be  multiplied  by  the  given  pressure  and  divided  by 

100.     For  65-10.  steam,  for  example,  the  multiplier  is,  in  fact,  — 5-  —  .65,  the 

100 

same  as  for  column  7.  Otherwise  regarded,  the  values  in  column  8  may 
be  taken  as  percentages  of  the  work  for  steam  as  admitted  for  the  whole  of 
the  stroke,  whatever  the  initial  pressure  may  be. 

It  is  apparent  that  the  multipliers  for  columns  7  and  8,  are  simply  equal 
to  the  respective  total  pressures  divided  by  100,  since  the  multiplier  for 
loo-lb.  pressure  is  taken  as  i.  These  multipliers  must  obviously  be  in  the 
direct  ratio  of  the  pressures;  and,  of  course,  the  multipliers  for  column  5 
must  be  in  the  inverse  ratio  of  the  pressures. 


COMPOUND   STEAM   ENGINE. 


849 


COMPOUND   STEAM-ENGINE. 

The  compound  steam-engine,  consisting  of  two  cylinders,  is  reducible  to 
two  forms,  in  which,  first,  the  steam  from  the  first  cylinder  is  exhausted 
direct  into  the  second  cylinder,  as  in  the  Woolf  engine;  and,  second,  the 
steam  from  the  first  cylinder  is  exhausted  into  an  intermediate  reservoir, 
whence  the  steam  is  supplied  to,  and  expanded  in,  the  second  cylinder,  as  in 
the  "  receiver-engine."  In  the  Woolf  engine,  according  to  the  original  type, 
the  pistons  of  the  two  cylinders  move  together,  and  make  the  same  strokes 
simultaneously,  the  steam  from  the  top  and  the  bottom  of  the  first  cylinder, 
being  exhausted  into  the  bottom  and  the  top,  respectively,  of  the  second 
cylinder.  In  the  receiver-engine,  the  pistons  are  connected  to  cranks  on 
one  shaft,  at  right  angles  to  each  other. 

It  is,  in  the  first  place,  assumed  that  there  is  no  clearance  at  either  end 
of  either  cylinder;  that  there  is  no  frictional  resistance  nor  other  hindrance 
in  the  engine  to  the  flow  of  steam;  that  the  pressure  of  expanding  steam 
is  inversely  as  the  volume;  that  the  expansion  of  the  steam  is  continued  in 
both  cylinders  to  the  end  of  the  stroke;  and  that  in  the  second  cylinder 
the  steam  acts  against  a  perfect  vacuum  on  the  other  face  of  the  piston.  It  is 
assumed,  further,  that  no  intermediate  fall,  or  "  drop "  of  pressure  takes 
place  between  the  first  and  second  cylinders ;  that  is,  that  the  initial  pressure 
in  the  second  cylinder  is  equal  to  the  final  pressure  in  the  first  cylinder; 
also,  that  there  is  no  loss  of  steam  by  condensation  within  the  cylinders. 

Taken  generally,  the  work  done  by  expansion  into  the  second  cylinder, 
is  that  due  to  the  increase  of  volume  of  the  steam  in  this,  the  second  stage. 

WOOLF  ENGINE — IDEAL  DIAGRAMS. 

The  diagrams  of  pressure  of  the  Woolf  engine  are  produced  in  Fig.  334, 
as  they  would  be  described  by  the  indicator,  according  to  the  arrows.  In 
these  ideal  diagrams,  pq  is  the  atmo- 
spheric line,  mn  the  straight  line  of 
perfect  vacuum,  cd  the  straight  line 
of  admission;  the  terminal  lines,  me 
and  ng,  straight  and  perpendicular 
to  the  atmospheric  line,  dg  the  hyper- 
bolic curve  of  expansion  in  the  first 
cylinder,  and^  the  consecutive  expan- 
sion-line of  back  pressure  for  the  re- 
turn stroke  of  the  first  piston,  and  of 
positive  pressure  for  the  steam-stroke 
of  the  second  piston.  At  the  point  h, 
at  the  end  of  the  stroke  of  the  second 
piston,  the  steam  is  exhausted  into  the 
condenser,  and  the  pressure  falls  to  the 
level  of  perfect  vacuum,  mn. 

The  diagram  pertaining  to  the  second 
cylinder,  below  the  curve  gh,  is  char- 


Fig.  334. — Woolf  Engine : — Ideal  Indicator 
Diagrams. 


acterized  by  the  absence  of  any  specific  period  of  admission;  the  whole  of 
the  steam-line  gh  being  expansive,  and  generated  by  the  expansion  of  the 
initial  body  of  steam  contained  in  the  first  cylinder  into  the  second.  The 


850 


STEAM   ENGINE — COMPOUND  CYLINDERS. 


initial  volume — the  volume  of  the  first  cylinder — is,  however,  gradually  re- 
duced by  the  advancing  movement  of  the  first  piston,  by  which  the  initial 
steam  is  gradually  driven  into  the  second  cylinder;  until,  when  the  stroke 
is  completed,  the  whole  of  the  steam  is  transferred  from  the  first,  and  is 
shut  into  the  second  cylinder.  The  first  cylinder,  then,  acts  as  a  collaps- 
ible head  to  the  second  cylinder  for  each  steam-stroke  of  the  latter;  the 
second  cylinder  being,  at  the  beginning  of  the  stroke,  augmented  by  the 
capacity  of  the  first;  and  at  the  end  of  the  stroke,  reduced  to  its  normal 
dimensions.  The  final  pressure  and  volume  of  the  steam  in  the  second 
cylinder  are,  consequently,  the  same  as  if  the  whole  of  the  initial  steam  had 
been  admitted  at  once  into  the  second  cylinder,  and  then  expanded  to  the 
end  of  the  stroke;  the  hypothetical  period  of  admission  being  such  a  frac- 
tion of  the  stroke  of  the  second  cylinder  as  would  represent  the  ratio  of 
the  volume  of  the  first  cylinder,  which  is  the  volume  of  the  steam  admitted, 
to  that  of  the  second  cylinder. 

The  net  work  of  the  steam,  according  to  the  supposed  distribution  in 
one  cylinder  only,  would  be  the  same  as  in  the  two  cylinders  compounded. 
Construct  a  combined  diagram,  with  a  continuous  expansion-line,  by 
piecing  the  first  upon  the  second  of  the  two  diagrams,  Fig.  334,  page  849, 
representing  the  work  as  if  it  were  done  in  one  cylinder  equal  in  capacity 
to  the  second  of  the  compound  cylinders.  For  this  purpose,  the  first 
diagram  is  to  be  contracted  to  a  length  bearing  the  same  proportion  to  the 
length  of  the  second  diagram,  as  the  volume  of  the  first  cylinder  to  that  of 


c d 


Fig-  335-—  Woolf  Engine :— Ideal  Diagrams, 

reduced. 


Fig-  336.— Woolf  Engine :— Ideal  Diagrams, 
combined. 


the  second  cylinder— in  this  example,  one-third.  It  is  necessary  to  do  so  in 
order  to  reduce  the  two  elements  of  the  combined  diagram  to  the  same  hori- 
zontal scale  for  measurement.  When  the  strokes  of  the  cylinders  are  equal 
to  each  other,  the  capacities  are  in  the  simple  ratio  of  their  areas.  In  this 
example,  therefore,  the  length  of  the  first  diagram  is  to  be  reduced  to  one- 
third  of  the  length  of  the  second.  For  this  purpose,  l&ghmn,  Fig.  335, 
annexed,  be  the  diagram  from  the  second  cylinder,  over  which  the  original 
diagram  from  the  second  cylinder  is  shown  in  dotting.  Draw  gg"'  parallel 
to  the  base,  and  mhc  perpendicular  to  the  base,  equal  to  the  height  of  the 
first  diagram;  set  off  g'"g"  equal  to  one-third  of  the  base,  and  upon  this 


WOOLF   ENGINE — IDEAL  DIAGRAMS.  851 

reduced  base  g'"g"  complete  the  first  diagram  cdg",  by  drawing  cd  parallel 
to  the  base,  and  equal  to  one-third  of  it,  and  the  expansion-curve  dg".  For 
the  lower  part  of  the  figure,  the  line  of  back  pressure  g"h  may  be  described 
by  the  method  of  ordinates,  repeated  from  the  curve  gh.  Thus  the  contracted 
diagram  cdg"h  is  completed.  To  combine  the  first  diagram,  thus  reduced 
to  uniformity  of  scale,  with  the  second  diagram,  let  again  ghmn,  Fig.  336, 
annexed,  be  the  second  diagram,  and  describe  the  first  diagram  as  reduced, 
in  a  reversed  position  cdg"h',  at  the  head  of  the  second  diagram,  the  same 
letters  of  reference  being  used.  Finally,  continue  the  hyperbolic  expansion- 
line  dg"  to  the  end  of  the  stroke  at  h.  Then  the  area  gg*ht  is  equal  to 
the  axeB.ggffft',  and,  when  substituted  for  it,  completes  the  regular  indicator 
diagram  cdhmn,  with  a  continuous  expansion-line  dg"h.  The  substitution 
is,  in  fact,  necessary,  since  the  lower  part  of  the  first  diagram  partly  overlaps 
the  second  diagram. 

According  to  this  combination,  the  upper  part  of  the  diagram,  Fig.  336, 
above  the  curve  gh,  namely,  cdhg,  represents  the  diagram,  as  contracted,  for 
the  first  cylinder,  modified  in  form,  but  unaltered  in  area;  and  the  lower 
part,  below  the  curve  gh,  remains  unaltered,  both  in  form  and  in  area,  as 
the  diagram  for  the  second  cylinder.  The  combined  diagram,  as  a  whole, 
exactly  measures  the  whole  net  work  done  in  both  cylinders,  and  is  such  as 
would  be  formed  by  admitting  and  expanding  the  same  quantity  of  steam 
in  one  cylinder  having  the  dimensions  of  the  second  cylinder,  with  the 
period  of  admission,  cd,  equal  to  one-third  of  the  capacity  of  the  first 
cylinder,  or  one-ninth  of  the  capacity  or  the  stroke  of  the  second  cylinder. 

It  follows  further,  that  the  work  effected  by  expansion  into  the  second 
cylinder  of  the  Woolf  engine, — that  is,  the  total  work  arising  from  expansion 
against  the  second  piston,  plus  the  gain  of  work  in  the  first  cylinder  by  the 
gradual  reduction  of  back  pressure  in  accordance  with  the  expansion, — is 
equal  to  that  which  would  be  effected  by  delivering  the  whole  of  the  steam 
into  the  second  cylinder  before  expansion  is  commenced,  as  in  the  receiver- 
engine.  By  this  distribution,  the  upper  part  of  the  combined  diagram, 
Fig.  336,  cut  off  by  the  horizontal  line  gg",  would  measure  the  net  work  of 
the  first  cylinder,  as  there  would  be  a  uniform  back  pressure  equal  to  ng 
on  the  piston;  and  the  lower  part  of  the  diagram,  below  gg",  would  mea- 
sure the  work  of  the  second  cylinder,  with  a  period  of  admission  equal 
to  gg", — the  capacity  of  the  first  cylinder  at  the  pressure  ng, — and  with 
expansion  to  the  end  of  the  stroke. 

To  exemplify  the  foregoing  conclusions,  under  the  conditions  originally 
stated,  suppose  that  the  steam  is  admitted  to  the  first  cylinder  at  a  total 
initial  pressure  of  63  Ibs.  per  square  inch;  that  the  areas  of  the  first  and 
second  cylinders  are  respectively  i  and  3  square  inches,  and  that  the 
common  length  of  stroke  is  6  feet.  The  steam  being  cut  off  in  the  first 
cylinder  at  one-third  of  the  stroke  for  the  period  of  admission,  cd,  Fig.  335, 
page  850,  it  is  expanded  to  three  times  its  initial  volume,  and  to  one-third 
of  the  initial  pressure,  namely,  ng,  equal  to  21  Ibs.,  at  the  end  of  the  stroke. 
The  steam  is  admitted  to  the  second  cylinder  at  the  same  pressure,  ng, 
and  is  expanded  there  to  three  times  the  volume  it  acquired  in  the  first 
cylinder,  or  to  3  x  3  =  9  times  the  initial  volume  in  the  first  cylinder.  At 
the  same  time,  the  pressure  is  reduced  in  the  second  cylinder  to  one-third 
of  the  final  pressure  in  the  first  cylinder,  or  to  one-ninth  of  the  initial  pres- 
sure there,  namely,  to  7  Ibs.  per  square  inch,  measured  by  mh,  Fig.  335. 


852  STEAM   ENGINE — COMPOUND   CYLINDERS. 

The  work  of  the  compound  engine  may  be  calculated  from  the  combined 
diagram,  Fig.  336,  page  850;  regarding  the  upper  part  of  the  figure,  above 
the  line  ggff,  as  the  net  work  of  the  first  cylinder,  according  to  the  equivalent 
distribution  mentioned  at  page  851,  where  the  action  is  compared  to  that  of 
a  receiver-engine;  and  the  lower  part,  below  gg",  as  the  work  of  the  second 
cylinder,  with  a  period  of  admission  equal  to  gg",  and  an  expansion  to  the 
end  of  the  stroke.  For  the  first  section,  the  total  work,  over  the  base  «»",  is 
calculated,  and  the  work  of  the  pressure,  ng,  as  back  pressure,  on  the  same 
base,  is  deducted  from  it  to  give  the  net  work.  Now,  the  total  pressure, 
nc,  calculated  on  the  area  of  the  second  cylinder,  is  63  Ibs.  *  3  square 
inches  =  189  Ibs.;  and  the  period  of  admission,  cd,  is  one-ninth  of  the 
stroke,  or  2/3  foot.  The  total  initial  work  is,  then,  189  x  2/3  =  126  foot- 
pounds, and  the  total  work,  with  an  expansion  of  three  times  for  the  stroke, 
gg",  2  feet,  is 

126  x  (i  +hyp  log  3)  =  264.4236  foot-pounds. 

The  work  of  the  back  pressure,  ng,  or  21  x  3  =  63  Ibs.,  into  the  stroke,  gg", 
or  2  feet,  is  63  Ibs.  x  2  feet=  126  foot-pounds,  and  the  net  work,  above 
the  line  gg",  is  264.4236  -  126  =  138.4236  foot-pounds. 

For  the  second  section,  according  to  the  equivalent  distribution,  the  initial 
work  is  equal  to  that  of  the  back  pressure  on  the  first  piston,  which  has  just 
been  calculated,  namely,  126  foot-pounds,  and  the  work  for  an  expansion  of 
three  times,  through  the  stroke,  nm,  is  found  by  what  is  only  a  repetition  of 
the  calculation  for  the  upper  section,  to  be  264.4236  foot-pounds. 

The  sum  of  the  two  sections  is  the  total  net  work  of  the  two  cylinders; 
thus:— 

Upper  section, 138.4236  foot-pounds. 

Lower  section, 264.4236         „ 


Total  net  work, 402.8472         „ 

Otherwise,  the  combined  diagram,  Fig.  336,  represents  the  whole  of  the 
work  as  if  it  were  done  in  one  cylinder  equal  in  capacity  to  the  second 
cylinder — assumed,  in  this  instance,  to  have  the  same  diameter  and  stroke. 
The  period  of  admission,  cd,  is  one-ninth  of  the  stroke,  or  6  -=-  9  =  ^  foot ; 
the  initial  pressure  being  63  Ibs.  x  31  =  89  Ibs.,  and  the  initial  work 
189  x  YZ  =  126  foot-pounds.  The  whole  work  of  the  stroke  is,  therefore, 

126  x  (i  4-  hyp  log  9)  =  126  x  3.1972  -  402.8472  foot-pounds; 

as  was  calculated  before. 

i 

RECEIVER-ENGINE — IDEAL  DIAGRAMS. 

The  hypothetical  distribution  which  has  been  described  for  the  Woolf 
engine,  according  to  which  all  the  steam  with  which  the  second  cylinder  is 
charged,  is  supposed  to  be  admitted  into  the  second  cylinder  before  expan- 
sion begins,  is  that  which  actually  takes  place  in  the  receiver-engine, — the 
second  general  combination  of  the  compound  engine, — in  which  the  pistons 
of  the  two  cylinders  are  connected  to  cranks  at  right  angles  to  each  other, 
on  the  same  shaft,  with  an  intermediate  receiver.  The  receiver  is  occupied 
by  steam  exhausted  from  the  first  cylinder,  and  it  supplies  steam  to  the 


RECEIVER-ENGINE — IDEAL   DIAGRAMS. 


853 


second  cylinder,  in  which  it  is  cut  off  and  then  expanded  to  the  end  of  the 
stroke.  On  the  assumption  that  the  initial  pressure  in  the  second  cylinder 
is  equal  to  the  final  pressure  in  the  first  cylinder,  and,  of  course,  equal  to 
the  pressure  in  the  receiver,  the  volume  cut  off  in  the  second  cylinder  must 
be  equal  to  the  volume  of  the  first  cylinder,  for  the  second  cylinder  must 
admit  as  much  at  each  stroke  as  is  discharged  from  the  first  cylinder. 

For  illustration,  suppose  again  that  the  areas  and  capacities  of  the  first 
and  second  cylinders,  with  the  same  length  of  stroke,  are  as  i  to  3,  and  that 
the  steam  is  cut  off  iat  one-third  of  the  stroke,  and  equally  expanded  in  both 
cylinders,  the  ratio  of  expansion  in  each  cylinder  being  thus  equal  to  the 
ratio  of  the  capacities  of  the  cylinders.  With  this  distribution,  the  volume 
admitted  to  the  second  cylinder  is  equal  to  the  volume  discharged  from  the 
first  cylinder,  and  there  is  no  intermediate  fall  of  pressure.  The  ideal 
diagrams  of  pressure  which  would  thus  be  formed  are  shown  in  juxtaposi- 
tion in  Fig.  337.  Here,  pq  is  the  atmospheric  line,  cd  is  the  line  of 
admission,  and  hg  the  exhaust-line  for  the  first  cylinder,  both  of  them  being 


-XO 


Fig.  337. — Receiver-Engine : — Ideal  Indicator 
Diagrams. 


Fig.  338. — Receiver-Engine: — Ideal  Diagrams, 
reduced  and  combined. 


parallel  to  the  atmospheric  line;  and  dg  is  the  expansion-curve.  In  the 
region  below  the  exhaust-line  of  the  first  cylinder,  between  it  and  the  line 
of  perfect  vacuum,  the  diagram  of  the  second  cylinder  is  formed;  hi,  the 
second  line  of  admission,  coincides  with  the  exhaust-line  hg  of  the  first 
cylinder,  and  thus  shows  that,  in  the  ideal  diagrams,  there  is  no  intermediate 
fall  of  pressure.  The  line  of  perfect  vacuum,  ol,  is  parallel  to  the  atmos- 
pheric line,  and  ik  is  the  expansion-curve.  The  arrows  indicate  the  order 
in  which  the  diagrams  are  formed. 

The  expansive  working  of  the  steam,  though  clearly  divided  into  two 
consecutive  stages,  is,  as  in  the  Woolf  engine,  essentially  continuous  from 
the  point  of  cut-off  in  the  first  cylinder  to  the  end  of  the  stroke  of  the 
second  cylinder,  where  it  is  delivered  to  the  condenser;  and  the  first  and 
second  diagrams  may  be  placed  together  and  combined  to  form  a  continu- 
ous diagram.  For  this  purpose,  take,  as  was  done  for  the  Woolf  engine, 
the  second  diagram  as  the  basis  of  the  combined  diagram,  namely,  hiklo, 
Fig.  338,  adding  the  atmospheric  line,/^.  The  period  of  admission,  hi,  is 
one-third  of  the  stroke,  and  as  the  ratios  of  the  cylinders  are  as  i  to  3,  hi 


854  STEAM   ENGINE— COMPOUND  CYLINDERS. 

is  also  the  proportional  length  of  the  first  diagram  as  applied  to  the  second. 
Produce  oh  upwards,  and  set  off  oc  equal  to  the  total  height  of  the  first 
diagram  above  the  vacuum  line;  and,  upon  the  shortened  base  hi,  and  the 
height  he,  complete  the  first  diagram  with  the  steam-line  cd,  and  the 
expansion-line,  di. 

By  this  construction,  the  regular  indicator  diagram  cdklo  is  formed,  as 
applied  to  the  second  cylinder,  and  it  measures  the  whole  net  work  done 
in  both  cylinders : — the  upper  section,  cdth,  being  the  measure  of  the  net 
work  done  in  the  first  cylinder,  and  the  lower  section,  hiklo,  being  the 
measure  of  the  work  of  the  second  cylinder. 

Resuming  the  data  supplied  for  exemplifying  the  Woolf  engine,  with 
reference  to  the  ideal  diagrams  for  the  receiver-engine,  Fig.  337,  let  the 
areas  of  the  first  and  second  cylinders  be  respectively  i  and  3  square  inches, 
the  stroke  6  feet,  and  the  initial  pressure  in  the  first  cylinder  63  Ibs.  per 
square  inch.  The  steam  being  cut  off  at  one-third  of  the  stroke  of  the  first 
cylinder,  the  final  pressure  is  21  Ibs.,  and  the  total  initial  work  therein  is 
equal  to  ocKcd  =  b$  Ibs.  x  2  feet=  126  foot-pounds;  for  the  whole  stroke 
the  total  work  is 

126  x  (i  +hyp  log  3)  =  126  x  2.0986  =  264.4236  foot-pounds, 

the  same  as  was  found  for  the  Woolf  engine.  For  the  second  cylinder,  the 
initial  pressure  is  21  Ibs.  x  3  square  inches  =  63  Ibs.,  and  the  initial  work  is 
represented  by  oh  x  hi,  which  is  equal  to  63  Ibs.  x  2  feet  =  126  foot-pounds. 
For  the  whole  stroke,  the  total  work  is 

i26x(i  +hyp  log  3)=i26x  2.0986  =  264.4236  foot-pounds. 

The  work  of  the  back  pressure,  oh,  on  the  first  piston,  which  is  continued 
for  the  whole  of  the  stroke,  is  to  be  deducted  from  this  total  work;  it  is 
represented  by  the  rectangle  ohxhg,  equal  to  21  Ibs.  x  6  feet  =126  foot- 
pounds. Then  (264.4236-126=)  138.4236  foot-pounds  is  the  net  or  effec- 
tive work  for  the  second  cylinder  for  one  stroke.  This,  the  work  for  the 
second  cylinder,  is  to  be  added  to  the  work  for  the  first  cylinder,  and  the 
sum,  402.8472  foot-pounds,  is  the  united  work  for  one  stroke  of  the  two 
cylinders. 

This  is  the  same  quantity  of  work  as  was  calculated  from  the  Woolf 
diagrams. 

It  is  obvious  that  the  two  combined  diagrams,  Figs.  336  and  338,  pages 
850  and  853,  are  identical  in  form  and  development. 

WORK  OF  STEAM  AS  AFFECTED  BY  INTERMEDIATE  EXPANSION. 

That  the  work  of  expanding  steam  is  to  be  calculated  from  the  expansion 
upon  a  moving  piston  only,  is  obvious  enough  when  it  is  considered  that 
the  steam  may  expand  into  an  intermediate  receiver,  and  into  intermediate 
passages,  without  doing  any  work  on  a  piston,  whilst  at  the  same  time  the 
pressure  falls  or  "  drops  "  as  the  volume  is  enlarged.  Under  these  circum- 
stances, the  second  cylinder  receives  the  steam  at  a  lower  pressure  and  in 
larger  volume  than  it  has  when  there  is  no  intermediate  expansion  and  fall 
of  pressure;  and  there  is  less  work  done,  whilst  the  ratio  of  active  expan- 


INTERMEDIATE   EXPANSION   IN   THE  WOOLF  ENGINE.        855 

sion  is  necessarily  reduced.  If  the  second  cylinder,  however,  be  enlarged 
in  capacity,  in  proportion  to  the  enlargement  of  the  volume  of  the  steam 
and  the  fall  of  pressure,  by  intermediate  expansion,  the  ratio  of  expansion, 
and  the  work  done  in  it,  would  remain  the  same. 

Whilst,  then,  there  is  no  reduction  of  work  consequent  on  intermediate 
expansion  of  the  steam,  provided  that  the  ratio  of  expansion  originally 
designed  be  maintained  by  means  of  a  second  cylinder  of  suitably  large 
capacity;  there  is  actually  a  reduction  of  work,  or  loss  of  effect,  by  such 
intermediate  expansion,  when  the  capacity  of  the  second  cylinder  remains 
the  same. 

INTERMEDIATE  EXPANSION  IN  THE  WOOLF  ENGINE. 

To  proceed  with  the  investigation  of  such  loss  by  intermediate  expansion 
as  is  suffered  in  the  Woolf  engine,  take  the  example  of  Woolf  engine  already 
treated,  with  the  same  proportions  and  dimensions,  and  suppose  that  the 
total  capacity  of  the  passages  from  the  first  to  the  second  cylinder  is  one- 
third,  or  33  T/3  per  cent,  of  the  capacity  of  the  first  cylinder.  The  ideal 
diagrams  Fig.  335,  and  the  combined  diagram  Fig.  336,  page  850, 
are  reproduced,  partly  in  dot-lining,  in  Figs.  339  and  340  annexed, 
the  same  letters  of  reference  being  applied.  To  these  are  added  the 
modifications  introduced  by  the  intermediate  fall  of  pressure.  The  admis- 
sion and  expansion  of  the  steam  in  the  first  cylinder  are  indicated,  as 


Fig.  339.— Woolf  Engine :— Diagrams  showing 
intermediate  fall  of  pressure. 


nt. 


Fig.  340. — Woolf  Engine : — The  same  diagrams 
reduced  and  combined. 


before,  by  the  straight  line  cd,  representing  an  initial  pressure  of  63  Ibs.  per 
square  inch,  and  the  curve  dg,  or  d g* ',  with  a  terminal  pressure,  ng,  or  n'g", 
of  2 1  Ibs.  But  when  the  exhaust  is  opened,  at  the  end  of  the  stroke,^  or  g"t 
the  steam  expands  into  the  intermediate  space,  and  occupies  a  total  volume 
equal  to  i  J/s  or  Vs  times  the  capacity  of  the  first  cylinder,  before  the  second 
piston  commences  its  stroke.  The  final  pressure  is,  at  the  same  time, 
reduced,  in  the  inverse  ratio,  to  ^ths  of  21  Ibs.,  or  15.75  Ibs.  from  ng  to  ng\ 
or  from  ifg*  to  n"g5,  Fig.  340.  With  this  lower  pressure,  and  the  aug- 
mented initial  volume,  i T  /3  times  the  capacity  of  the  first  cylinder,  the 


856 


STEAM   ENGINE — COMPOUND   CYLINDERS. 


steam  expands  into  the  second  cylinder,  and  acquires  a  final  volume  by 
expansion  equal  to  the  capacity  of  the  second  cylinder  plus  the  intermediate 
space,  or  3  I/3  times  the  capacity  of  the  first  cylinder.  Hence  the  ratio  of 

expansion  into  the  second  cylinder  is  ^-^-=2.5;  and  the  final  pressure 

1     /3 

nth",  is  (15.75  Ibs.  -=-2.5=)  6.3  Ibs.  per  square  inch.  The  actual  curve 
of  expansion,  g4/i",  is,  like  the  normal  curve  gh,  an  elongated  hyperbolic 
curve — an  elongation  of  the  hypothetical  expansion-curve,  g5ti",  which  flows 
from  the  augmented  initial  volume,  as  illustrated  in  the  annexed  Fig.  341, 
in  which  those  curves  are  reproduced,  and  in  which  the  augmented 
initial  volume  is  measured  by  the  extension,  nn'"ri,  of  the  base-line; 

this  extension  comprises  n'  ri", 
the  capacity  of  the  first  cylin- 
der, and  n'"n,  the  capacity  of 
the  intermediate  space,  one- 
third  of  riri"  ;  making  together 
i  */3  times  the  capacity  of  the 
first  cylinder.  At  the  end 

Fig.  34i.-Woolf  Engine  ^Expansion  Curves  for  first  and      °f    *e    Stl?ke    <>f    the    SCCOnd 

second  cylinders.  cylinder,  when  the  piston  has 

arrived  at  m,  the  first  piston 

has  arrived  at  n",  leaving  the  clear  interval  of  intermediate  space,  ri"n, 
open  to  the  second  cylinder,  which,  added  to  the  capacity  nm  of  this 
cylinder,  makes  ri"nm,  the  final  volume  of  the  steam  expanded  into  the 
second  cylinder,  equal  to  3  J  /3  times  the  capacity  of  the  first  cylinder.  Of 
this  total  volume,"  the  section  n'"n",  iT/3  times  the  capacity  of  the  first 
cylinder,  is  the  hypothetical  period  of  admission,  composed  of  the  inter- 
mediate space  ri"n,  and  the  capacity  of  the  first  cylinder  nn" ' ;  the  remain- 
ing section  ri'm  being  the  hypothetical  period  of  expansion.  The  curve  of 
back-pressure,  g'  h'",  on  the  first  piston,  has  the  same  initial  and  final  pres- 
sures as  the  expansion-curve  g4  h". 

In  piecing  the  first  and  second  diagrams,  to  form  the  combined  diagram, 
Fig.  340,  the  triangular  area  of  positive  pressure  on  the  first  piston,  g*gsh'", 
is  replaced  by  the  equal  triangular  area  g4  gs  h",  and  thus  the  united 
work  of  the  two  cylinders  is  indicated  by  the  seven-sided  diagram  cdg" 
g5  h"m  n. 

The  effect  of  the  intermediate  fall  of  pressure  in  reducing  the  performance 
of  the  expanding  steam,  is  clearly  shown  by  the  combined  diagram,  Fig.  340, 
in  which  the  section  g"  h  of  the  normal  expansion-curve  dg"  /i,  is  replaced 
by  the  lower  expansion-curve  g5  h",  with  the  vertical  line  g"gs  denoting  the 
intermediate  fall  of  pressure.  The  four-sided  area  g"g*h"h  expresses  the 
net  loss  of  useful  expansive  work  caused  by  the  intermediate  fall :  being  the 
balance  of  loss  after  deducting  the  gain  by  the  reduction  of  back-pressure 
on  the  first  piston,  measured  by  the  area  g"g5  ti"  h',  from  the  loss  of  pres- 
sure on  the  second  piston,  measured  by  the  area,  gg4  h"  h,  shown  in  both 
the  Figs.  339  and  340. 

The  loss  of  work  by  expansion  of  steam  in  the  Woolf  engine,  into  an 
intermediate  space  between  the  first  and  second  cylinders,  may  be  shown 
and  calculated  in  the  same  way  for  other  volumes  of  intermediate  space — 
say,  one-half  more,  and  as  much  more  as  the  capacity  of  the  first  cylinder. 
Take" all  four  cases,  as  follows: — 


INTERMEDIATE  EXPANSION  IN  THE  RECEIVER-ENGINE.        857 

Intermediate  Space.  Ratios  of  Expansion.  Combined  Ratio. 

istcaser-Nil (  ist  cylinder  r  to  3 

(ad       „       i  to  3  i  to  9 

ad  case:-  V3  capacity  of  ISt  cylinder  i  "tcylinder  '  '°  3 

)  2d       „       i  to  2.5         i  to  7.5 

ist  cylinder  i  to  3 

2d       „       i  to  2  V3      i  to  7 

i  to  6 


4thcase:-i  |  ist  cylinder  i  to  3 

\2d  „  I  tO  2 


Applying  these  ratios  to  the  initial  steam  admitted  to  the  first  cylinder,  the 
total  initial  work  is,  as  before,  126  foot-pounds,  and  the  total  net  work  for 
one  stroke  of  the  two  cylinders  is  as  follows  :l — 

Foot-pounds. 

ist  case: — 126  x  (i  +hyp  log  9)  or  3.1972  =  402.8472 
2d  case: — 126  x  (i  +  hyp  log  7.5)  or  3.0149  =  379.8774 
3d  case: — 126  x  (i  +hyp  log  7)  or  2.9459  =  371.1834 
4th  case: — 126  x  (i  +hyp  log  6)  or  2.7918  =  351.7668 

INTERMEDIATE  EXPANSION  IN  THE  RECEIVER-ENGINE. 

With  respect  to  the  loss  by  intermediate  expansion  and  fall  of  pressure 
in  the  receiver-engine,  take  examples  based  on  the  same  data  as  have  been 
applied  to  the  discussion  of  the  Woolf  engine;  and  suppose,  in  the  first 
instance,  that  the  steam  is  expanded  in  the  receiver  into  iJ/3  times,  or 
four-thirds  of,  its  volume  when  exhausted  from  the  first  cylinder,  the  pres- 
sure being  proportionally  reduced  to  three -fourths  of  the  final  pressure  in 
the  first  cylinder,  prior  to  its  being  admitted  into  the  second  cylinder. 
With  this  modification,  the  action  of  the  steam  is  represented  diagram- 
matically  in  Figs.  342  and  343  annexed;  the  first  for  the  two  cylinders 
separately,  the  second  being  the  combined  diagram.  These  diagrams  are 
fundamentally  the  same  as  the  ideal  or  normal  diagrams,  Figs.  337  and  338, 
page  853,  constructed  for  the  receiver-engine  without  any  intermediate 
fall  of  pressure,  and  the  same  letters  of  reference  apply  to  the  same  parts. 
The  admission-line  cdt  one-third  of  the  stroke,  and  the  expansion-curve  dg, 
for  the  first  cylinder,  Fig.  337,  are  the  same  as  in  the  normal  diagram, 
showing  an  expansion  of  three  times;  the  initial  pressure,  oc,  being  63  Ibs. 
per  square  inch,  and  the  final  pressure,  Ig,  being  equal  to  21  Ibs.  on  the 
square  inch  area  of  piston.  At  the  end  of  the  stroke,  the  pressure  of  the 
steam,  as  it  is  exhausted  into  the  receiver,  falls  one-fourth,  to  15.75  Ibs., 
measured  by  lg't  which  gives  the  level  of  the  back  pressure  on  the  first 
piston,  g'h',  parallel  to  gh.  The  steam  exhausted  from  the  first  cylinder  is 
at  the  same  time  expanded  to  i*/3  times  the  capacity  of  the  cylinder.  A 
volume  of  steam  ix/3  times  the  first  cylinder  must,  therefore,  be  admitted 

1  These  calculations,  as  well  as  others  which  are  quoted  in  this  section  on  compound 
engines,  are  detailed  at  length  in  a  work  on  the  steam  engine,  by  the  author,  now  in 
course  of  preparation  h 


858 


STEAM   ENGINE — COMPOUND   CYLINDERS. 


to  the  second  cylinder,  of  the  reduced  pressure  15.75  Ibs.  per  square  inch. 
The  length  h'  /',  set  off  on  the  line  h'g',  the  measure  of  the  enlarged  volume, 
is  the  period  of  admission  into  the  second  cylinder,  and  it  is  ix/3  times  hi, 
or  iI/3  thirds  of  the  length  of  the  stroke  h'g',  or  2^  feet.  The  point  /',  in 
fact,  obviously  lies  in  the  normal  curve  of  expansion  ik;  and  from  this 
point  to  the  end  of  the  stroke,  at  k,  the  two  curves  of  expansion,  namely, 
the  normal  curve  ik,  and  the  new  curve  i'k,  so  far  as  it  extends,  are  iden- 
tical, with  a  terminal  pressure,  Ik,  of  7  Ibs.  per  square  inch.  The  outline 
of  the  combined  diagram  is,  then,  cdini'klo. 


-SD 


Fig.  342.— Receiver-engine  .-—Diagrams  showing 
Intermediate  Fall  of  Pressure. 


Fig.  343. — The  same  diagrams  reduced  and 
combined. 


The  rate  of  expansion  in  the  second  cylinder,  according  to  this  distribu- 
tion, is  not  so  high  as  in  the  first  cylinder — the  initial  volume  being 
ir/3  times  the  first  cylinder,  and  the  final  volume  being  the  capacity  of  the 
second  cylinder,  or  3  times  the  first  cylinder.  The  ratio  of  expansion  is, 

therefore,  — =t-  =2.25. 

On  the  first  diagram,  Fig.  342,  it  appears  that,  whilst  there  is  a  gain  of 
net  work  to  the  first  cylinder  when  compared  with  the  normal  performance, 
measured  by  the  drop  of  pressure,  gg',  for  the  whole  stroke  gh;  there  is, 
on  the  contrary,  a  loss  of  work  to  the  second  cylinder,  measured  by  the 
same  drop  of  pressure,  hh',  for  the  area  hii'h'.  The  net  loss  is  directly 
indicated  on  the  combined  diagram,  Fig.  343,  in  which  the  gain  in  the  first 
cylinder  is  measured  by  the  rectangle  hinh',  and  the  loss  in  the  second 
cylinder  by  the  trapezoid  hii'h'.  The  difference  of  these,  the  small  tri- 
angular area  ii'n,  is  the  measure  of  the  net  loss. 

Taking  four  cases  for  comparison,  corresponding  to  those  calculated  for 
the  Woolf  engine,  the  results  of  calculation  are  as  follows  z1 — The  augmented 
initial  volumes  for  expansion  in  the  second  cylinder,  and  the  actual  ratios 
of  expansion  in  the  two  cylinders,  are, — 

1  These  calculations,  as  well  as  others  which  are  quoted  in  this  section  on  compound 
engines,  are  detailed  at  length  in  a  work  on  the  steam  engine,  by  the  author,  now  in 
course  of  preparation. 


WORK  OF  THE  WOOLF  ENGINE. 

859 

ist  case 

Augmented  Initial 
Volume  in  Parts  of  the                  Ratios  of  Expansion.                  Combined  Ratio. 
First  Cylinder. 

(  i  st  cylinder,...  i  to  3 

'  (  2d      do.     ...i 

to  3 

i  to  9 

2d  case... 
3d  case... 
4th  case 

.   1  1/3  .             fist  cylinder,...  i 
(  2d      do.     ...i 

to  3 
to  2.25 

i  to  6.75 
i  to  6 

i  to  4.5 

i#                /istcylinder,...i 

to  3 

tO  2 

to  3 
to  1.5 

'  (  2d      do.     ...i 
(  i  st  cylinder,...  i 

•'(2d      do.     ...i 

The  net  works  are  calculated  in  terms  of  the  initial  work,  126  foot-pounds, 
and  the  combined  ratios  of  expansion,  with  an  allowance  for  the  net  work 
acquired  by  the  first  cylinder  due  to  the  intermediate  fall  of  pressure:  — 

foot-pounds. 

ist  case:  —  126  x(i  +  hyp  log  9)  or  3.1972  =402.8472 
2d  case:  —  126  x  (i^+hyp  log  6.75)  or  3.1595  =398.0970 
3d  case:—  126  x  (i'/3  +  hyp  log  6)  or  3.1251  -  393.7542 


4th  case:  —  126 


log  4.5)    or  3.0041  =  378.5166 


By  comparison,  it  appears  that  the  loss  of  work  by  intermediate  fall  of 
pressure,  is  less  in  the  receiver-engine  than  in  the  Woolf  engine;  being  only 
6  per  cent,  of  loss  in  the  former,  as  against  12.7  per  cent,  in  the  latter, 
when  the  pressure  falls  to  half  the  final  pressure  in  the  first  cylinder. 

WORK  OF  THE  WOOLF  ENGINE,  WITH  CLEARANCE. 
Let  Figs.  344  and  345  represent  the  diagrams  of  pressure  from  the  first 
and  second  cylinders  of  a  Woolf  engine  having  the  same  dimensions,  pro- 


Fig.  344. — Woolf  Engine : — Diagrams  with 
Clearance. 


Fig.  345. — Woolf  Engine : — The  same  diagrams 
reduced  and  combined. 


portions,  and  letters  of  reference,  as  in  the  preceding  examples,  with  the 
addition  of  a  clearance  at  each  end  of  the  first  cylinder,  measured  by  cc' 


86O  STEAM   ENGINE — COMPOUND  CYLINDERS. 

or  mm',  equal  to  7  per  cent,  of  the  stroke,  or  .42  foot.  The  rectangular 
clearance  space  cc'm'm,  measures  the  passive  work  of  the  clearance,  or  the 
product  of  the  pressure  m  c  by  the  period  of  the  clearance  cc'. 

As  the  steam  is  cut  off  at  a  third,  or  33^3  per  cent,  of  the  stroke,  the 

actual  ratio  of  expansion  is  (see  page  828)   I0°+  t.  =  2.653.     The  initial 

o«5  /3   '    i 

pressure  being  63  Ibs.,  as  before,  the  final  pressure,  ng,  is  — ^-  =  23.75  Iks.; 

"    Do 

and  the  final  volume,  taking  the  working  capacity  of  the  first  cylinder  as  i, 
is  i +  (7  per  cent.),  or  1.07.  If  there  be  no  more  clearance,  or  no  inter- 
mediate space,  between  the  first  and  second  cylinders,  the  initial  volume  for 
expansion  into  the  second  cylinder  is  1.07;  and  the  final  volume  is  3.  The 

ratio  of  expansion  in  it  is,  therefore,  — 3.  =2.804;  and  the  final  pressure, 

23.75  I>07 

m  h,  is     g      =8.47  Ibs.  per  square  inch. 

In  view  of  these  pressures,  it  is  apparent  that,  whilst  the  work  of  admis- 
sion during  the  period  c  d  is  the  same  as  it  was  when  there  was  no  clearance, 
namely,  126  foot-pounds,  the  work  by  expansion  is  greater,  for  the  final 
pressures  are  respectively  as  follows : — 

With  no  Clearance.  With  Clearance. 

In  the  first  cylinder, 21  Ibs.  per  sq.  inch.      23.75  Ibs.  per  sq.  inch. 

In  the  second  cylinder,...   7         „         „  8.47 

If  it  were  practicable  to  construct  the  compound  cylinder  so  that  there 
should  not  be  any  intermediate  clearance,  the  employment  of  a  smaller 
cylinder,  as  a  prefix  to  a  given  cylinder,  for  receiving  the  charge  of  steam 
to  be  expanded  through  both  cylinders,  would  have  the  economical  effect 
of  reducing  the  percentage  of  end-clearance,  measured  in  parts  of  the 
larger  cylinder.  But  it  is  not  practicable  to  do  so;  and  it  remains  to  trace 
the  influence  of  intermediate  space  combined  with  the  initial  clearance  of 
the  first  cylinder,  on  the  action  and  work  of  the  steam  in  the  second  cylinder. 

Take,  as  before,  four  cases,  and  suppose  that  the  volume  of  the  inter- 
mediate space,  including  what  is  technically  the  clearance  of  the  second 
cylinder,  is  a  simple  fraction  of  the  capacity  of  the  first  cylinder  plus  its 
clearance,  7  per  cent,  or  of  1.07  times  the  capacity  of  the  first  cylinder,  as 
follows : — 

for  the  ist,  2d,  3d,  4th  case, 

the  intermediate  spaces  are, 

o,  x/3,  YZ,  i,  part  of  the  capacity  of  the  first 

cylinder  plus  its  clearance;  or  they  are, 

o,  -357?          -535?        I-°7    °f    tne    capacity    of    the    first 

cylinder.  Add  to  these  1.07,  the  capacity  of  the  first  cylinder  plus  its 
clearance;  and  the  sums  are  the  total  initial  volumes  for  expansion  in  the 
second  cylinder, 

1.07,  1-427,  1.605,  2-I4>  times  the  capacity  of  the  first 
cylinder.  Again,  to  the  same  values  of  the  intermediate  space,  add  3,  the 
capacity  of  the  second  cylinder;  and  the  sums  are  the  final  volumes  by 
expansion  in  the  second  cylinder, 

3'°>  3-357>        3-535?        4-°7  times  the  capacity  of  the  first 


WORK   OF   THE   WOOLF   ENGINE.  86l 

cylinder.  The  ratios  of  expansion  in  the  second  cylinder  are  the  quotients 
of  the  final  by  the  initial  volumes : — 

2.804,  2.352,  2. 202,  1.902,  ratios  of  expansion.  The 
intermediate  falls  of  pressure  are,  in  parts  of  the  final  pressure  in  the  first 
cylinder, 

o,  i^,  J/3,  y2  of  the  final  pressure;  or,  putting 

the  final  pressure  equal  to  23.75  Ibs.,  as  was  found,  they  are 

0,  5'94>          7-92,        11.87    IDS-    Per    square    inch.       The 
initial  pressures  for  expansion  in  the  second  cylinder  are, 

1,  24,  2/3J  y*  of  tne  nna-l  pressure  in  the  first 
cylinder;  or 

23-75?        I7-8i,        I5-83,        11.87  Ibs.  per  square  inch;   and  the 
final  pressures  in  the  second  cylinder  are, 

8.47,          7.57,          7.19,          6.24  Ibs.  per  square  inch. 

The  combined  ratios  in  the  four  cases  are  as  follows : — 

ist  case:— ist  ratio  of  expansion, i  to  2.653          COMBINED  RATIO. 

2d  do.  i  to  2.804  i  to  7.438 

2d  case: — ist  ratio  of  expansion, i  to  2.653 

2d  do.  i  to  2.352  i  to  6.241 

3d  case: — ist  ratio  of  expansion, i  to  2.653 

2d  do.  i  to  2.202  i  to  5.843 

4th  case: — ist  ratio  of  expansion, i  to  2.653 

2d  do.  i  to  1.902  i  to  5.046 

The  initial  work  of  the  steam  of  63  Ibs.  total  pressure,  admitted  into  the 
first  cylinder,  for  2  feet  of  the  stroke,  and  with  a  clearance  of  7  per  cent., 
or  .42  feet,  is  as  follows: — 

Work  done  on  the  piston, 63  Ibs.  x  2  feet      =126  foot-pounds. 

Work  done  in  the  clearance,...  63  Ibs.  x  .42  foot  =    26.46     „ 


Total  initial  work  of  the  steam,  63  Ibs.  x  2.42  feet=  152.46     „ 

This  sum  is  the  initial  work  on  which  the  work  by  expansion  is  calculated ; 
whilst  it  is  26.46  foot-pounds  in  excess  of  the  initial  work  done  on  the 
piston.  The  total  work  is,  then,  calculated  as  follows : — 

NET  WORK  IN 

ist  case: — 152.46  x  (i  +hyp  log  7.44)  or  3.0069  =  458.27      FOOT-POUNDS. 
less,  work  in  initial  clearance, 2 6. 46         431.81 

ad  case: — 152.46  x  (i  -i-hyp  log  6.24)  or  2.8310  =  431.47 

less 26.36         405.11 

3d  case: — 152.46  x(i+hyp  log  5.84)  or  2.7647  =  421.35 

less 26.36         394-99 

4th  case: — 152.46  x  (i  +hyp  log  5.05)  or  2.6194  =  399.29 

less 26.36         372.93 


862 


STEAM  ENGINE — COMPOUND  CYLINDERS. 


The  calculations  have  been  made  in  this  form  for  the  sake  of  comparison 
with  those  that  were  made  for  the  work  when  there  was  no  clearance  (page 
857).  They  can  be  made  more  directly  by  means  of  the  formula  (  7  )  at 
page  828.  The  reductions  of  net  work,  in  the  2d,  3d,  and  4th  cases,  are 
successively  6.2,  8.6,  and  13.7  per  cent,  of  the  work  in  the  ist  case. 

WORK  OF  THE  RECEIVER-ENGINE,  WITH  CLEARANCE. 

For  the  work  of  receiver-engines,  with  clearance,  taken  at  7  per  cent.,  at 
each  end  of  the  stroke  of  each  cylinder,  the  annexed  Fig.  346  shows  the 
diagrams  of  pressure,  using  the  same  data  and  letters  of  reference  as  in 
Fig.  342,  p.  858,  with  the  clearance  measured  by  cc',  hh',  or  oo,  equal  to  7  per 
cent,  of  oL  The  steam  being  cut  off  at  x/3d,  the  actual  ratio  of  expansion  in 


the  first  cylinder  is,  as  for  the  Woolf  engine,  p.  860, 


1-  = 


the  final  pressure,  Ig,  is 


5, 
•* 


33  7s  +7 


=  2.653; 


=  23.75  Ibs.,  which  is  also  the  pressure  in  the 


receiver,  when  there  is  no  intermediate  fall  of  pressure.  The  same  is  the 
initial  pressure  oh,  in  the  second  cylinder,  with  the  clearance  oo'  .  The 
volume  admitted  into  the  second  cylinder  is  equal  to  the  capacity  of  the 


Fig.  346. — Receiver-engine: — Diagrams  with 
Clearance. 


Fig.  347. — Receiver-engine : — The  same  diagrams 
reduced  and  combined. 


first  cylinder  plus  its  clearance,  or  to  one-third  of  the  capacity  of  the 
second  cylinder  plus  its  clearance;  that  is,  to  one-third  of  107  per  cent., 
or  352/3  per  cent.,  which  consists  of  the  clearance,  7  per  cent.,  and  (352/s 
-7  =  )  282/3  per  cent,  of  the  stroke  of  the  second  cylinder.  The  steam 
admitted  into  the  second  cylinder  thus  occupies  less  than  one-third  of  the 
stroke,  by  42/3  per  cent.,  as  indicated  by  the  length  of  the  period  of  admis- 
sion, hi,  in  the  diagram.  As  the  steam  is  expanded  from  the  capacity  of  the 
first  cylinder  plus  its  clearance,  to  that  of  the  second  cylinder  plus  its  clear- 
ance, the  ratio  of  expansion  in  the  second  cylinder,  is  necessarily  equal  to 

the  ratio  of  the  capacities  of  the  two  cylinders,  which  is  3 ;  and  T°°  '  =  3 ; 
and  the  final  pressure,  Ik,  is  23^5  =  7.92  Ibs.  per  square  inch. 


WORK  OF  THE  RECEIVER-ENGINE.  863 

The  combined  diagram,  Fig.  347,  shows  a  dislocated  expansion-line,  in  two 
parts :  dg  for  the  first  cylinder,  and  ik  for  the  second  cylinder.  The  first 
part,  dg,  is  extended  continuously  to  the  end  of  the  stroke  at  k',  and  shows 
the  loss  of  work  caused  by  the  excess  of  the  volume  of  clearance  of  the 
second  cylinder  over  that  for  the  first,  as  measured  by  the  area  of  the  strip 
igk'k. 

For  the  other  three  cases,  of  intermediate  falls  of  pressure,  respectively 
}^  th,  J/3d,  and  J^  of  the  final  pressure  in  the  first  cylinder,  the  relations 
are  as  follows: — 

For  the     ist,  2d,  3d,  4th  case 

the  augmented  initial  volumes  for  expansion  in  the  second  cylinder  are, 

i,  iJ/3,  i^,  2        times  the  capacity  of  the 

first  cylinder  plus  the  clearance;  or  they  are 

1.07,         1.427,         1.605,  2<I4  times  the  capacity  of  the 

first  cylinder.     The  final  volumes  by  expansion  in  the  second  cylinder  are 
equal  to  the  capacity  of  the  second  cylinder  plus  its  clearance,  or  to 

3.21,  3.21,  3.21,          3.21   times  the  capacity  of  the 

first  cylinder;  and  the  ratios  of  expansion  in  the  second  cylinder  are 

3.00,  2.25,  2.00,  1.50. 

The  intermediate  falls  of  pressure  are 

o,  i^,  x/3,  YZ    the    final   pressures  in    the 

first  cylinder;  and  they  are  actually 

o,  5.94,  7.92,         11.87  Ibs.  per  square  inch.     The 

pressure  in  the  reservoir,  and  the  initial  pressure  in  the  second  cylinder,  are 
23-75,         I7-8i,         15.83,         11.87  Ibs.  per  square  inch;   and 
the  final  pressures  in  the  second  cylinder  are 

7.92,  7.92,  7.92,  7.92  Ibs.  per  square  inch. 

To  calculate  the  works  done  in  the  four  cases : — First,  the  normal  net 
work  of  the  first  cylinder,  above  the  level  of  the  terminal  pressure,  hig, 
common  to  all  the  cases.  The  total  initial  work  done  on  the  piston  is,  as  was 
found  (p.  86 1),  126  foot-pounds,  and  the  work  of  the  clearance  26.46  foot- 
pounds. The  sum,  152.46  foot-pounds,  is  the  initial  work  on  which  the 
work  by  expansion  2.653  times  is  calculated;  thus, 

Foot-pounds. 

152.46  x(i+hyp  log  2.653)  or  1.9757  =  301.21 
Less  work  in  initial  clearance, 26.46 


Total  work  on  piston,  above  the  vacuum  line, 

Deduct  work  of  back  pressure,  oh  x  hg,  or  | 

23. 75  foot-pounds  x  3  inches  x  2  feet, J 

Normal  net  work  on  the  first  piston, 132.25 

Next,  the  works   gained  to  the  first   piston  by  the  intermediate  falls  of 
pressure. 

For  the       ist,  2d,  3d,  4th  cases, 

the  work  is  expressed  by  the  rectangles, 

o,         5.94  Ibs.  x  3  x  2,     7.92  Ibs.  x  3  x  2,     1 1.87  Ibs.  x  3  x  2, 
or,  o,  35-64,  47-52,  71.22  foot-pounds. 


864  STEAM   ENGINE — COMPOUND   CYLINDERS. 

Third,  for  the  work  done  in  the  second  cylinder,  the  initial  work  for  expan- 
sion must  be  the  same  as  that  for  the  first,  namely,  152.46  foot-pounds; 
there  being  the  same  quantity  of  steam.  The  clearance  is  .42  foot,  and  the 
passive  work  in  the  clearance, 

for  the     ist,  2d,  3d,  4th  case, 

is  23.75  Ibs.  x3x.42,     17-81  x  3  x. 42,     15. 83  x  3  x. 42,     n.87x3x.42, 

or  29.92,  22.44,  T9-95>  x4-95  foot-pounds. 

The  works  in  the  second  cylinder  are  calculated  from  these  data,  with  the 
ratios  of  expansion,  as  follows : — 

Foot-pounds. 

ist  case:— 152.46  x  (i+hyp  log  3)= 3J9-93 

less  the  work  in  clearance,  29.92         290.01 

2d  case: — 152.46  x  (i  +hyp  log  2.25)  = 276.10 

less  the  work  in  clearance,  22.44         253-66 

3d  case: — 152.46  x  (i  +hyp  log  2)  = 258.13 

less  the  work  in  clearance,  !9-95         238.18 

4th  case: — 152.46  x  (i  +hyp  log  1.5)=  214.28 

less  the  work  in  clearance,  I4-95         I99'33 


The  total  net  work  in  both  cylinders  for  one  stroke,  is  found  by  adding 
together  the  three  portions  of  work  for  each  case : — 

NET  WORK  RATIO 

in  Foot-pounds,     of  Net  Work. 

i  st  case : — first  cylinder  above  final  pressure,     132.25 
intermediate, o.oo 

132.25 
second  cylinder, 290.01     422.26,    as    roo 

2d  case: — first  cylinder  above  final  pressure,     132.25 
intermediate, 35-64 

167.89 
second  cylinder, 253.66     421.55,   as   99.8 

3d  case: — first  cylinder  above  final  pressure,     132.25 
intermediate, 47-52 


179.77 
second  cylinder, 238.18     417.95,   as   99.0 


4th  case:— first  cylinder  above  final  pressure,    132.25 
intermediate, 71.22 

203.47 
second  cylinder, J99-33     402.80,   as   95.4 


WORK  OF  THE  RECEIVER-ENGINE.  865 

Here  it  is  seen  that  the  reduction  of  the  quantity  of  work  performed  for 
one  stroke  of  the  pistons,  by  intermediate  falls  of  pressure,  does  not 
exceed  i  per  cent,  when  the  fall  amounts  to  one-third  of  the  final  pressure 
in  the  first  cylinder;  and  it  is  less  than  5  per  cent,  even  when  the  fall 
amounts  to  half  the  final  pressure.  The  proportional  reduction  of  work 
is  something  less  with  clearance, — as  in  this  instance, — than  without  clear- 
ance, as  exemplified  at  page  859. 

The  work  for  one  stroke  may  be  calculated  in  terms  of  the  combined 
ratios  of  expansion  for  the  two  cylinders ;  making  allowances  for  the  loss  by 
clearance,  and  the  gain  to  the  first  cylinder  by  the  intermediate  fall  of 
pressure.  The  total  initial  work  for  expansion  is,  as  was  found  (page  863), 
152.46  foot-pounds;  and  the  ratios  of  expansion  are  as  follows: — 

ist  case:— first  ratio  of  expansion, i  to  2.653      COMPOUND  RATia 

second          do.  i  to  3.000        i  to  7.959 

2d  case: — first  ratio  of  expansion, i  to  2.653 

second          do.  i  to  2.250        i  to  5.969 


3d  case: — first  ratio  of  expansion, , i  to  2.653 

second          do.  i  to  2.000        i  to  5.306 

4th  case: — first  ratio  of  expansion, i  to  2.653 

second          do.  i  to  1.500        i  to  3.979 


With  respect  to  the  first  case,  it  is  obvious  from  an  inspection  of  the 
combined  diagram,  Fig.  347,  page  862,  that  the  calculation  of  the  work  in 
terms  of  the  initial  work  for  expansion,  and  the  total  ratio  of  expansion, 
covers  the  whole  area  of  the  diagram,  including  the  clearance-areas,  thus : — 

152.46  x  (i  +hyp  log  7.959)  or  3.0743  =  468.71  foot-pounds. 

From  this  is  to  be  deducted  the  work  of  the  initial  clearance  in  the  first 
cylinder,  26.46  foot-pounds,  represented  by  the  area  cc '  o"  o;  and  also  the 
work  of  the  excess  of  clearance  in  the  second  cylinder,  over  and  above 
that  of  the  clearance  in  the  first  cylinder,  calculated  on  che  pressure  in  the 
receiver.  As  the  clearance  of  the  second  cylinder  is,  like  that  of  the  first, 
7  per  cent,  of  the  stroke,  or  .42  feet,  the  volumes  of  the  respective  clear- 
ances are  in  the  ratio  of  the  capacities  of  the  cylinders,  or  as  3  to  i,  and 
are  measured  by  the  spaces  oo'  and  oo"  on  the  diagram.  The  clearance 
steam  of  the  first  cylinder,  therefore,  when  transferred  to  the  second  cylinder, 
fills  only  one-third  of  its  clearance  space,  measured  by  oo",  and  of  the 
pressure  oh;  and  additional  steam  from  the  receiver,  of  the  same  pressure, 
is  required  to  fill  the  remaining  two-thirds  of  the  clearance  of  the  second 
cylinder,  or  .42  x  2/$  =  .28  foot,  measured  by  o" o' .  The  work  of  the  two 
clearances,  to  be  deducted,  is,  then,  as  follows : — 

Work  of  clearance  of  first  cylinder,  cc' o" o*  63  Ibs.  x   )         ,    ,  r  , 

3  in.  x  .14  foot,  or  63  Ibs.  x  ,'  in.  x  .42  foot }  =  26'^6  foot-pounds. 

Excess  of  clearance  of  second  cylinder,  h' o' o" ,   \  , 

23.75  Ibs.  x3  in.x.28  foot (  = 

Work  of  clearances, 46.41         do. 

55 


STEAM   ENGINE — COMPOUND   CYLINDERS. 

The  gross  work  of  the  diagram  being,  as  above, 468.71  foot-pounds. 

The  work  of  clearances  to  be  deducted  is 46.41         do. 

Net  work  for  one  stroke, 422.30         do. 

For  the  2d  case,  the  gross  work  is — 

152.46  x  (i  +hyp  log  5.969)  or  2.7866  =  424.84  foot-pounds. 
Deduct  the  work  of  the  clearances,  as  above,. . .  49.41         do. 


378.43         do. 

To  this  is  to  be  added  the  compensatory  gain  by  the  fall  of  the  pressure 
in  the  reservoir,  which  is  equal  to  5.94  Ibs.  per  square  inch.  It  is  to  be 
multiplied  into  the  length  of  the  stroke  of  the  first  cylinder,  for  the  reduction 
of  back-pressure,  and  the  clearance  of  the  second  cylinder  for  the  saving  of 
passive  work  in  the  clearance.  The  stroke  is,  as  reduced,  2  feet;  the 
clearance  is  .42  foot,  and  the  sum  of  these  is  2.42  feet.  Then  the  work 
of  the  gain  is, 

5.94  Ibs.  x  3  in.  x  2.42  feet         =   43.12  foot-pounds, 
which  is  to  be  added  to  the  remainder,  above,. . .378.43         do. 

making  the  net  work  for  one  stroke 421.55         do. 

The  calculations  of  net  work  are  similarly  performed  for  the  3d  and  4th 
cases,  and  they  are  all  brought  together  for  the  four  cases  for  comparison, 
as  follows : — 

Foot-pounds. 

ist  case: — 152.46  x  (i  +hyp  log  7.959)  or  3.0743      =  468.71 

deduct  for  clearances, 46.41      422.30 


2d  case: — 152.46  x  (i  +  hyp  log  5.969)  or  2.7866      =424.84 
deduct  for  clearances, 46.41 

378.43 
add  for  fall  of  receiver-pressure  5.94  Ibs.  x 

3  in.  x  2.42  feet, 43.12      421.55 

3d  case: — 152. 46  x  (i  +  hyp  log  5.306)  or  2.6688      =406.87 
deduct  for  clearances, 46.41 

360.46 
add  7.92  Ibs.  x  3  in.  x  2.42  feet 57-5°      4J7.96 


4th  case: — 152.46  x  (i  +  hyp  log  3.979)  or  2.3810     =363.01 
deduct  for  clearances, , 46.41 

316.60 
add  11.87  Ibs.  x  3  in.  x  2.42  feet, 86.18      402.78 

The  net  works  thus  obtained  are  the  same  as  those  that  were  deduced  from 
the  diagrams  treated  separately  (page  864). 


WORK  IN  WOOLF  AND   RECEIVER   ENGINES  COMPARED.       867 


COMPARATIVE  WORK  OF  STEAM  IN  THE  WOOLF  ENGINE  AND  THE 
RECEIVER-ENGINE. 

It  has  been  shown  that  the  work  of  steam  in  the  compound  engine,  when 
there  is  no  clearance  and  no  intermediate  fall  of  pressure,  is  the  same  in 
amount,  whether  performed  on  the  Woolf  system  or  the  receiver-system; 
but  that,  when  there  is  an  intermediate  fall  of  pressure,  with  the  enlarge- 
ment of  volume  by  which  it  is  accompanied,  the  work  done  on  the  receiver- 
system  is  greater  than  that  on  the  Woolf  system;  that  is  to  say,  the  reduction 
of  work  by  fall  of  pressure  is  less  rapid  with  the  receiver  than  on  the  Woolf 
system.  This  is  apparent  in  the  following  comparative  note  of  the  perfor- 
mances, from  which  it  also  appears  that,  whilst  the  receiver-engine  does 
more  work,  it  expands  the  steam  to  a  less  number  of  times  than  the  Woolf 
engine : — 

WOOLF  ENGINE  (no  clearance).  RECEIVER-ENGINE  (no  clearance). 

Ratio  of  Expansion.         Net  Work.  Ratio  of  Expansion.  Net  Work. 

ist  case: — 9.0  402.85  ft.-pds 9.0  402.85  ft.-pds. 

2d  case: — 7.5  379-88  6.75 398.10 

3d  case:— 7.0  37*-i8  6.0  393-75 

4thcase:— 6.0  35x-77  4-5   378.52 

In  fact,  it  was  found  that  the  reduction  of  work  in  the  4th  case,  when  the 
pressure  fell  to  one-half  intermediately,  was  about  13  per  cent,  in  the 
Woolf  engine,  and  only  6  per  cent,  in  the  receiver-engine.  The  apparent 
anomaly  that  the  engine  in  which  the  greater  expansion  of  steam  takes 
place,  performs  a  less  net  work,  is  explained  by  the  fact  that  in  the  former, — 
the  Woolf  engine, — much  of  the  initial  work  of  the  steam  for  the  second 
cylinder  is  lost  in  the  intermediate  space;  whilst,  in  the  latter, — the  receiver- 
engine, — there  is  no  loss  of  this  kind. 

By  the  addition  of  clearance  to  each  cylinder,  equal  to  7  per  cent,  of  the 
stroke  at  each  end,  the  actual  ratios  of  expansion  are  sensibly  reduced 
as  compared  with  the  ratios  without  clearance, — in  the  Woolf  engine,  from 
9  to  7.4  when  there  was  no  intermediate  fall  of  pressure,  and  from  6  to  5 
when  there  was  a  fall  of  one-half.  In  the  receiver-engine,  the  reduction  of 
ratio  is  less  than  in  the  Woolf  engine: — it  is  from  9  to  8  when  there  is 
no  fall,  and  from  4.5  to  4  when  there  is  a  fall  of  one-half.  Thus,  the 
effect  of  the  addition  of  clearance  is  clearly  to  reduce  the  net  expansion. 
At  the  same  time,  it  increases  the  net  work  done,  as  appears  from  the 
following  statement: — 

WOOLF  ENGINE— 7  %  clearance.  RECEIVER-ENGINE— 7  %  clearance. 

Ratio  of  Expansion.  Net  Work.  Ratio  of  Expansion.  Net  Work. 

istcase: — 7.44  431.71  ft.-pds 7.96  422.3oft.-pds. 

2d  case:— 6.24  405.11     „       5-97  421.55     » 

3d  case:— 5.84  394-99     »       5-3*  417.96     „ 

4thcase:— 5.05  372-93     »       3-98  402.78     „ 

Taking  only  the  4th  case : — in  the  Woolf  engine,  the  net  work  is  raised,  by  the 
addition  of  clearance,  from  352  to  373  foot-pounds;  and,  in  the  receiver- 
engine,  from  378.5  to  403  foot-pounds. 

With  clearance,  as  without  clearance,  it  is  found  that  the  reduction  of 
net  work,  by  intermediate  fall  of  pressure,  is  less  in  the  receiver-engine, 


868  STEAM   ENGINE — COMPOUND   CYLINDERS. 

where  it  is  only  4^  per  cent.,  with  clearance,  than  in  the  Woolf  engine, 
where  it  amounts  to  about  14  per  cent,  when  the  pressure  falls  interme- 
diately one-half. 

As  the  combined  ratios  of  expansion  in  the  receiver-engine  are  less,  for 
each  case,  than  in  the  Woolf  engine;  so  the  terminal  pressures  of  the 
expanded  steam  in  the  second  cylinder,  on  passing  to  the  condenser,  are 
greater  in  the  receiver-engine  than  in  the  Woolf  engine : — 

For  the  ist,  2d,  3d,  4th  case,  with  7  per  cent,  clearance, 

the  terminal  pressures  in  the  second  cylinder  are,  for  the  WToolf  engine, 

8.47,  7.57,          7.19,         6.24  Ibs.  per  square  inch, 

and  for  the  receiver-engine, 

7.92,  7.92,          7.92,         7.92. 

In  the  first  case,  the  terminal  pressure  in  the  Woolf  engine  is  greater 
than  in  the  receiver-engine;  for  there  was  no  intermediate  space  assumed 
in  the  former,  whilst  clearance-space  for  the  second  cylinder  was  assumed 
in  the  latter;  but,  in  the  other  cases,  the  terminal  pressures  in  the  former 
fall  consecutively  below  that  of  the  first  case.  They  also  fall  below  those 
of  the  latter,  which  remain  constant  for  all  the  cases.  This  constancy  of 
terminal  pressure  in  the  second  cylinder  of  the  receiver-engine,  simply 
follows  from  the  fact  that  the  terminal  volume  of  the  expanded  steam  is 
always  the  same, — that  of  the  second  cylinder  plus  the  clearance, — what- 
ever be  the  intermediate  fall  of  pressure;  whilst  in  the  Woolf  engine,  on  the 
contrary,  the  terminal  volume  is  equal  to  that  of  the  second  cylinder, 
increased  by  the  volume  of  the  intermediate  space,  and  the  terminal 
pressure  must  be  less  as  the  terminal  volume  is  increased. 

As  the  terminal  pressure  in  the  receiver-engine  is  thus  shown  to  be,  in 
all  practical  cases,  greater  than  in  the  Woolf  engine,  other  conditions  being 
the  same,  it  directly  follows  that  the  work  performed  in  expanding  from  a 
given  initial  pressure  to  the  several  terminal  pressures,  must  be  greater 
in  the  receiver-engine  than  in  the  Woolf  engine. 

It  may  be  gathered  from  these  arithmetical  deductions  that  the  receiver- 
engine  is  an  elastic  system  of  compound  engine,  in  which  considerable 
latitude  is  afforded  for  adapting  the  pressure  in  the  receiver  to  the  demands 
of  the  second  cylinder,  without  considerably  diminishing  the  effective  work 
of  the  engine.  In  the  Woolf  engine,  on  the  contrary,  it  is  clearly  of  much 
importance  that  the  intermediate  volume  of  space  between  the  first  and 
second  cylinders,  which  is  the  cause  of  an  intermediate  fall  of  pressure, 
should  be  reduced  to  the  lowest  practicable  amount. 

Supposing  that  there  is  no  loss  of  steam  in  passing  though  the  engine,  by 
cooling  and  condensation,  it  is  obvious  that  whatever  steam  passes  through 
the  first  cylinder,  must  also  find  its  way  through  the  second  cylinder, 
neither  more  nor  less.  By  varying,  therefore,  in  the  receiver-engine,  the 
period  of  admission  in  the  second  cylinder,  and  thus  also  the  volume  of 
steam  admitted  for  each  stroke,  the  steam  will  be  measured  into  it  at  a 
higher  pressure  and  of  a  less  bulk,  or  at  a  lower  pressure  and  of  a  greater 
bulk:  the  pressure  and  density  naturally  adjusting  themselves  to  the 
volume  permitted  to  escape  from  the  receiver  into  the  cylinder.  With  a 
sufficiently  restricted  admission,  the  pressure  in  the  receiver  may  be  main- 
tained at  the  pressure  of  the  steam  as  exhausted  from  the  first  cylinder. 
On  the  contrary,  with  a  wider  admission,  the  pressure  in  the  receiver  may 


FORMULAS  AND   RULES   FOR   COMPOUND   ENGINES.  869 

fall  or  "  drop  "  to  three-fourths,  or  even  one-half  of  the  pressure  of  the 
exhaust  steam  from  the  first  cylinder. 

There  is  a  means  of  counterbalancing  the  loss  of  performance  by  inter- 
mediate fall  of  pressure,  by  so  enlarging  the  second  cylinder  as  to  effect 
the  same  ultimate  ratio  of  expansion  behind  the  pistons,  as  would  be 
effected  in  the  originally  designed  engine  if  there  were  no  intermediate 
fall.  For  example,  when  the  capacities  of  the  first  and  second  cylinders 
are  as  i  to  3,  and  the  steam  is  cut  off  in  each  at  one-third  of  the  stroke, 
without  any  intermediate  fall,  the  steam,  if  there  be  no  clearance,  is  expanded 
into  nine  times  its  initial  volume.  But,  when  there  is  an  intermediate  fall 
of  pressure,  of,  say,  one-fourth  of  the  final  pressure  in  the  first  cylinder, 
involving  an  increase  of  volume  of  steam  in  the  ratio  of  3  to  4,  the  second 
cylinder  must  be  correspondingly  enlarged  in  the  ratio  of  3  to  4,  in  order 
to  contain  the  charge  of  steam  for  expansion,  when  cut  off,  as  before,  at 
one-third  of  the  stroke.  By  such  enlargement  of  the  second  cylinder,  in 
the  ratio  of  the  intermediate  enlargement  of  the  steam,  the  same  ultimate 
ratio  of  expansion  is  secured,  and  an  equivalent  performance  is  effected. 
Such  a  remedy,  when  specially  applied  for  the  purpose  of  counterbalancing 
ineffective  expansion  of  steam,  involves  the  employment  of  enlarged 
cylinders,  and  entails  the  objections  of  increased  weight,  bulk,  and  cost 
of  machinery.  It  would  be  more  useful  as  a  remedy,  when  applied  to 
the  Woolf  engine,  than  to  the  receiver-engine. 

FORMULAS  AND  RULES  FOR  CALCULATING  THE  EXPANSION  AND  THE 
WORK  OF  STEAM  IN  COMPOUND  ENGINES. 

In  view  of  the  preceding  discussions  of  the  expansive  working  of  steam 
in  compound  cylinders,  the  following  algebraic  symbols  are  used : — 

a  =  the  area  of  the  first  cylinder  in  square  inches. 
a'  =  ihe  area  of  the  second  cylinder  in  square  inches. 
r  =  the  ratio  of  the  area  of  the  second  cylinder  to  that  of  the  first  cylinder. 
L  =  the  length  of  the  stroke  in  feet,  supposed  to  be  the  same  for  both  cylinders. 
/=the  period  of  admission  to  the  first  cylinder,  in  feet,  excluding  clearance. 
c=ihe  clearance  at  each  end  of  the  cylinders,  in  parts  of  the  stroke,  in  feet. 
L'  =  the  length  of  the  stroke  plus  the  clearance,  in  feet. 
/'  =  the  period  of  admission  plus  the  clearance,  in  feet. 
s  =  the  length  of  a  given  part  of  the  stroke  of  the  second  cylinder,  in  feet. 
P  =  the  total  initial  pressure  in  the  first  cylinder,  in  pounds  per  square  inch, 

supposed  to  be  uniform  during  admission. 

P'  =  the  total  pressure  at  the  end  of  the  given  part  of  the  stroke,  s. 
p  =  the  average  total  pressure  for  the  whole  stroke. 
R  =  the  nominal  ratio  of  expansion  in  the  first  cylinder,  or  L-?-/. 
R'  =  the  actual  ratio  of  expansion  in  the  first  cylinder,  or  L' -=-/'. 
R"  =  the  actual  combined  ratio  of  expansion  behind  the  pistons,  in  the  first  and 

second  cylinders  together. 

R'"  =  the  actual  ratio  of  expansion,  or  number  of  volumes  into  which  the  steam 

occupying  the  first  cylinder  at  the  end  of  the  stroke,  is  expanded  in  the 

second  cylinder  at  the  end  of  any  part  of  the  return  stroke,  s : — the 

special  initial  volume,  or  the  capacity  of  the  first  cylinder,  being  =  i. 

n  =  the  ratio  of  the  final  pressure  in  the  first  cylinder  to  any  intermediate  fall 

of  pressure  between  the  first  and  second  cylinders. 

N  =  the  ratio  of  the  volume  of  the  intermediate  space  in  the  Woolf  engine, 
reckoned  up  to,  and  including  the  clearance  of,  the  second  piston,  to 
the  capacity  of  the  first  cylinder  plus  its  clearance. 
•z£/  =  the  whole  net  work  in  one  stroke,  in  foot-pounds. 


8/0  STEAM   ENGINE — COMPOUND  CYLINDERS. 

Formulas  and  rules  may  be  constructed  on  the  basis  of  the  combined 
ratios  of  expansion  behind  the  two  pistons : — the  combined  ratio  being  the 
product  of  the  actual  ratios  of  expansion  in  the  first  and  the  second  cylinders. 
When,  as  usually  happens  in  practice,  intermediate  expansion  takes  place 
between  the  cylinders,  if  the  ratio  of  this  expansion  be  multiplied  into  the 
combined  ratio  of  expansion  behind  the  pistons ;  or,  if  the  three  individual 
ratios  of  expansion  in  the  first  and  second  cylinders,  and  in  the  intermediate 
space,  be  multiplied  together,  the  product  is  the  ratio  of  total  expansion  of 
the  steam  within  the  engine,  to  the  end  of  the  stroke  of  the  second  cylinder, 
when  it  is  discharged  into  the  condenser.  For  example,  if  the  steam  be 
expanded  to  three  times  its  volume  in  the  first  cylinder,  twice  in  the  second 
cylinder,  and  one-and-a-half  times  in  the  intermediate  space ;  the  combined 
ratio  of  expansion  behind  the  pistons  is  the  product  of  the  first  and  second 
of  these ;  that  is, — 

In  first  cylinder, i  to  3,      or  3 

In  second  cylinder, „ i  to  2,      or  2 


Combined  ratio  of  expansion  behind  )  ,  ,. 

pistons, }    Ito6>      Or6 

Intermediate  expansion, i  to  1.5,  or  1.5 


Ratio  of  total  expansion, i  to  9,      or  9 

Or,  the  individual  ratios  may  be  placed  consecutively  thus : — 

In  first  cylinder, i  to  3,      or  3 

Intermediate  expansion, i  to  1.5,   or  1.5 

In  second  cylinder, i  to  2,      or  2 


Ratio  of  total  expansion, i  to  9,      or  9 

Conversely,  when  the  total  expansion  is  given,  the  expansion  behind  the 
pistons  may  be  calculated  by  dividing  the  total  ratio  by  the  ratio  of  inter- 
mediate expansion.  Thus,  if  the  total  ratio  be  9,  and  the  intermediate 

ratio  be  1.5,  the  combined  ratio  behind  the  pistons  is  -2-  =  6;   or  i  to  6. 

Generally,  if  the  total  ratio  be  divided  by  any  one  of  the  individual  ratios, 
the  quotient  is  the  product  or  combination  of  the  two  others. 

Further,  if  two  ratios  be  equal  to  each  other,  the  combined  ratio  is  equal 
to  the  square  of  one  of  them;  and,  conversely,  the  square  root  of  a  given 
ratio  is  the  value  of  two  elementary  ratios,  which  when  combined  yield  the 
given  ratio.  Thus,  if  there  be  two  ratios,  each  equal  to  3,  then  3  x  3,  or 
32  =  9,  the  combined  ratio  formed  by  those  two.  Conversely,  (^9  =  )  3 
is  the  value  of  two  elementary  ratios  which,  if  combined,  form  the  ratio  9. 
Similarly,  the  two  equal  ratios  which,  when  combined,  form  the  ratio  7,  are 
each  equal  to  *J  7  =  2.65;  the  square  of  2.65,  or  2.65  x  2.65,  being  equal 
to  7. 


FORMULAS  AND   RULES   FOR   COMPOUND   ENGINES.  871 

To  find  the  actual  ratio  of  expansion  in  the  first  cylinder. — This  is  found 
by  the  formula,  page  828,  when  the  stroke,  the  period  of  admission,  and  the 
clearance  are  given.  It  is  equal  to 

7  =  R' ('5) 

That  is  to  say,  the  actual  ratio  of  expansion  in  the  first  cylinder  is  equal  to 
the  quotient  of  the  length  of  stroke  plus  the  clearance  divided  by  the  period 
of  admission  plus  the  clearance.  For  example,  if  the  steam  be  cut  off  at 
one-third  of  the  stroke,  and  the  clearance  be  7  per  cent.,  the  length  of  stroke 
being  equal  to  i;  then  the  stroke  plus  the  clearance  is  equal  to  1.07,  and 

the  period  of  admission  is  equal  to  .3333  +  .07  =.4033;  and  I'°'  =  2.653, 
the  actual  ratio  of  expansion.  -4°33 

To  find  the  ratio  of  Intermediate  Expansion. — According  to  the  assumption 
that  the  volume  of  a  given  weight  of  steam  is  inversely  as  its  elasticity,  or 
its  pressure  per  square  inch,  the  enlargement  of  volume,  or  expansion,  may 
be  deduced  from  the  pressures  before  and  after  expansion.  Thus,  if  the 
pressure  be  reduced  from  20  Ibs.  to  15  Ibs.  per  square  inch,  the  volume, 
inversely,  is  enlarged  in  the  ratio  of  15  to  20;  and,  if  the  initial  volume  be 


20 


taken  as  i,  then,  by  proportion,  15  :  20  :  :  i  :  1.33;  or  i  x  —  =  1.33.     Thus, 

in  reducing  the  pressure  ^th,  the  volume  is  enlarged  ^d,  and  the  ratio  of 
expansion  is  1.33. 

By  the  notation,  n  is  the  ratio  of  the  final  pressure  in  the  first  cylinder  to 
the  intermediate  fall  of  pressure  between  the  first  and  second  cylinders ;  or, 
it  is  the  denominator  of  the  fractional  part  of  the  final  pressure,  expressing 
the  fall  of  pressure.  When  the  fall  is  ^th,  therefore,  n  =  ^;  and  the 
remaining  pressure  is  ^ths,  and  is  as  3,  or  n-i.  The  pressures,  then, 
before  and  after  the  fall,  are  as  ;/ :  n  —  i ;  and,  inversely,  the  volumes  are  as 
n  -  i  :  n.  Taking  the  capacity  of  the  first  cylinder  with  its  clearance,  as  i, 

the  expanded  volume  is  found  by  the  proportion  n  -  i  :  n  \  :  i  :     n    ;  and 
the  ratio  of  intermediate  expansion  is  equal  to  n  ~  I 

(16) 


n-  i 


Substituting  4  for  n  in  this  expression,  the  ratio  of  expansion  in  the  preceding 
example,  is  -i— =  A=  1.33,  as  already  found. 

It  is  necessary,  in  the  receiver-engine,  thus  to  reckon  backwards, — from 
the  observed  pressures  to  the  volumes, — in  order  to  find  the  intermediate 
ratio  of  expansion,  since  the  volume  of  the  receiver  affords  no  evidence 
whatever  of  the  amount  of  expansion  between  the  first  and  second  cylinders. 

The  same  process  may,  of  course,  be  applied  in  the  Woolf  engine,  to  find 
the  intermediate  expansion;  but  the  ratio  of  this  expansion  is,  otherwise, 
exactly  and  directly  determinable  by  the  volume  of  the  intermediate  space. 
The  ratio  of  the  capacity  of  the  first  cylinder  plus  the  clearance,  to  the 
intermediate  space,  is,  by  the  notation,  as  i  to  N ;  and  the  sum  of  these, 


8/2  STEAM   ENGINE— COMPOUND  CYLINDERS. 

or  the  enlarged  volume,  is  as  (i  +  N).     The  ratio  of  intermediate  expansion 
is,  therefore,  as  i  to  (i  +N);  or  it  is 

(i+N).    (17) 

To  find  the  value  of  N  in  terms  of  the  intermediate  fall  of  pressure: — 
The  intermediate  ratio  was  found,  in  terms  of  the  ratio  of  pressure  n,  to  be 

11    ;  and  i+N  =  -^-;  so  that 


n-  i  n  —  i 


That  is,  the  volume  of  the  intermediate  space  relative  to  that  of  the  first 
cylinder  plus  its  clearance,  is  equal  to  the  quotient  of  the  final  pressure  in 
the  first  cylinder,  divided  by  the  reduced  pressure  after  the  fall,  minus  i. 
For  example,  the  final  and  reduced  pressures  being  20  Ibs.  and  15  Ibs. 
respectively,  20^-15  =  1.33;  and  1.33-1  =.33,  which  is  the  value  of  N, 
the  intermediate  space,  relative  to  the  capacity  of  the  first  cylinder  plus  its 
clearance,  taken  as  i.  In  this  calculation,  the  actual  values  of  the  pressures 
have  been  used,  instead  of  their  relative  values  as  indicated  in  the  above 
expression  (18);  but  the  result  is  the  same,  for,  putting  for  n  the  ratio  4, 

then,  -*—  =  -=1.33;  and  I-33-I=-33>  as  before. 
4~  J     3 

The  capacity  of  the  intermediate  space  in  the  Woolf  engine,  is  found  by 
multiplying  that  of  the  first  cylinder  plus  its  clearance  by  the  ratio  N. 

To  find  the  Ratio  of  Expansion  in  the  second  cylinder.  —  In  the  Woolf 
engine.  This  would  be  expressed  by  the  ratio  of  the  capacity  of  the  first 
cylinder  to  that  of  the  second  cylinder,  if  there  were  no  clearances  nor 
other  intermediate  space.  With  clearances  and  intermediate  space,  the 
ratio  of  expansion  in  the  second  cylinder  is  less  than  that,  and  is  equal 
to  the  ratio  of  the  capacity  of  the  first  cylinder  plus  its  clearance  plus 
the  intermediate  space,  to  the  capacity  of  the  second  cylinder  plus  the 
intermediate  space,  this  last  being  taken  to  include  the  clearance  of  the 
first  and  second  cylinders.  Taking  the  capacity  of  the  first  cylinder  plus  its 
clearance  as  i,  that  of  the  intermediate  space  is  N.  The  capacity  of  the 
second  cylinder,  with  its  clearance,  is  expressed  by  the  ratio  r;  without 
clearance,  it  is  less  than  r  by  as  much  in  proportion  as  the  capacity  of  the 
cylinder  is  less  than  the  cylinder  plus  the  clearance,  or  as  L  is  less  than 

(L  +  c\  or  L'.    The  reduced  ratio  is,  then,  r  x  —  ;  and  the  ratio  of  expansion 

JLj 

in  the  second  cylinder  is  as  (i  +  N)  is  to  (r  x  —  )  +  N;  or 

JL< 


(rx     y 
Ratio  of  expansion  in  second  cylinder  =  -  —  —  ................  (  J  9  ) 

That  is,  the  ratio  of  the  first  to  the  second  cylinder  is  multiplied  by  the 
length  of  stroke,  and  divided  by  this  length  plus  the  clearance;  and  the 
ratio  of  the  intermediate  space  is  added  to  the  quotient,  making  a  sum,  say,  A. 


FORMULAS  AND  RULES   FOR  COMPOUND   ENGINES.          8/3 

To  the  ratio  of  the  intermediate  space  is  added  i,  making  a  sum,  say,  B.  Sum 
A  is  divided  by  sum  B,  and  the  quotient  is  the  ratio  of  expansion  in  the 
second  cylinder.  For  example,  let  r=3,  ^"  =  .333,  L  =  6  feet,  and  L'  =  6 
plus  7  per  cent,  of  6,  or  6.42.  Then 

(3*  ^-)  +  -333 

6-42  _       - 

i  +.333  '353> 

the  ratio  of  expansion  in  the  second  cylinder. 

In  the  receiver-engine.  The  actual  ratio  of  expansion,  in  the  second 
cylinder,  is  not  affected  by  clearance,  assuming,  of  course,  that  the  per- 
centage of  clearance  is  the  same  as  in  the  first  cylinder.  When  there  is  no 
intermediate  fall  of  pressure,  the  ratio  of  expansion  is  simply  that  of  the 
first  and  second  cylinders,  or  r.  But,  with  an  intermediate  fall,  this  ratio  is 

reduced  as  the  ratio  of  intermediate  expansion  is  increased,  namely  -  —  • 
and  it  is  as  this  ratio  inversely,  or, 

Ratio  of  expansion  in  second  cylinder  =  r  x  -  =  '  -  '—  ......  (  20  ) 

n  n 

For  example,  putting  the  ratio  of  the  cylinders,  r  =  3,  and  the  ratio,  «,  of 
the  intermediate  fall  to  the  final  pressure  in  the  first  cylinder,  =  4,  as  before; 

then,  \4~  I)  3  =  3^ij  =  2.25,  the  actual  ratio  of  expansion  in  the  second 

4  4 

cylinder. 

To  find  the  total  actual  Ratio  of  Expansion  as  well  as  the  combined  actual 
Ratio  of  Expansion  behind  the  two  pistons.  The  total  actual  ratio  of  expan- 
sion is,  as  was  stated  (page  870),  the  product  of  the  ratios  of  the  three 
consecutive  expansions:  —  in  the  first  cylinder,  in  the  intermediate  space, 
and  in  the  second  cylinder. 

For  the  Woolf  engine.  The  expressions  of  these  expansions  are  numbered 
(15),  (17),  and  (19),  and  their  product  is  as  follows:  — 


L> 

;or, 


Total  actual  ratio  of  expansion  =      x  (r     +  N),    or  R'(       +  N)  .  .  .  (21) 

/  -I—I  JL/ 

That  is  to  say,  the  ratio,  r,  of  the  first  to  the  second  cylinder  is  to  be 
multiplied  by  the  length  of  stroke,  and  divided  by  this  length  plus  the 
clearance  ;  and  the  ratio-value  of  the  intermediate  space,  N,  is  added  to  the 
quotient.  The  sum  is  then  multiplied  by  the  actual  ratio  of  expansion  in 
the  first  cylinder,  and  the  product  is  the  total  actual  ratio  of  expansion. 
For  example,  let  the  steam  be  cut  off  at  a  third  of  the  stroke  of  the  first 
cylinder,  with  a  clearance  of  7  per  cent.  \  let  the  ratio,  r,  of  the  cylinders  be 
3,  and  the  ratio-value,  N,  of  the  intermediate  space,  .333  or  ^d.  Then, 
the  stroke  of  the  first  cylinder  being  =  i,  the  actual  ratio  of  expansion  in 


8/4  STEAM   ENGINE  —  COMPOUND   CYLINDERS. 

it,  R',  as  was  exemplified  at  page  871,  is  1.07-=-  .4033  =  2.653.  The  modified 
ratio  of  the  cylinders  is  3  x  —  —  =  2.804;  and  2.804  +  -333  =  3-  13  7-  Finally^ 
2.653x3.137  =  8.322,  the  total  actual  ratio  of  expansion.  It  may  be 

observed,  that  the  fraction,  -i-,  above   employed,  is  equivalent   to  the 

6  I-°7 

fraction,  ^  —  -,  employed  for  the  same  purpose,  in  the  example,  page  873. 

The  combined  actual  ratio  of  expansion  behind  the  pistons,  in  the  Woolf 
engine,  is  the  product  of  the  first  and  third  of  the  above-cited  expressions, 
namely  (15)  and  (19),  or, 


/  \ 

(22) 


That  is  to  say,  the  product,  as  above  found,  for  the  total  expansion,  is  to  be 
divided  by  the  ratio-value  of  the  intermediate  space  plus  i  ;  the  quotient  is 
the  combined  actual  ratio  of  expansion  behind  the  pistons.  For  example, 
resuming  the  data  of  the  preceding  example,  the  final  product  expressing 
the  total  actual  ratio  of  expansion,  was  found  to  be  8.322;  and  the  divisor 

to  be  applied  to  it,  is  i  +  .333  =  1.333.    Then,    '^22  =  6.242,  the  combined 

o»5«5 
actual  ratio  of  expansion  behind  the  pistons. 

For  the  Receiver-engine.  The  total  actual  ratio  of  expansion  is  the  product 
of  the  expressions  of  the  three  consecutive  expansions,  numbered  (15),  (i  6), 
and  (20);  their  product  is  as  follows:  — 

L'        n        (n  —  i  )  r        L'        -n  ,  ,       \ 

xxL-1  ~*'*f     ...................  (23) 


That  is  to  say,  the  ratio,  r,  of  the  first  and  second  cylinders  is  to  be  multi- 
plied by  the  actual  ratio  of  expansion,  R',  in  the  first  cylinder.  The  product 
is  the  total  actual  ratio  of  expansion.  For  example,  making,  as  before, 
r=  3,  and  R'  =  2.653,  the  product  (3  x  2.653  =  )  7-959?  *s  the  total  actual  ratio^ 
of  expansion. 

The  product  of  the  first  and  third  of  the  above  three  expressions,  namely 
(15)  and  (20),  gives  the  value  of  the  combined  actual  ratios  of  expansion 
behind  the  pistons;  thus, 


That  is  to  say,  the  ratio  of  the  first  and  second  cylinders  is  multiplied  by 
the  actual  ratio  of  expansion  in  the  first  cylinder,  and  by  the  ratio  of  the 
intermediate  fall  of  pressure  to  the  final  pressure  in  the  first  cylinder  minus  i; 
and  the  final  product  is  divided  by  this  ratio  simply.  The  quotient  is  the 
combined  actual  ratio  of  expansion  behind  the  pistons.  For  example, 
resuming  the  product  in  the  last  preceding  example,  and  taking  ;/,  the  ratio 

of  the  intermediate  fall  of  pressure  =4;  then  3  x  2.653  x  ini  =  7.959  x  3 

4  4 

=  5-9^9,  the  required  ratio. 


FORMULAS  AND   RULES   FOR  COMPOUND   ENGINES.          8/5 

To  find  the  Work  done  in  the  two  cylinders  of  compound  engines  —  The 
Woolf  engine.  It  has  already  been  stated  that  the  formula  (  5  ),  page  828, 
for  the  work  of  steam  expanded  in  one  cylinder,  applies  also  to  the  work  of 
steam  in  the  Woolf  engine,  when  the  combined  actual  ratio  of  expansion 
behind  the  pistons  in  the  two  cylinders,  is  given.  Thus,  the  net  total  work 
for  one  stroke  of  the  two  pistons,  quoting  that  formula,  is, 


(25) 


RULE  i.  To  find  the  net  work  done  by  steam  in  the  two  cylinders  of  a 
Woolf  engine,  for  one  stroke,  with  a  given  combined  actual  ratio  of  expansion.  — 
To  the  hyperbolic  logarithm  of  the  combined  actual  ratio  of  expansion 
behind  the  two  pistons,  add  i  ;  multiply  the  sum  by  the  period  of  admission 
to  the  first  cylinder  plus  the  clearance,  in  feet;  and  from  the  product  subtract 
the  clearance.  Multiply  the  area  of  the  first  piston,  in  square  inches,  by 
the  initial  pressure  in  pounds  per  square  inch,  and  by  this  remainder.  The 
product  is  the  net  work  in  foot-pounds. 

For  example,  let  the  2d  case,  pages  860,  86  1,  be  calculated  by  this 
rule:  —  a  P  =  i  x  63  =  63  Ibs.,  /'  =  2.42  feet,  c  =  .42  foot,  and  R"=6.24. 
Then, 


.42  (i+  hyp  log  6.24)  -.42] 

.42  x  2.8310)  -  .42] 
=  63  (6.85  1  -  .42)  =  405.20  foot-pounds, 


=  6$  [2.4 

=  63  [  (2. 


as  was  before  calculated,  allowing  for  small  errors  of  approximation. 

The  Receiver-engine.  —  A  complete  formula  for  the  work  of  the  receiver- 
engine  necessarily  comprises  three  elements  :  —  First,  the  expression  of  the 
gross  work,  including  the  work  of  the  clearances;  second,  the  deduction 
for  the  passive  work  of  the  clearances;  third,  the  addition  for  the  gain  of 
work  by  the  reduction  of  the  back  pressure  on  the  first  piston  when  there 
is  an  intermediate  fall  of  pressure.  Beginning  with  the  first  case,  pages 
863,  864,  in  which  there  is  no  intermediate  fall  of  pressure,  the  total  initial 
work  of  the  steam  admitted  to  the  first  cylinder  is  expressed  by  a  P  /'; 
whence  the  total  work  with  expansion  is 

gR")  ....................  (26) 


This  measures  the  total  area  of  the  diagram,  Fig.  347,  page  862,  including  the 
clearances.     The  work  of  the  clearance  of  the  first  cylinder,  cc'  o"o,  is 


The  work  of  the  clearance  of  the  second  cylinder  is  the  rectangle  hh'o'o, 
which  includes  the  section  hoo"  of  the  first  clearance;  and,  deducting  this, 
the  remainder,  which  is  the  rectangle  h'  o'  o"  ,  is  to  be  added  to  the  first 
clearance.  To  express  this  remainder  algebraically,  the  volumes  of  the 
first  and  second  clearances,  oo"  and  od  ',  are  in  the  ratio  of  the  areas  of  the 
cylinders,  or  as  i  to  r,  and  the  volume  of  the  difference,  o'  '  o  ',  is  as  c  (r  —  i). 
The  height,  d  '  h'  ',  is  the  final  pressure  in  the  first  cylinder,  and  is  equal  to  the 


876  STEAM   ENGINE  —  COMPOUND   CYLINDERS. 

initial  pressure  divided  by  R',  the  actual  ratio  of  expansion  in  the  first 
cylinder;  or, 

P 

TT/* 

Therefore  the  work  of  the  excess,  o"or,  of  the  second  clearance  is, 


and  the  two  works  of  the  clearances  are  together, 


to  be  deducted  from  the  gross  work  by  expansion  (26).    Whence  the  equation 
for  the  net  work,  in  the  first  case  :  — 


or 


K. 
)],  .........  (27) 


when  there  is  no  intermediate  fall  of  pressure. 

Before  reducing  this  formula  to  a  rule,  it  may  be  remarked  that  it  gives 
values  which  approximate  closely  to  the  true  values,  for  cases  in  which  there 
are  intermediate  falls  of  pressure  —  such  cases  as  usually  occur  in  practice;  — 
and,  for  ordinary  practical  purposes,  the  results  of  the  application  of  this 
formula  will  be  sufficiently  near  to  exactness.  It  was  found,  in  fact  (page  864), 
that  the  reductions  of  work  by  intermediate  falls,  as  compared  with  the 
work  done  when  there  was  no  fall,  were  as  follows:  —  When  the  pressure 
falls  to 

24,         2^,         y?,  of  the  final  pressure  in  the  first  cylinder, 
the  reduction  of  work  is, 

0.2,         i.o,        4.6  per  cent,  of  that  in  the  first  case. 

The  intermediate  fall  of  pressure  is  rarely  so  much  as  two-thirds;  and  even 
with  this  fall  the  reduction  of  work,  it  is  seen,  only  amounts  to  i  per  cent. 
The  slightness  of  the  reduction  results  from  the  fact,  as  was  before  explained, 
that  though  the  actual  ratio  of  expansion,  with  intermediate  falls,  is  less 
than  when  there  is  no  intermediate  fall,  yet  the  loss  of  work  by  such  reduc- 
tion of  expansion  is  practically  compensated  by  the  gain  of  net  work  on  the 
first  piston  by  the  fall  of  back  pressure  against  it. 

Adopting,  then,  the  formula  (27)  as  applicable  for  all  cases  of  receiver- 
engines  arising  in  practice,  it  is  required  only  to  give  the  actual  ratio  of 
expansion  in  the  first  cylinder,  and  to  multiply  this  ratio  by  the  ratio  of  the 
capacities  of  the  two  cylinders,  to  arrive  at  the  ratio  of  expansion  to  be 
employed  in  the  formula.  This  is  literally  the  actual  combined  ratio  of 
expansion  for  the  first  case,  without  intermediate  fall  of  pressure,  as  was 
found  (page  865),  represented  by  R"  in  the  formula  (27). 


FORMULAS   FOR  COMPOUND   ENGINES.  8/7 

RULE  2.  To  find  the  net  work  done  by  steam  in  the  two  cylinders  of  a 
receiver-engine  for  one  stroke,  with  a  given  actual  ratio  of  expansion  in  the  first 
cylinder.  —  Multiply  the  actual  ratio  of  expansion  in  the  first  cylinder  by  the 
ratio  of  the  two  cylinders,  and  to  the  hyperbolic  logarithm  of  the  compound 
ratio  add  i;  multiply  the  sum  by  the  initial  period  of  admission  to  the  first 
cylinder,  plus  the  clearance,  in  feet  (product  A).  Divide  the  ratio  of  the 
two  cylinders,  minus  i,  by  the  actual  ratio  of  expansion  in  the  first  cylinder; 
add  i  to  the  quotient,  and  multiply  the  sum  by  the  initial  clearance  in  feet 
(product  B).  Subtract  product  B  from  product  A,  giving  the  remainder  C. 
Multiply  the  area  of  the  first  cylinder,  in  square  inches,  by  the  total  initial 
pressure  in  pounds  per  square  inch,  and  by  the  remainder  C.  The  product 
is  the  net  work  in  foot-pounds  for  one  stroke. 

This  rule  is  applicable  to  any  of  the  four  cases,  page  865  :  a=  i  square 
inch,  P  =  63  Ibs.  per  square  inch,  ^  =  .42  foot,  /'=2.42  feet,  R'  =  2.653, 
R"  =  7.959,  r=3,  and  hyp  log  ^'  =  2.0743.  Then,  on  the  model  of  the 
given  formula  (27), 

w  =  63  [2.42  (i  +  2.0743)  -  .42  (i  + 


=  63  (7.440  -  .737)  -  422.29  foot-pounds, 

as  was  before  calculated  for  the  first  case.  Or,  following  the  wording  of 
the  rule:  —  The  combined  actual  ratio  of  expansion  is  7.959,  of  which  the 
hyperbolic  logarithm  is  2.0743;  adding  i  to  this,  the  sum,  3.0743,  is  multi- 
plied by  2.42,  the  initial  period  of  admission  plus  the  clearance,  and 
3.0743  x  2.42  =  7.440  (product  A).  Again,  the  ratio  of  the  cylinders  is  3, 
and  3-1  =  2;  the  actual  ratio  of  expansion  in  the  first  cylinder  is  2.653, 
and  24-2.  653  =  .  75  4.  Adding  i  to  this  quotient,  the  sum  is  multiplied  by 
the  initial  clearance  .42,  or  1.754  x  .42  =  .737  (product  B).  The  difference 
of  products  A  and  B  is  (7.440  -  .737  =  )  6.703,  and  this,  multiplied  by  63  Ibs., 
the  initial  pressure  per  square  inch,  and  by  i,  the  area  of  the  piston  in  square 
inches,  gives 

6.703  x  63  x  i  =  422.29  foot-pounds, 

the  work  of  one  stroke. 


8;8 


STEAM  ENGINE— COMPOUND. 


COMPRESSION   OF   STEAM   IN   THE   CYLINDER. 

The  work  expended  in  compressing  such  exhaust  steam  as  is  not  per- 
mitted to  escape  during  the  return-stroke  of  the  piston,  and  is  shut  into  the 
cylinder  against  the  retiring  piston,  is  to  be  reckoned  against  the  quantity 
of  steam  thus  reclaimed.  For  every  phase  of  the  distribution  there  is  a  par- 
ticular period  of  compression,  by  the  adoption  of  which  the  resulting  effi- 
ciency of  the  steam,  for  a  given  distribution,  is  raised  to  a  maximum.  The 
method  of  determining  the  best  period  of  compression  will  be  given  in  the 
author's  work  on  The  Steam  Engine.  The  following  table,  No.  298,  contains  the 
best  periods  of  compression  for  several  periods  of  admission,  with  7  per  cent, 
clearance,  and  for  several  back  exhaust-pressures.  It  is  seen,  by  the  table, 
that,  the  more  expansively  the  steam  is  worked,  the  greater  should  be  the 
period  of  compression — that  is,  the  exhaust  port  should  be  closed  the  earlier 
in  the  course  of  the  return-stroke;  and  that  the  greater  the  proportion  of 
back-pressure  to  initial-pressure,  the  less  should  be  the  period  of  compression. 

Table  No.  298. — COMPRESSION  OF  STEAM  IN  THE  CYLINDER. 
BEST  PERIODS  OF  COMPRESSION  : — Clearance  7  per  cent. 


Total  Back-pressure,  in  Percentages  of  the  Total  Initial  Pressures. 

Cut-off  in 

Percentages 

2/^j 

5 

IO 

15 

2O 

25              30 

35 

of  the 

Stroke. 

Periods  of  Compression,  in  parts  of  the  Stroke. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

10 

65 

57 

44 

32 









15 

58 

52 

40 

29 

23 





— 

2O 

52 

47 

37 

27 

22 







25 

47 

42 

34 

26 

21 

17 





30 

42 

39 

32 

25 

2O 

16 

14 

12 

35 

39 

35 

29 

23 

19 

15 

13 

II 

40 

36 

32 

27 

21 

18 

14 

13 

II 

45 

33 

30 

25 

20 

17 

14 

12 

10 

5° 

30 

27 

23 

18 

16 

13 

12 

IO 

55 

27 

24 

21 

17 

15 

13 

II 

9 

60 

24 

22 

19 

15 

14 

12 

II 

9 

65 

22 

2O 

17 

15 

14 

12 

IO 

8 

70 

19 

17 

16 

14 

14 

12 

10 

8 

75 

17 

16 

14 

13 

12 

II 

9 

8 

NOTES  TO  TABLE. — i.  For  periods  of  admission,  or  percentages  of  back-pressure,  other 
than  those  given,  the  periods  of  compression  may  be  readily  found  by  interpolation. 

2.  For  other  clearances,  the  values  of  the  tabulated  periods  of  compression  are  to  be  altered 
in  the  ratio  of  7  to  the  given  percentage  of  clearance. 


PRACTICE    OF    EXPANSIVE   WORKING   OF 

STEAM. 


ACTUAL  PERFORMANCE  OF  STEAM  IN  THE  STEAM-ENGINE. 

In  working  steam  expansively,  the  practical  performance  is  affected  by 
several  circumstances.  There  is  the  influence  of  the  wire-drawing  of  the 
steam  during  its  admission  into  the  cylinders;  of  the  needful  opening  of  the 
exhaust  passages  before  the  end  of  the  stroke,  for  the  escape  of  the  steam 
from  the  cylinder;  of  the  back  exhaust  pressure  on  the  piston,  and  the 
closing  of  the  exhaust  passage  before  the  end  of  the  return-stroke,  with  the 
consequent  shutting  in  and  compression  by  the  piston  of  a  portion  of  the 
exhausting  steam.  These  influences  have  been  analyzed  and  measured 
by  the  author.  He  concluded  that,  when  the  cylinders  were  liberally  pro- 
portioned, first,  the  possible  loss  by  early  exhaust  was  of  no  importance,  and 
that  the  early  release  was,  on  the  contrary,  beneficial,  in  facilitating  a  complete 
exhaust  during  the  return-stroke;  second,  that  the  loss  by  wire-drawing  was 
of  little  or  no  moment,  and  that,  as  wire-drawing  was,  to  some  degree,  equiva- 
lent to  an  earlier  cut-off,  it  might  even  prove  advantageous  in  point  of 
economy;  third,  that  the  loss  by  back  exhaust  pressure  in  excess  of  the 
atmospheric  resistance  in  non-condensing  engines,  in  good  practice,  is  of 
little  or  no  importance.  These  conclusions  were  based  upon  the  performance 
of  locomotives,  fitted  with  the  link-motion,  and  worked  with  steam  of  100  Ibs. 
effective  pressure  per  square  inch  in  the  boiler;  but  they  are  applicable  to 
all  classes  of  steam-engine.1 

The  only  obstacle  to  the  working  of  steam  advantageously  to  a  high 
degree  of  expansion  in  one  cylinder,  in  general  practice,  is  the  condensation 
to  which  it  is  subjected,  when  it  is  admitted  into  the  cylinder  at  the  begin- 
ning of  the  stroke,  by  the  less  hot  surfaces  of  the  cylinder  and  the  piston ; 
the  proportion  of  which  is  increased  with  the  ratio  of  expansion,  so  that  the 
economy  of  steam  by  expansive  working  ceases  to  increase  when  the  period 
of  admission  is  reduced  down  to  a  certain  fraction  of  the  stroke,  and  that, 
on  the  contrary,  the  efficiency  of  the  steam-  is  diminished  as  the  period  of 
admission  is  reduced  below  that  fraction.  The  initial  condensation  here 
pointed  out,  is  succeeded  by  the  re-evaporation  of  a  portion  of  the  condensed 
steam  during  the  later  portion  of  the  period  of  expansion ;  because,  as  the 
pressure  falls,  the  temperature  of  the  steam,  and  of  the  water  which  it  con- 
tains, also  falls,  until  it  ultimately  descends  below  the  actual  temperature  of 
the  cylinder,  when  the  heat  of  the  cylinder  is  absorbed  by  the  water,  and 

1  See  Railway  Machinery,  1855,  pp.  69-99;  also  a  paper  on  "  The  Expansive  Working 
of  Steam  in  Locomotives,"  in  the  Proceedings  of  the  Institution  of  Mechanical  Engineers, 
1852,  pp.  60-82,  and  109-128. 


880  PRACTICE   OF  EXPANSIVE  WORKING  OF   STEAM. 

evaporation  takes  place.  The  author,  in  1851,  experimentally  demonstrated 
the  existence  of  this  condensation  in  the  cylinders  of  locomotives.  Its 
reality  and  importance  are  now  thoroughly  understood  and  admitted.1  He 
deduced  from  his  experiments  that,  in  jacketless  cylinders,  imperfectly  pro- 
tected, the  quantity  of  steam  condensed  amounted  to  from  n  to  42  per 
cent,  of  the  whole  of  the  steam  admitted  to  the  cylinders,  according  to  the 
period  of  admission,  ranging  from  75  to  12  per  cent,  of  the  stroke.2 

The  author  also  deduced  that,  on  the  contrary,  when  the  cylinders  of 
locomotives  were  thoroughly  protected  and  heated  in  the  smoke-box,  there 
was  no  evidence  to  prove  that  initial  condensation  took  place  in  the  cylinders, 
to  any  important  extent,  within  the  limits  of  the  expansive-working  that  was 
practised.  By  the  application  of  a  jacket  of  steam  from  the  boiler,  to  the 
cylinder,  a  material  increase  in  the  efficiency  of  the  steam  has,  in  most  cir- 
cumstances, been  effected.  But,  it  is  incontestable  that  the  jacket,  though 
it  diminishes,  does  not  wholly  prevent  initial  condensation  of  the  steam 
admitted. 

By  the  compounding  of  cylinders,  steam  may  be  worked  more  expansively, 
and  with  a  greater  degree  of  efficiency,  than  in  a  single  cylinder;  for, 
obviously,  the  fluctuations  of  temperature  which  give  rise  to  the  condensa- 
tion that  interferes  with  the  action  of  steam  worked  expansively,  are  divided 
and  reduced  to  one-half,  in  each  cylinder,  of  what  they  amount  to  when  the 
whole  of  the  expansive  action  is  confined  to  one  cylinder. 

DATA  OF  THE  PRACTICAL  PERFORMANCE  OF  STEAM. 

Single- Cylinder  Condensing  Engines: — Steam-jacketted  and  covered. — The 
following  data  are  reduced  from  the  recorded  performances  of  the  engines  :3 — 

1  The  author  was  the  first,  so  far  as  he  is  aware,  to  discover  and  demonstrate  the  exist- 
ence of  initial  condensation  in  steam-cylinders,  and  to  prove  that  it  increases  rapidly  and 
to  a  formidable  extent  as  the  ratio  of  expansion  is  increased.     See  his  paper  on  "Ex- 
pansive Working  of  Steam  in  Locomotives,"  in  the  Proceedings  of  the  Institution  of 
Mechanical  Engineers,  1852,  page  109.     See  also  Railway  Machinery,  1855: — "When 
steam  is  admitted  to  the  cylinder  while  the  latter  is  comparatively  cold,  a  very  sensible 
condensation  of  the  steam  takes  place  during  admission,  which  continues  to  a  certain 
extent  during  expansion.     The  heat  thereby  separated  is  absorbed  by  the  material  of  the 
cylinder,  and  raises  its  temperature.     A  portion  of  this  heat  passes  off,  and  is  irrecoverably 
lost ;  the  remainder  is  re-absorbed  by  the  precipitated  steam  during  the  expansion  of  the 
existing  steam,  if  the  expansion  be  long  enough  continued — that  is,  until  the  temperature 
of  the  latter  has  fallen  below  that  of  the  cylinder.     This  is  clearly  proved  by  indicator- 
diagrams  taken  at  very  slow  speeds,  on  which  occasions,  the  cylinder  is  cold  enough  to 
exhibit  these  operations  in  high  relief." — page  84. 

2  In  condensing  engines,  the  loss  by  initial  condensation  may  be  much  greater  than 
40  per  cent.,  for  which  examples  will  here  be  given.     Mr.  Sutcliffe  has  followed  the  same 
method  of  analysis  in  stationary  engines,  and  in  the  seventh  edition  of  Hopkinson  on  the 
Indicator,  published  in  1875,  he  appears  to  have  precisely  adopted  the  conclusions  and 
even  the   language  of  the   author.      "  The   initial   condensation,"  he   says,   page   298, 
"relatively  to  the  initial  measure  of  steam  used,  and  the  pressure  of  steam  found  at  the  end 
of  the  stroke,  is  greater  as  the  cut-off  is  earlier;  by  the  diagrams  referred  to,  and  others 
from  the  same  engines  [referring  to  the  Corliss  engines  at  Saltaire],  we  find  the  initial  con- 
densation, relatively  to  the  terminal  vario-thermal  line,  to  be  as  follows  : — 

"  At  7.4  expansions  =27.0  per  cent. 
9.04        ,,         =36.67  per  cent. 
11.4  „          =46.67  per  cent." 

8  For  Nos.  I,  2,  3,  4,  Proceedings  of  the  Institution  of  Mechanical  Engineers,  1862,  1867, 
1868.  For  No.  5,  Report  of  the  American  Commission  on  the  Vienna  Exhibition,  vol.  iii., 
page  23. 


SINGLE-CYLINDER   CONDENSING  ENGINES. 


881 


Table  No.  299. — WORK  OF  EXPANDED  STEAM: — SINGLE-CYLINDER 
CONDENSING  ENGINES. 


1 

Actual 
Ratio 
of 
Expan- 
sion. 

Weight  of  Steam  per 
I.H.P.  as  per  Indicator. 

Coal 
Con- 
sumed 

I.H.rp. 

Total 
Initial 
Pressure 
at 
Cut-off. 

As  cut  off. 

As  expanded 

I 

2 

3 
4 

5 

Corliss,  Saltaire,  

5.20 
6.62 
4.08 

3.31 
6.60 
6.30 
6.50 
4.90 

Ibs. 
14.51 

15-43 
17.28 

17.83 
13.12 
14.66 
14.00 
14.28 

Ibs. 
16.03 
20.78 
20.97 
19.05 
17.20 
16.73 
18.28 
15.58 

Ibs. 
2.5 

Ibs. 
34^ 
55 
51 
50 

44 
43 
50 
47 

46~ 

15 

19.75 
23.25 
27.25 
30.5 
17-5 
23 
29 

50 

Allen  Engine 

Do 

— 

Do  

Crossness       Pumping  ) 
Engine                  ..  I 

Do. 
Do. 

Crossness  averages,  

East  London  Pumping 
Engines  :  — 
72-inch  cylinder 

6.07 

i-93 

2.80 
3-62 
4.38 
5.10 
2.81 
3-66 
3.65 

IO.O 

14.27 

16.24 
14.25 
12.92 
12.25 
1  1.  60 
13-95 
11-33 
I4.8I 

16.95 

15-74 
15.22 
14.91 
13.58 
1377 
16.93 
14.44 
14.83 

2.2 

Do 

Do             

— 

Do.            

Do 

8o-inch  cylinder 

— 

QO        do 

™        *r                 

100      do 

Sulzer  Engine  (Corliss  / 
srear),  .                    .  \ 

3-3 

With  regard  to  No.  i,  the  Corliss  engine  at  Saltaire,  it  is  stated  by  Mr. 
Hopkinson  that  6.96  Ibs.  of  water  were  evaporated  per  pound  of  coal.  For 
No.  3,  with  good  boilers,  allow,  say,  8^  Ibs.  of  steam  per  pound  of  coal. 
For  Nos.  5  and  6,  the  quantities  consumed  were  observed.  Then,  the 
following  are  the  actual  quantities  of  steam  consumed,  compared  with  the 
quantities  indicated : — 


Steam  Consumed 
per  I.H.P.  per  Hour. 

I.  Saltaire, 17.4    Ibs. 

3.  Crossness  (estimated)  18.7      „ 

4.  East  London  : — 

72-inch  cylinder  (ratio  3.62), 20.72  „ 

80  „  21.38  „ 

90  „  ,  18.82  „ 

loo  „  20.08  „ 

5.  Sulzer, 19.6  „ 


More  than  Sensible 
Steam  cut  off. 

20  per  cent. 


60 


35 


Compound    Condensing  Engines,  with   and  without  steam-jackets. — The 
following  data  are  deduced  from  particulars  supplied  by  the  constructors : — 


56 


882 


PRACTICE   OF   EXPANSIVE  WORKING  OF   STEAM. 


Table  No.  300. — WORK  OF  EXPANDED  STEAM  : — COMPOUND 
CONDENSING  ENGINES. 


Actual  Ratio 
of 

Expansion. 

Weight  of  Steam  per 
I.  H.  P.  as  per  Indicator. 

Coal 

Con- 
sumed 

I.H.rp. 

Total 
Initial 
Pres- 
sure at 
Cut-off. 

As  cut  off. 

As  ex- 
panded. 

40.  Day,  Summers,  &  Co.,  Receiver,  ) 
Marine,  steam  -jacketted,  ) 

(  1st  cyl.  1.715 
(  2d  cyl.  2.  220 

Ibs. 
14.74 
10-75 

Ibs. 
I7.OO 
11.99 

Ibs. 
2.IO 

Ibs. 
49K 

Both    3.807 

5#.  John  Elder  &  Co.,  Receiver,  ) 
Marine,  steam-jacketted,  j 

j  1st  cyl.  1.850 
(  2dcyl.  1.852 

H-45 
13.21 

14.85 
14.85 

1.61 

56 

Both    3.426 

6.  J.  &E.Wood,  Receiver,  Station-  ) 
ary  no  jackets  .  \ 

j  istcyl.  4.010 
j  2d  cyl.  1.857 

10.94 
13-34 

10.77 
12.03 

2.I41 

85K 

Both    7.446 

7.  Donkin,  Woolf,  Stationary,  2d  ) 
cylinder  only  steam-jacketted,  ...  } 

\  istcyl.  5.269 
1  2d  cyl.  2.  590 

10.09 
II.  12 

19.16 

1.9 

5t# 

Both  13.650 

8.  Donkin,  Woolf,  Stationary,  ) 
steam-jacketted,  .  .  ...  .  .  .  .  j 

(  istcyl.  2.486 
j  2d  cyl.  3.  22  1 

I3.I8 
13.87 

17.85 

— 

50^ 

Both    8.007 

9.  Donkin,  Woolf,  Stationary,  with-  ) 
out  steam  in  jackets,  \ 

(  istcyl.  3.165 
|  2d  cyl.  3.221 

15-59 
18.73 

I9.OI 

— 

48^ 

Both  10.200 

10.  Thomson,  Woolf,  Stationary,  ) 
steam-jacketted  \ 

(  istcyl.  2.985 
J2d  cyl.  3.384 

10.84 
12.71 

15.27 

— 

36^ 

Both  10.  100 

1  This  quantity  is  the  result  of  an  estimate.  The  actual  quantity  of  coal  consumed  per 
indicator  horse-power  was  2.67  Ibs.,  from  which  one-fifth,  estimated  as  for  general  heating 
purposes,  was  deducted,  leaving  2.14  Ibs.  per  indicator  horse-power,  as  consumed  by  the 
engine. 


The  quantities  of  steam  consumed  from  the  boiler  were  observed  in  each 
trial,  except  for  the  first  three,  and  were  as  follows : — 


WOOLF  ENGINES. 


Steam  consumed 
per  I.H.P. 

7.  Donkin,  2d  cylinder,  steam-jacketted,...  20.55  Ibs. 

8.  „        steam-jacketted, 22.51    „ 

9.  „        no  steam  in  jacket, 32.72    „ 

10.  Thomson,  steam-jacketted, 20.93    „ 


More  than  Sensible 
Steam  cut  off. 

103  per  cent.  more. 


I  10 

93 


Single- Cylinder  Engines,  steam-jacketted,   non-condensing. — The   trials  of 


SINGLE-CYLINDER   NON-CONDENSING  ENGINES. 


883 


portable  engines  at  Cardiff,  in  I872,1  afford  various  examples,  from  which 
the  following  deductions  are  made : — 


Table  No.  301. — WORK  OF  EXPANDED  STEAM: — PORTABLE 
ENGINES. 


CONSTRUCTORS. 

Actual 
Ratio  of 
Expansion. 

Weight  of  Steam  per 
I.H.P.,  as  per  Indicator. 

Total 
Initial  Pres- 
sure at 
Cut-off. 

As  cut  off. 

As  expanded 

II 

12 
13 
14 

II 

17 

Marshall,  Sons,  &  Co.,  
Davey,  Paxman,  &  Co...... 

Brown  &  May,  

4.8 
5.0 

R6J3.S 

2.4 

3.8 
2.7 
5.0 

Ibs. 
16.87 
H-93 
20.52 

25.32 
18.54 
20.08 
16.28 

Ibs. 
29.82 
26.45 

28.87 
30.27 

23-93 
22.13 
29.98 

Ibs. 
741080 

73 
73 

52 
72.5 
77-2 
63.0 

Tasker,  

Reading  Engine  Works,  .  .  . 
Turner 

Ashby  &  Co.,  

The  quantities  of  water,  as  steam,  actually  consumed  per  indicator  horse- 
power, are  subjoined;  together  with  the  effective  mean  pressures  in  the 
cylinders.  And,  to  make  a  comparison  with  what  the  performance  would 
amount  to  if  the  atmospheric  pressure  were  removed,  as  if  the  steam  were 
condensed,  one  atmosphere,  or  1 5  Ibs.  per  square  inch,  is  added  to  the  effec- 
tive mean  pressure,  as  given  in  the  second  last  column,  with  the  weight  of 
steam  per  total  indicator  horse-power  accruing,  in  the  last  column : — 


Effective 
Mean 
Pressure. 

Steam 
Consumed 
per  I.H.P. 

More  than 
Sensible 
Steam 
cut  off. 

Effective 
Mean 
Pressure 
plus  15  Ibs. 

Steam  Con- 
sumed per 
Total  I.H.P. 

Ibs. 

Ibs. 

per  cent. 

Ibs. 

Ibs. 

II 

3L25 

25.9 

54 

46.25 

174 

12 

33-9 

29.6 

99 

48.9 

20.7 

13 

29.2 

3L8 

55 

44-2 

21.2 

H 

297 

37-9 

5o 

44-7 

25.2 

15 

37-0 

24.1 

30 

52.0 

17.2 

16 

36.24 

27-6" 

37 

51.2 

19-3 

17 

20.4 

43-2 

165 

35-4 

24.7 

Single-  Cylinder  Engines,  completely  covered  and  heated;  non-condensing.— 
Average  results  of  the  trials  of  the  "  Great  Britain  "  locomotive  made  in 
1850,  analyzed  by  the  author  in  1852,  and  published  in  1856,2  are  given  in 
the  following  table,  No.  302.  The  cylinders  are  "inside,"  being  placed 


1  The  Trials  of  Portable  Steam-engines  at  Cardiff.     Report  by  the  Judges,  1872. 

2 Railway  Machinery,  1856,  page  80.  See  also  a  paper  "On  the  Behaviour  of  Steam 
in  the  Cylinders  of  Locomotives  during  Expansion,"  by  D.  K.  Clark,  in  the  Minutes  of 
Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  Ixxii.  1882-83;  page  275. 


884 


PRACTICE  OF   EXPANSIVE   WORKING  OF   STEAM. 


within  the  smoke-box,  and  totally  surrounded  by  the  atmosphere  of  hot 
burnt  gases.  The  cylinders  are  18  inches  in  diameter,  with  a  stroke  of 
24  inches;  and  8  feet  driving  wheels.  The  ratios  of  expansion  are  here 
reckoned  in  terms  of  the  whole  of  the  stroke,  though  they  were  not  so 
reckoned  in  the  original  investigation. 

Table  No.  302. — WORK  OF  EXPANDED  STEAM — "  GREAT  BRITAIN  " 

LOCOMOTIVE. 


Weight  of 

Steam  per 

Total  Initial 

Notch. 

Cut-off. 

Actual  Ratio 
of  Expansion. 

Indicator 
Horse-power, 
as  per  Indi- 

Pressure per 
Square  Inch  at 
Cut-off. 

cator. 

No. 

per  cent. 

ratio. 

Ibs. 

Ibs. 

ISt 

67 

1.45 

28.97 

79-4 

3d 

50 

1.90 

24.52 

72.6 

5th 

29 

2.94 

19.74 

67.4 

The  general  effect  of  the  observations  was  that  there  was  no  material 
degree  of  initial  condensation  of  the  steam  in  the  cylinders  of  the  "  Great 
Britain,"  confirmatory  of  the  results  of  the  author's  experiments  with  the 
well-protected  and  heated  inside  cylinders  of  locomotives  on  the  Edinburgh 
and  Glasgow  Railway.1  Here  follows  a  statement  of  steam  consumed  per 
indicator  horse-power,  taking  the  initial  quantities  cut  off  for  the  quantities 
actually  consumed;  showing  the  relative  quantities  that  would  have  been 
due  if  the  atmospheric  pressure  had  been  removed,  as  in  a  condensing 
engine: — 


Notch. 

Effective 
Mean  Pressure 
per  Square 
Inch. 

Steam 
Consumed 
per  I.  H.  P. 

Effective 
Mean  Pressure 
plus  15  Ibs.  per 
Square  Inch. 

Steam 
Consumed  per 
Total  I.H.P. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

ISt 

68.1 

28.97 

83.1 

23-7 

3d 

53-5 

24.52 

68.5 

19.2 

5th 

32.1 

19.74 

47.1 

13-5 

American  Marine  Engines? — Mr.  Emery  tested  the  condensing  engines 
of  the  U.S.  steamers  Bache  and  Rush,  having  compound  cylinders  fully 
steam-jacketted  and  lagged,  and  the  Dexter  and  Dallas,  having  single 
cylinders  felted  and  lagged  only.  The  Bache  was  tried  in  four  ways : — with 
and  without  steam  in  the  jackets,  using  both  cylinders,  and  using  only  the 
second  cylinder.  The  following  are  the  best  results  of  performance  for 
each  series  of  trials  :— 

1  Railway  Machinery,  page  8.2,  &c. ;  and  the  table  in  the  same  book,  page  151. 

2  Journal  of  the  Franklin  Institute,  February  and  March,  1875.     See  also  notices  of  the 
trials  in  the  Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xl.  page  292,  and  vol.  xli. 
page  296. 


AMERICAN   MARINE   ENGINES. 


885 


Table  No.  303. — WORK  OF  EXPANDED  STEAM  : — AMERICAN 
MARINE  ENGINES — CONDENSING. 


STEAMER. 

Actual 
Ratio  of 
Expan- 
sion. 

Indicator 
Horse- 
power. 

Weight  of  Steam 
per  I.  H.  P.  per  Hour. 

Total 
Initial 
Pressure 
per 
Square 
Inch. 

r     ?y 
ndicator, 

First 
Cylinder. 

Actually 
Con- 
sumed. 

18 
19 

20 
21 

22 

23 

24 

Bache  (Woolf). 
Without  steam  in  jackets  :  — 
First  cylinder,  

3-75 
9.15 

2-73 
6.66 

^\ 
5.63) 

6.91 
16.85 

3-77 
9.19 
2.86 
6.98 
2-35 
5-73 
2-34  £ 
5-7M 
2.09 
5.10 
1.74 
4.24 

11.82 
7.62 
5-32 

12.62 
8.57 

5-ii 
2.18 

2.46 

6.22 

1.60 
4-03 

4.46 

349 
2.08 

5.07 
3.13 

2.32 

55-93 

77.06 

46.40 

46.40 

7745 
99.18 
110.50 
104.03 
102.26 
134.53 

47.24 

71.75 
89.14 

54.84 
74.62 
116.01 
66.74 

266.5 
168.7 

185.9 
292.4 
196.2 

138.0 
221.4 
234.3 

Ibs. 

1541 
15.67 
15.37 

14.55 
I4.IO 
14.97 
15.85 
15.85 
14.85 
17.27 

21.03 
17.76 
17-35 

16.42 
15.58 
16.25 
24.05 

I7.I 
19.7 

16.2 

I6.3 
20.3 

19.2 
20.1 
22.8 

Ibs. 

23.76 
23.04 
23.21 

25.11 
20.71 
20.33 
20.37 
21.97 
22.38 
21.17 

35.08 
29.62 
26.25 

27.11 

24.09 
23.15 
34.03 

18.4 
22.1 

23.9 
23.9 
3L8 

26.7 
26.9 
3LO 

Ibs. 

about 
90 

90 
90 

90 
90 
90 
90 
90 
90 
90 

90 
90 
90 

90 
90 
90 
27 

82.3 
50.2 

80.4 

53-6 

47 
394 

Both  cylinders,.. 

First  cylinder,  

Both  cylinders 

First  cylinder  . 

Both  cylinders,  

With  steam  in  jackets  :  — 
First  cylinder 

Both  cylinders,.. 

First  cylinder,  

Both  cylinders,  

First  cylinder 

Both  cylinders  .. 

First  cylinder,  

Both  cylinders 

First  cylinder 

Both  cylinders,.     .        .... 

First  cylinder,  

Both  cylinders 

First  cylinder, 

Both  cylinders,  

Without  steam  in  jacket  :  — 
Second  cylinder  only,  
Do.            do  

Do.            do.       .    . 

With  steam  in  jacket  :  — 
Second  cylinder  only 

Do.            do  

Do.            do  

Do.            do. 

Rush  (Receiver). 
Steam-jacketted  :  — 
First  cylinder, 

Both  cylinders,  

First  cylinder,  

Both  cylinders, 

Dexter  (single  cylinder). 
Felted  and  lagged,  

Do.          do 

Do.          do  

Dallas  (single  cylinder). 
Felted  and  lagged,  

Do           do        

Do          do 

886 


PRACTICE   OF   EXPANSIVE  WORKING  OF   STEAM. 


French  Stationary  Engines^ — M.  Hallauer  reports  the  results  of  experi- 
ments on  a  24-inch  single-cylinder  engine,  worked  by  steam  superheated 
150  degrees,  and  lagged  and  felted — yielding  135  indicator  horse-power; 
and  his  own  experiments  on  a  Woolf  engine,  with  steam-jacketted  cylin- 
ders : — 

FRENCH  STATIONARY  ENGINES — CONDENSING. 


Actual 
Ratio  of 
Expansion. 

Steam 
Consumed 
per  I.  H.  P. 
per  Hour. 

Total 
Initial  Pres- 
sure in 
Cylinder. 

25 

Hirn  (superheated  steam  in  ) 
single  cylinder)  ( 

4 

Ibs. 
15-5 

Ibs. 
60 

26 

Leloutre  (  Woolf  engine,  ) 
steam-jacketted)  ( 

12 

24.83 



GENERAL  DEDUCTIONS  FROM  THE  DATA  OF  THE  ACTUAL 
PERFORMANCE  OF  STEAM. 

Single  Cylinders,  with  steam-jackets;  condensing. — The  analysis  of  diagrams 
from  the  Allen  engine,  No.  2,  page  88 1,  indicates  that  the  expansion-ratio 
6.62  was  better  than  the  ratios  4.08  and  3.31.  Mr.  C.  T.  Porter  maintains 
that  the  ratio  8  is  best  for  the  Allen  engine.  For  the  Crossness  engines, 
No.  3,  the  ratio  6  appears  to  be  the  best;  though  perhaps  the  Corliss,  No.  i, 
with  the  ratio  5.2,  is  fully  as  good  as  the  Crossness.  The  Sulzer  (Corliss 
gear),  No.  5,  with  the  ratio  10,  is  not  so  efficient  as  these  others.  Of  the 
East  London  engines,  No.  4,  the  72-inch  engine  appears  to  have  greater 
efficiency  when  expanding  5.10  times  than  for  less  ratios,  and  of  the  expan- 
sion-ratios for  the  four  observed  consumptions  of  water  per  indicator  horse- 
power, the  highest  ratio,  3.66,  gives  the  greatest  efficiency.  Again,  the 
Bache  marine  engine,  No.  21,  page  885,  yielded  the  highest  observed 
efficiency  with  a  ratio  5.11,  but,  by  plotting,  it  is  easily  shown  that  the 
efficiency  is  practically  the  same  for  a  ratio  of  6. 

Non- Condensing. — The  portable  engines,  page  883,  supply  examples: — 

For  initial  pressures  between  70  Ibs.  and  80  Ibs., -omitting  No.  13  as  unequal. 


No. 

Total 
Maximum 
Pressure. 

Expansion- 
ratio. 

Water  per 
Normal  I.  H.  P. 
per  Hour. 

Water  per 
Total  I.H.P., 
if  Atmosphere 
were  removed. 

Ibs. 

Ibs. 

Ibs. 

16 

77.2 

2.7 

27.6 

19-3 

15 

72.5 

3-8 

24.1 

17.2 

ii 

74  to  80 

4.8 

25.9 

17.4 

12 

73 

5.0 

29.6 

20.7 

For  lower  initial  pressures. 

14 

f 

2.4 

37-9 

25.2 

17 

63 

5.0 

43-2 

24.7 

1  "Recherches  Experimentales  Sur  les  Machines  a  Vapeur."     By  MM.  Hallauer  and 
Dwelshauver-Dery;  Bidletin  de  la  Societe  Industrielle  de  Mulhouse,  1877,  page  190. 


GENERAL  DEDUCTIONS.  887 

It  is  seen  that  the  highest  efficiency  is  attained  with  an  expansion-ratio  of 
3.8,  whether  against  or  without  atmospheric  resistance. 

Single  Cylinders  without  steam-jackets;  condensing. — The  most  favourable 
results  of  Nos.  20^  23,  24,  and  25,  are  here  abstracted: — 


No. 
2O  

Expansion-ratio. 
.    ^.32    ., 

Water  Consumed 
per  I.H.P.  per  Hour. 

26.25  Ibs 

23    .. 

•>  ^~. 
4..4O 

23  Q 

,.    -3.4.0    . 

*yy 

..    23.Q 

24.   . 

e  o? 

26  7 

,.    -\.\-\   . 

26  Q 

2?    .; 

..  4.n  .. 

.  18.62  —  steam 

It  appears  that,  using  ordinary  steam,  expansion-ratios  within  the  limits 
of  3^  and  4^  are  practically  of  equal  efficiency,  and  that  they  give  the 
highest  efficiency.  The  same  ratios  probably  apply  to  the  use  of  super- 
heated steam,  of  which  there  is  only  one  result,  No.  25. 

Non- Condensing. — The  results  from  the  cylinder  of  the  "Great  Britain" 
locomotive,  page  884,  show  that  the  highest  ratio  of  expansion  that  was 
tried,  namely  2.94,  gave  the  highest  efficiency.  A  greater  ratio  of  expansion 
would  probably  have  given  a  still  greater  efficiency. 

Compound  Cylinders  with  steam-jackets;  condensing. — Receiver-engines. — 
Comparing  the  marine  engines,  Nos.  40  and  50,  it  appears  that  the  ratio  of 
expansion,  3.426,  gave  more  efficiency  than  3.807.  But  in  the  Bache  and 
the  Rush,  Nos.  19  and  22,  it  appears  that  a  ratio  of  from  6  to  7  was 
most  efficient.  With  Nos.  40  and  5  a,  the  total  initial  pressure  was  49%  Ibs. 
and  56  Ibs.  absolute  per  square  inch;  but  in  Nos.  19  and  22,  it  was  90  Ibs. 
and  82  Ibs. 

Woolf  Engines. — The  ratio  13.65  for  No.  7  was  better  than  the  ratio 
8.007  f°r  No.  8,  requiring  20.55  Iks.  an(^  22-5I  Ibs.  of  water  per  indicator 
horse-power  respectively.  No.  10,  with  a  ratio  10.1,  required  20.93  Ibs.  of 
water;  whilst  No.  26,  with  a  ratio  of  12,  required  24.83  Ibs.  The  most 
efficient  ratio  of  expansion  is  most  probably  about  10. 

Proportional  Ratios  of  Expansion  in  the  first  and  second  cylinders — Receiver- 
engines. — Comparing  Nos.  40,  5  a,  and  22,  the  best  action  is  seen  to  be 
obtained  with  equal  ranges  of  expansion ;  for  No.  5  a  is  better  than 
No.  4  a,  and  No.  22,  in  which  the  ratios  are  equal,  being  each  2}^,  is 
the  best. 

Woolf  Engines. — Nos.  7,  8,  and  10  form  a  curious  series: — 

•XT.  Expansion-ratio,  Total  Expansion-  Steam  consumed 

in  First  Cylinder.  ratio.  per  I.H.P. 

8 2.486 8.007  22.51  Ibs.  or7i  per  cent,  more 

than  per  diagram. 

10 2.985  10.100 20.93  „  or  93        „ 

7  5-269 13-650 20.55  »  or  I03       „ 

It  appears  that,  as  the  initial  proportion  of  steam  condensed  in  the 
first  cylinder  increases, — as  shown  by  the  percentages  in  the  last  column,— 
the  efficiency  increases.  The  best  performances  of  the  Bache,  No.  19,  are 
made  with  a  total  expansion  of  from  6  to  9,  having  from  2^  to  3^  expan- 


888 


PRACTICE   OF   EXPANSIVE  WORKING  OF   STEAM. 


sion-ratios  in  the  first  cylinders,  and  2^  in  the  second.  No.  26,  with  a  total 
expansion-ratio  12,  consumed  24.83  Ibs.  of  steam  per  indicator  horse-power 
per  hour;  and  the  Bache,  No.  19,  for  an  expansion-ratio  of  16.85,  consumed 
25.11  Ibs., — the  highest  rate  of  consumption  recorded  for  Woolf  engines. 

Compound  Engines  without  steam  in  jackets;  condensing.  —  The  only 
example  of  receiver-engine  is  No.  6,  which  is  not  provided  with  steam- 
jackets.  Nos.  9  and  18  are  examples  of  Woolf  engines.  The  results  are 
here  brought  together : — 


No. 

Total  Initial 
Pressure  per 
Square  Inch. 

Actual  Ratios 
of  Expansion. 

Steam  Con- 
sumed per 
I.H.P.  per 
Hour. 

Coal  per 
I.H.P.  per 
Hour. 

6.  Receiver  

Ibs. 
QC  i/ 

(  4.0IO  J 

3  1-857  ( 

_ 

2.  14  Ibs. 
(estimated, 

9.  Woolf.  

°5/z 
48^ 

'  7446  3 

(  3-165  ) 
3  3.221  f 

32.72 

see  note 
to  table, 
page  882). 

1  8. 

QO 

(  10.200  ) 

f  375  "^ 

3  2.44  f 

21  ?6 

QO 

te) 

(  2.73  •) 

3  2.44  1 

23  OA 

QO 

(.  6.66  J 

(-2.31  } 
5  2.44  1 

23.21 

te) 

In  the  Woolf  engines,  it  appears  from  the  results,  that  the  highest  effi- 
ciency is  obtained  by  an  expansion-ratio  of  about  3  in  the  first  cylinder,  with 
a  total  ratio  of  7.  In  the  receiver-engine  No.  6,  the  ratio  in  the  first 
cylinder  was  carried  to  4,  with  a  total  ratio  of  7^,  with  an  apparently 
excellent  result. 


CONCLUSIONS  ON  THE  ACTUAL  PERFORMANCE  OF  STEAM. 

For  the  development  of  the  highest  efficiencies  of  steam,  as  used  in  the 
steam-engine,  the  steam-jacket  or  other  means  for  protecting  the  steam 
from  the  cooling  and  condensing  action  of  the  cylinder,  must  be  employed. 
The  superheating  of  steam  prior  to  its  introduction  into  the  cylinder  is 
probably  the  most  efficient  means  that  may  be  employed  for  this  pur- 
pose. The  application,  to  the  cylinder,  of  hot  gases — hotter  than  the 
steam — is  probably  the  next  best  means;  and  next  comes  the  steam- 
iacket. 


CONCLUSIONS   ON   THE   PERFORMANCE  OF   STEAM.  889 

The  importance  of  sustaining  the  temperature  of  steam  expanded  in  a 
cylinder, — preventing  its  falling  low  and  leading  to  the  cooling  of  the 
cylinder,— is  strikingly  proved  by  the  foregoing  hypothetical  calculations 
of  the  consumption  of  steam  per  indicator  horse-power  in  non-condensing 
cylinders,  on  the  assumption  that  the  resistance  of  the  atmosphere  is  removed, 
— likening  the  conditions  to  those  of  condensing  engines.  In  the  cases  of 
the  portable  engines  and  the  locomotive,  the  consumptions,  on  this  supposi- 
tion, amounted  only  to  17.2  Ibs.  per  indicator  horse-power  per  hour,  for 
the  portable  engine  (No.  15),  with  an  expansion-ratio  of  3.8;  and  to  14.8  Ibs. 
for  the  locomotive  (No.  18),  with  an  expansion-ratio  of  2.94.  These  results 
are  below  anything  that  has  been  recorded  of  single-cylinder  condensing 
engines  for  the  same  ratios  of  expansion;  and  their  superiority  is  due 
doubtless  to  the  fact  that  the  temperature  of  non-condensing  cylinders 
never  falls  below  212°. 

It  is  deducible  from  the  results,  that  the  compound  steam-engine  is  more 
efficient  than  the  single-cylinder  engine,  and  that,  of  the  two  kinds  of 
compound  engines,  the  receiver-engine  is  more  efficient  than  the  Woolf 
engine.  The  reasons  for  the  superiority  of  the  receiver-engine  have  been 
partly  pointed  out  in  the  comparative  analysis,  page  867.  There  is  another 
reason  in  the  fact  that  whilst  the  temperature  of  the  first  cylinder  of  the 
receiver-engine  never  falls  below,  nor  even  down  to,  that  of  the  receiver, 
which  stands  at  a  constant  pressure  and  temperature;  in  the  Woolf  engine, 
on  the  contrary,  the  average  temperature  in  the  first  cylinder  must  be  that  of 
the  steam  expanding  into  the  second  cylinder,  which  falls  continuously  with 
the  expansion. 

The  most  efficient  ratios  of  expansion,  together  with  the  quantities  of 
steam,  or  water,  from  the  boiler,  consumed  per  indicator  horse-power  per 
hour, — deduced  from  the  foregoing  analyses, — are  placed  for  comparison  in 
the  table  No.  304. 

It  is  scarcely  necessary  to  observe,  that  the  evidence  of  the  initial 
condensation  of  steam  during  the  period  of  its  admission  into  the  cylinder, 
is  of  great  importance,  and  that,  clearly,  there  is  a  wide  margin  for  economy 
in  the  employment  of  steam  for  the  production  of  power.  Mr.  Bramwell, 
in  an  excellent  and  interesting  paper  on  marine  engines,  in  I872,1  showed 
that  the  average  consumption  of  coal  per  indicator  horse-power  per  hour, 
by  steam-ships  with  compound  engines  in  long  sea  voyages,  varied  from 
2.8  Ibs.  to  1.7  Ib.  in  nineteen  steamers,  for  which  the  average  consumption 
amounted  to  2.11  Ibs.  The  foregoing  deductions  are  consistent  with,  and 
are  corroborated  by,  these  facts. 

In  the  same  paper,2  Mr.  Bramwell  states  that  in  H.M.S.  Briton,  fitted  with 
compound  engines  on  the  system  of  Mr.  E.  A.  Cowper,  the  steam  was 
heated  within  a  steam-jacket,  on  its  passage  from  the  first  to  the  second 
cylinder,  and  that  the  consumption  of  coal,  at  nearly  maximum  power,  was 
1.98  Ibs.  per  indicator  horse-power  per  hour,  and  that,  at  a  third  of  the 
power,  the  consumption  of  coal  was  as  low  as  1.30  Ibs.  This  evidence  is 
confirmatory  of  the  conclusion  that  the  work  of  steam  is  most  efficiently 
developed  when  it  is  previously  superheated. 

1  "  On  the  Progress  effected  in  the  Economy  of  Fuel  in  Steam  Navigation,  considered 
in  Relation  to  Compound-Cylinder  Engines  and  High-Pressure  Steam,"  in  the  Proceedings 
of  the  Institution  of  Mechanical  Engineers,  1872. 

2  Page  153. 


890 


PRACTICE  OF   EXPANSIVE  WORKING  OF   STEAM. 


From  the  foregoing  and  other  evidence  discussed  in  the  author's  work 
on  the  Steam- Engine,  the  following  summary  table  has  been  prepared, 
showing  the  most  economical  results  of  performance  of  single-cylinder  and 
compound-cylinder  steam-engines  under  various  conditions.  These  results 
are  not  put  forward  as  final;  but  simply  to  indicate  the  directions  in  which 
the  best  action  of  the  steam-engine  may  be  obtained. 

Table  No.  304. — PRACTICAL  PERFORMANCE  OF  STEAM-ENGINES: — THE 
MOST  EFFICIENT  RATIOS  OF  EXPANSION,  AND  THE  QUANTITIES  OF 
WATER  CONSUMED  FROM  THE  BOILER  PER  INDICATOR  HORSE-POWER. 


DESCRIPTION  OF  CYLINDERS. 

Most  Efficient 
Ratio  of 
Expansion. 

Steam,  or 
Water  from  the 
Boiler,  con- 
sumed per 
I.H.P.  per 
Hour. 

initial  volume 

pounds. 

Single  cylinder,  with  steam-jacket,  condensing  :  — 

Thoroughly  steam-jacketted  j  short  ^tordS"*' 

4 
6 

21 
20.6 

Only  side-iacketted  \  c£ng  stro^e"" 

3-2 

21.7 

'          J                                (  Short  stroke.... 

5 

23 

Single  cylinder,  with  steam-jacket,  non-condensing, 
Single  cylinder,  without  jacket,  condensing  :  — 

4 

25 

Long  stroke  

4-5 

20 

Short  stroke.... 

4.25 

25 

Single  cylinder,  without  jacket,  condensing,  steam  ) 

superheated     j 

1  J  /'•  2 

Single  cylinder,  without  jacket,  non-condensing,  ) 

i8'/ 

cylinder  well  protected            .    .       j 

/*> 

Compound  cylinder,  steam-jacketted,  condensing:  — 

Receiver     .                             .         

10 

15  to  16 

Woolf  

12 

14  to  18 

Compound  cylinder,  no  jackets,  condensing  :  — 

Receiver  

6% 

2"^ 

Woolf  

21 

FRICTIONAL  RESISTANCE  OF  STEAM  ENGINES. 
See  page  951. 


FLOW   OF  AIR  AND   OTHER   GASES. 


DISCHARGE  OF  GASES  THROUGH   ORIFICES. 

• 

Am. 

Gases  and  vapours  act  like  liquids  in  flowing  through  orifices  and  tubes, 
in  virtue  of  the  difference  of  the  inside  and  outside  pressures;  and  the 
velocity  of  flow  is  regulated  with  respect  to  the  fundamental  formula  for 
gravity,  page  279, 

v  =  8</T  .................................  (i) 

For  liquids,  the  height  through  which  the  water  falls,  to  the  orifice  of  flow, 
can  be  ascertained  by  direct  measurement;  whilst,  for  gases,  it  is  necessary 
to  find  the  height  for  calculation,  thus  :  —  The  head  due  to  the  difference  of 
pressures  per  square  inch  of  the  gas  or  vapour,  is  equal  to  the  height  of  a 
column  of  the  gas  inside,  i  inch  square,  of  which  the  weight  is  equal  to  the 
difference  of  pressure;  and  if  this  net  pressure  per  square  inch  be  divided 
by  the  weight  of  a  prism  of  the  gas,  i  inch  square  and  i  foot  high,  the 
quotient  is  the  height  in  feet,  of  the  equivalent  column  of  gas,  from  which 
the  velocity  of  flow  is  to  be  calculated. 

Flow  of  Air  through  an  orifice  due  to  small  differences  of  pressure. 

The  velocities  of  flow  due  to  small  differences  of  pressure  measured  by  a 
water-gauge,  are  given  by  the  expression,  — 


in  which  V  =  the  velocity  in  feet  per  second,  2  £-=64.4,  A  =  the  height  of 
the  column  of  water,  in  inches,  measuring  the  difference  of  pressure,  t  =  the 
temperature,  and  /  =  the  barometric  pressure  in  inches  of  mercury.  The 
quantity  773.2  is  the  volume  of  air  at  32°,  and  under  a  pressure  of  29.92 
inches,  when  that  of  an  equal  weight  of  water  at  32°  F.  is  taken  as  i.  The 
expression  may  be  reduced  to  the  simpler  form  — 

Velocity  of  Flow  of  Air  through  an  orifice. 
For  small  differences  of  pressure. 


For  the  temperature  t=  62°  F.,  the  formula  becomes,  by  substitution, 


(4) 


892  FLOW  OF  AIR   AND   OTHER   GASES. 

For  the  temperature  /  =  62°  F.,  and  the  pressure  /=  29.92  inches  of  mer- 
cury —  the  most  usual  values  —  the  formula  becomes, 

.............................  (5) 


These  values  must  be  multiplied  by  the  coefficients  pertaining  to  differ- 
ently formed  orifices,  which  are  given  by  Weisbach,  as  follows  :  — 

ORIFICE.  COEFFICIENT  OF  EFFLUX. 

Conoidal  mouth-piece,  of  the  form  of  the  contracted  vein,  ) 

with  effective  pressure  of  from  .23  to  i.i  atmosjjheres,  )  '9'    °  '99 

Circular  orifices  in  thin  plates,  ......................................  .56  to  .79 

Short  cylindrical  mouth-pieces,  ....................................  .81  to  .84 

The  same,  rounded  at  the  inner  end,  .............................  .92  to  .93 

Conical  converging  mouth-pieces,  .................................  .90  to  .99 

ANEMOMETER. 

When  a  current  of  air  flows  through  a  tube,  which  is  restricted,  or 
reduced  to  a  smaller  diameter  at  some  portion  of  its  length,  the  velocity 
of  the  current  is  accelerated  in  passing  into  the  restricted  portion,  and  is 
retarded  in  passing  out  of  it  into  the  tube  of  normal  diameter;  and,  if  the 
restriction  be  so  formed  as  to  accelerate  and  retard  the  current  without  shock, 
there  is  no  loss  of  head  in  the  operation.  M.  Arson1  employs  this  principle 
in  his  anemometer;  but  he  modifies  the  form  of  the  restriction,  in  so  far  that 
whilst  the  approach  to  the  restriction  is  gradually  contracted,  the  exit  from 
it  is  square,  so  that  the  current  passes  abruptly  from  the  orifice  into  the 
tube  of  full  bore.  The  pressure,  or  rather  the  degree  of  "  vacuum  "  at  the 
exit,  is  measured  by  a  water-gauge  attached  at  the  entering  angle  or  corner; 
and  the  difference  of  the  heights  (hv  —  ti)  due  respectively  to  the  external  or 
barometrical  pressure,  and  the  internal  pressure,  is  equal  to  the  difference 

of  the  heights  (  —  -  —  —\  due  to  the  normal  and  the  maximum  velocities  ; 

\2g       2g) 

that  is, 


M.  Arson  so  adjusts,  by  preference,  the  sectional  area  of  the  restriction, 
that  —  =  2  ^;  whence  the  difference  (hv-h\  becomes  equal  to  -^!.    The 

2g          2g  2g 

difference  (hv-ti)  is  measured  directly  by  the  water-gauge;  and  thus,  by 
simple  computation,  the  normal  velocity,  that  is,  the  velocity  of  the  wind 
blowing  through  the  tube  when  fairly  directed  towards  it,  may  be  determined. 

v2  v  2 

That  the  quantity  —  may  be  equal  to  2  —  ^  ,  the  sectional  areas  of  the  tube 

2g  2g 


and  the  restriction  must  be  as  i  to  /\/^7  or  i  to  1.414.  From  direct 
observation,  it  appears  that  the  results  obtained  by  the  formula,  say 
formula  (  2  ),  page  891,  are  to  be  reduced  by  the  coefficient  0.94,  to  give 
the  actual  velocities. 

1  Compte  Rendu  de  la  Societe  des  Ingenieurs  Civils,  1876,  page  505. 


OUTFLOW  OF   STEAM. 


OUTFLOW  OF  STEAM  THROUGH  AN  ORIFICE. l 

The  flow  of  steam  of  a  greater  pressure  into  an  atmosphere  of  a  less 
pressure,  increases  as  the  difference  of  pressure  is  increased,  until  the  out- 
side pressure  is  reduced  to  58  per  cent,  of  the  absolute  pressure  in  the 
boiler.  The  flow  of  steam  is  neither  increased  nor  diminished  by  reducing 
the  outside  pressure  below  58  per  cent,  of  the  inside  pressure,  even  to  the 
extent  of  a  perfect  vacuum.  In  flowing  through  a  nozzle  of  the  best  form, 
the  steam  expands  to  the  outside  pressure,  and  to  the  volume  due  to  this 
pressure,  so  long  as  it  is  not  less  than  58  per  cent,  of  the  inside  pressure. 
For  an  outside  pressure  of  58  per  cent.,  and  for  lower  pressures,  the  ratio 
of  expansion  is  i  to  1.624.  The  following  table  is  selected  from  Mr. 
Brownlee's  data  in  the  "  Report  on  Safety  Valves,"  to  exemplify  the  vary- 
ing discharges  under  a  constant  initial  pressure  in  the  boiler,  into  various 
outside  pressures.  The  formulas  by  means  of  which  the  results  of 
the  table  were  calculated  are  given  by  Mr.  Brownlee  at  page  30  of  the 
"  Report." 

Table  No.  305. — OUTFLOW  OF  STEAM  : — FROM  A  GIVEN  ABSOLUTE 
INITIAL  PRESSURE  INTO  VARIOUS  LOWER  PRESSURES. 

Initial  Pressure  in  Boiler,  75  Ibs.  per  square  inch. 


Absolute 

Weight  dis- 

Pressure in 
Boiler,  per 
Square 

Outside 
Pressure,  per 
Square  Inch. 

Ratio  of 
Expansion 
in  Nozzle. 

Velocity  of 
Efflux  at  Con- 
stant Density. 

Actual  velocity 
of  Efflux, 
Expanded. 

charged  per 
Square  Inch 
of  Orifice  per 

Inch. 

Minute. 

Ibs. 

Ibs. 

ratio. 

[eet  per  second. 

feet  per  second. 

pounds. 

75 

74 

.012 

227.5 

230 

16.68 

75 

72 

•037 

386.7 

4OI 

28.35 

75 

70 

.063 

490 

521 

35-93 

75 

65 

.136 

660 

749 

48.38 

75 

61.62 

.198 

736 

876 

53-97 

75 

60 

.219 

765 

933 

56.12 

75 

5o 

•434 

873 

1252 

64 

75 

45 

•575 

890 

1401 

65.24 

75 

5    43-46    ( 

|  (58  p.  cent.)    J 

1.624 

890.6 

1446.5 

65.3 

75 

15 

1.624 

890.6 

1446.5 

65-3 

75 

0 

1.624 

890.6 

1446.5 

65.3 

When,  on  the  contrary,  steam  of  varying  initial  pressures  is  discharged 
into  the  atmosphere, — pressures  of  which  the  atmospheric  pressure  is  not 


1  See  on  the  subject  of  the  efflux  of  steam,  Mr.  Wm.  Froude's  paper  on  the  "Dis- 
charge of  Elastic  Fluids  under  Pressure,"  in  the  Proceedings  of  the  Institution  of  Civil 
Engineers,  vol.  vi.,  1847,  page  356;  also,  Mr.  R.  D.  Napier's  account  of  his  experi- 
ments, 1866;  Dr.  Rankine,  in  The  Engineer,  November  and  December,  1869;  and  the 
"  Report  on  Safety- Valves,"  in  the  Transactions  of  the  Institution  of  Engineers  and  Ship- 
Builders  in  Scotland,  vol.  xviii.,  1874-75,  PaSe  13>  from  the  last  of  which  the  particulars 
given  in  the  text  are  derived;  also  Eli  W.  Blake,  in  The  Engineer,  December,  1869,. 
page  418;  Wilson  on  Elastic  Fluids,  in  Engineering,  vol.  xiii,  page  35,  &c.,  1872. 


894 


FLOW   OF   AIR  AND   OTHER   GASES. 


more  than  58  per  cent., — the  velocity  of  efflux,  at  constant  density,  that  is, 
supposing  the  initial  density  to  be  maintained,  is  given  by  the  formula, — 

^  =  3-5953  »J  h    (6) 

-z/=the  velocity  of  outflow  in  feet  per  minute,  as  for  steam  of  the  initial  density. 
^=the  height  in  feet  of  a  column  of  steam  of  the  given  absolute  initial  pressure, 

of  uniform  density,  the  weight  of  which  is  equal  to  the  pressure  on  the 

unit  of  base. 

The  lowest  initial    pressure   to   which   the  formula   applies,  when   the 
steam  is  discharged  into  the  atmosphere  at  14.7  Ibs.  per  square  inch,  is 

(14.7  x  I£^  =  )  25.37  Ibs.  per  square  inch.     A  number  of  examples  of  the 

58 

application  of  the  formula  are  given  in  table  No.  306,  for  initial  absolute 
pressures  of  from  25.37  Ibs.  to  100  Ibs.  per  square  inch. 

The  truth  of  the  formulas  is  confirmed  with  a  surprising  degree  of  exact- 
ness by  the  experiments  of  Mr.  Brownlee. 


Table  No.  306. — VELOCITY  OF  EFFLUX  OF  STEAM  INTO  THE  ATMOSPHERE. 


Absolute 
Initial 
Pressure 
per  Square 
Inch. 

Outside 
Pressure 
per  Square 
Inch. 

Ratio  of 
Expansion 
in  Nozzle. 

Velocity  of 
Efflux,  as  at 
Constant 
Density. 

Actual 
Velocity  of 
Efflux, 
Expanded. 

Weight  of 
Steam  dis- 
charged per 
Minute,  per 
Square  Inch. 

Ibs. 

Ibs. 

ratio. 

feet  per  second. 

feet  per  second. 

pounds. 

25-37 

14.7 

1.624 

863 

1401 

22.81 

3° 

14.7 

1.624 

867 

1408 

26.84 

40 

14.7 

1.624 

874 

1419 

35.18 

45 

14.7 

1.624 

877 

1424 

39-78 

5o 

14.7 

1.624 

880 

1429 

44.06 

60 

14.7 

1.624 

885 

1437 

52.59 

70 

14.7 

1.624 

889 

1444 

6l.07 

75 

14.7 

1.624 

891 

1447 

65.30 

90 

14.7 

1.624 

895 

1454 

77-94 

IOO 

14.7 

1.624 

898 

1459 

86.34 

FLOW   OF   AlR    THROUGH    PIPES   AND    OTHER    CONDUITS. 

Mr.  Hawksley1  states,  as  the  result  of  varied  experience,  that  the  formula 
put  forward  by  him  for  the  flow  of  water  in  pipes,  given  at  page  933,  may 
be  employed  also  for  the  flow  of  air  in  pipes.  It  is, 

•  (7) 


in  which  v  is  the  velocity  in  feet  per  second,  h  is  the  head  in  feet  of  air, 
d  is  the  diameter  in  feet,  and  /  is  the  length  in  feet.  But,  it  is  convenient 
to  express  the  head  in  inches  of  water.  Taking  the  density  of  water  as 


1  Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xxxiii.,  page  55- 


FLOW  OF  AIR   THROUGH   CONDUITS.  895 

815  times  that  of  air,  the  multiplier  (?^  =  )  68  is  to  be  placed  under  the 
sign,  when  ^  =  48  ^/ -_£,  and  by  reduction,— 

Flow  of  Air  through  pipes. 

(8) 


156,800 

•z/  =  velocity,  in  feet  per  second. 

h  =  the  head,  in  inches  of  water. 

</=the  diameter  in  feet. 

/  =  the  length  in  feet. 

£=the  perimeter  in  feet. 

a  =  the  sectional  area  in  square  feet. 

Q  =  V  x  a  =  the  quantity  of  air  discharged,  in  cubic  feet  per  second. 

H  =  the  horse-power  required. 

For  passages  or  conduits  of  irregular  forms,  as  shafts  and  air-ways  in 
mines  and  tunnels,  the  perimeter  and  the  sectional  area  become  factors  in 
the  formula:1 — 

Flow  of  Air  through  passages  of  any  form  of  section. 

(10) 

m 

h— .  \"/ 

633,000  a 

The  quantity  of  air  discharged  is  expressed  by  the  product  of  the  velocity 
in  formulas  (  8  )  and  (  10  ),  and  the  sectional  area,  or  by  (v  x  a);  whence, 
by  reduction, 

Quantity  of  Air  discharged 

from  a  pipe, 0  =  311  */ (12) 

v      / 

from  a  passage  of  any  form  of  section,  Q=  796  \/ ^—r (  J3  ) 

c  / 

The  effective  horse-power  expended  on  the  net  work  done  in  drawing 
air  through  a  pipe  or  other  passage,  is  expressed  by  the  product  of  the 
sectional  area  by  the  velocity  in  feet  per  second,  and  by  the  head  or 
"drag"  in  pounds  per  square  foot,  divided  by  550.  That  is  to  say,  H  = 

21 — ,  in  which  i  inch  of  water  is  taken  as  equivalent  to  a  pres- 

1  These  formulas  have  been  worked  out  from  Mr.  Hawksley's  fundamental  formula, 
by  the  author,  in  the  Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  xliv.,  page  90, 
in  their  application  to  tunnels;  and  also  in  Simms1  Practical  Tunnelling,  3<i  edition. 
Mr.  Hawksley  states  that  his  formulas  apply  with  exactness  to  the  shafts  and  air-ways 
of  mines. 


896 


FLOW   OF  AIR  AND   OTHER   GASES. 


sure  of  5.20  Ibs.  per  square  foot,  for  any  passage.     By  substituting  .7854^ 
for  a,  the  expression  is  adapted  for  pipes.     Reducing,  the  formulas  are,  — 

Effective  horse-power  in  net  work  of  discharge  of  air  from  straight  passages  :— 


from  a  passage  of  any  form  of  section,  H  = 
from  a  pipe, H  = 


tLOLJ.  =  -£-? 
106      106 


.......   (  1  4  ) 


Substitute  in  these  formulas,  the  values  of  ^,  ( 9 )  and  ( 1 1  ),  and  the 
horse-power  is  given  in  terms  of  the  velocity,  perimeter,  and  length : — 


from  a  passage  of  any  form  of  section,  H  = 


67,000,000 


from  a  pipe, 


H  = 


(16) 


21,200,000 


Flow  of  Compressed  Air  through  pipes. — According  to  the  experiments  of 
d'Aubuisson,  and  those  of  a  Sardinian  commission,  on  the  resistance  of  air 
through  long  conduits  or  pipes,  the  diminution  of  pressure  is  very  nearly 
directly  as  the  length,  and  as  the  square  of  the  velocity,  and  inversely  as 
the  diameter.  The  resistance  is  not  varied  by  the  density.  A  table  of  the 
loss  of  pressure  in  pipes,  for  a  length  of  1000  metres,  and  for  diameters  of 
from  10  to  35  centimetres,  at  velocities  of  from  i  to  6  metres  per  second,  is 
given  by  Mr.  Cornut,1  and  is  here  reproduced  in  English  measures.  The 
absolute  pressure  is  not  stated. 

Table  No.  307. — Loss  OF  PRESSURE  BY  FLOW  OF  AIR  IN  PIPES. 

Length  of  pipe,  1000  metres,  or  3280  feet. 


Diameter  of  Pipe,  in  Inches. 

Velocity  at  the 

4 

6 

8 

10                   12 

14 

Entrance  to  the  Pipe. 

Loss  of  Pressure  in  Ibs.  per  Square  Inch. 

metres  per 
second. 

feet  per 
second. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

I 

3-28 

.114 

.076 

.057 

.057 

.038 

.038 

2 

6.56 

.500 

•343 

.250 

.210 

.172 

•153 

3 

9.84 

1.183 

.800 

.592 

477 

•394 

•343 

4 

13.12 

2.060 

1-374 

1.030 

.840 

.687 

.600 

5 

16.40 

3.200 

2.160 

1.610 

1.290 

1.  100 

.923 

6 

19.68 

4.446 

2.964 

2.223 

1.778 

1.482 

1.280 

At  the  works  for  excavating  the  Mont  Cenis  Tunnel,  the  supply  of  com- 
pressed air  was  conveyed  in  a  cast-iron  pipe  7^  inches  in  diameter.  The 
loss  of  pressure,  and  leakage  of  air,  from  the  supply  pipes,  in  a  length  of 
i  mile  15  yards,  was  only  3^  per  cent,  of  the  head: — the  absolute  initial 
pressure  was  5.70  atmospheres,  and  it  was  reduced  to  5.50  atmospheres, 

1  See  M.  Cornut's  paper  on  "  Compressed-air  Machinery,"  Bulletin  de  la  Socitte  Indus- 
trielle  Minerale,  1866-67,  PaSe  2OI« 


ASCENSION   OF  AIR   BY  DIFFERENCE  OF  TEMPERATURE.       897 


whilst  there  was  an  expenditure  at  the  rate  of  64  cubic  feet  of  compressed 
air  per  minute.  In  the  middle  of  the  tunnel,  through  a  length  of  pipe  of, 
say,  20,000  feet,  or  3.80  miles,  the  absolute  pressure  only  fell  from  6  atmo- 
spheres to  5.7  atmospheres,  or  to  95  per  cent,  of  the  original  pressure. 

RESISTANCE  OF  AIR  TO  THE  MOTION  OF  FLAT  SURFACES. 

Mr.  James  C.  Fairweather1  found  by  experiment  that  the  law  of  the 
resistance  of  air  to  flat  vanes  moving  through  it  perpendicularly  to  their 
planes,  was  correctly  expressed  by  Dr.  Hutton: — "In  the  case  of  slow 
motion,  nearly  as  the  square  of  the  velocity;  but  gradually  increasing  more 
and  more  above  that  proportion  as  the  velocity  increases."  From  the 
results  given  in  table  No.  30 7 A,  it  may  be  inferred  that  the  resistance  per 
square  foot  increases  with  the  total  area  of  surface  exposed.  These  results 
are  considerably  in  excess  of  those  of  Colonel  Beaufoy,  heretofore  accepted. 

Table  No.  307^ — RESISTANCE  OF  AIR  TO  THE  MOTION  OF  FLAT  VANES. 

In  Ibs.  pressure  on  the  given  surface. 
(Reduced  from  Mr.  Fairweather's  Experiments.) 


Lineal 
Dimension. 

Area. 

5 

s 

10 

jeed,  in  fe< 
15 

:t  per  secon 
20 

d. 
25 

30 

side  of  square. 
741 
12.9 
18.58 

sq.  inches. 

54-85 
166.3 

345-2 

sq.  feet. 
.38 
I.I55 
2.40 

Ibs. 

•55 
i-3 
3-25 

Ibs. 
1.4 

5-5 
15.0 

Ibs. 

3'25 
13-6 

Ibs. 

57 

Ibs. 
9.4 

Ibs. 
14.0 

diameter  of 
circle. 

7.24 
12.65 
18.36 

41.15 
125.8 
264.8 

.286 

.875 
1.840 

•30 
.85 
2.4 

I.I5 

3.85 
1  0.0 

2.6 
9.1 

4.6 
16.4 

7-4 

10.9 

ASCENSION  OF  AIR  BY  DIFFERENCE  OF  TEMPERATURE. 

Mr.  Hawksley2  gives  the  following  formula  for  the  velocity  of  air  in  the 
up-cast  shaft  of  a  mine,  due  to  the  different  weights  of  the  columns  of  air, 
of  different  temperatures,  in  the  up-cast  and  down-cast  shafts,  comprising 
allowance  for  frictional  resistances. 


DJ 


7+368 


(18) 


T  =  the  temperature  of  the  air  in  the  up-cast  shaft,  in  Fahrenheit  degrees. 

t = the  temperature  of  the  air  in  the  down-cast  shaft. 
D  =  the  depth  of  the  shaft,  in  feet. 
m  —  the  periphery  of  the  air-course,  in  feet. 

j-=the  section  of  the  air-course,  in  square  feet. 

/=  the  length  traversed  by  the  current,  in  feet. 

z/=the  velocity  of  the  current,  in  feet  per  second. 

1  "  Resistance  of  Air,"  Proceedings  of  the  Royal  Society  of  Edinburgh,  1872-75 ;  vol.  viii., 
page  35 1. 

2  Proceedings  of  the  Institution  of  Civil  Engineers,    1847,   vol.   vi.,   page   192.     Mr. 
Hawksley  states,  vol.  xxx.,  page  304,  that  the  formula  tallies  exactly  with  the  results  of 
Mr.  Nicholas  Wood's  experiments  on  the  ventilation  of  collieries. 

57 


WORK   OF    DRY  AIR   OR   OTHER   GAS, 
COMPRESSED   OR   EXPANDED. 


In  this  investigation  of  the  work  of  air  by  compression  and  by  expansion, 
the  following  symbols  are  employed : — 

/  =  the  temperature  of  the  gas  in  Fahrenheit  degrees. 

T  =  the  absolute  temperature  of  the  gas  =  t+  46 1°. 

h  =  the  specific  heat  of  the  gas,  under  constant  pressure. 

h'  =  the  specific  heat  of  the  gas  under  constant  volume. 

K  =  the  specific  heat  of  i  pound  of  the  gas,  under  constant  pressure,  in  foot- 
pounds. 

K'  =  the  specific  heat  of  i  pound  of  the  gas,  under  constant  volume,  in  foot- 
pounds. 7  T£ 

7  =  the  ratio  of  the  specific  heats  of  a  gas  =  -,  =  —. 

J  =  the  mechanical  equivalent  of  a  unit  of  heat  =  772  foot-pounds. 
P  =  the  total  pressure  of  the  gas,  in  pounds  per  square  foot. 
p  =  the  total  pressure  of  the  gas,  in  pounds  per  square  inch. 
V  =  the  volume  of  the  gas,  in  cubic  feet. 
v  =  the  volume  of  clearance  at  each  end  of  the  cylinder. 
W  =  work  done  in  foot-pounds. 

It  was  stated,  page  349,  that  the  product  of  the  volume  and  the  pressure 
of  i  pound  of  a  gas,  is  equal  to  the  product  of  the  absolute  temperature 
and  a  constant  coefficient;  or, 

VP  =  *T,    (i) 

V/  =  *"T,    (2) 

in  which  a  and  a'  are  the  constants  to  be  used  respectively  for  the  pressures 
P  and  p.  The  values  of  a  have  already  been  given  for  several  gases,  in 
table  No.  114,  page  349,  at  pressures,  /,  per  square  inch.  For  the  values, 
a,  due  to  pressures,  P,  per  square  foot,  those  values  of  a'  are  to  be  multi- 
plied by  144;  or  the  values,  a,  may  be  deduced,  independently,  from  the 
respective  densities  of  the  gases.  The  two  series  of  values  of  the  constants 
a  and  a',  are  here  annexed  for  several  gases : — 

Constant  a,  -     Constant  a', 

One  Pound  of  Gas.  Formula  ( i ).  Formula  (  2  ). 

Hydrogen, 767.4  5-332O° 

Gaseous  steam, 85.4  59372 

Nitrogen, 54.72  38o27 

Olefiantgas, 53.98  37506 

Air, 53-15  36935 

Oxygen, 48.07  334o6 

Carbonic  acid  (ideal), 35.00  24322 

Do.          (actual), 34.76  24155 

Sulphuric  ether  vapour  (ideal), 20.49  14246 

Vapour  of  mercury  (ideal), 7.62  05296 


WORK   OF   DRY  AIR  AT  CONSTANT  TEMPERATURE.          899 

It  follows  from  the  general  equations,  as  was  stated  at  page  345,  that  the 
pressure  of  a  gas  varies  inversely  as  the  volume,  when  the  temperature  is 
constant;  and  that  the  product  of  the  pressure  and  the  volume  is  propor- 
tional to  the  absolute  temperature.  When  the  temperature  is  uniform, 
therefore,  the  product  of  the  pressure  and  volume  of  a  given  weight  of  a 
gas  is  also  uniform;  and,  if  the  gas  be  either  compressed  or  expanded,  it  follows 
the  hyperbolic  ratio.  The  resulting  hyperbolic  curve  of  expansion  or  of 
compression  is  called  an  isothermal  curve,  or  curve  for  constant  temperature. 

When  a  mass  of  a  gas  is  either  compressed  by  or  expanded  on  a  piston 
within  a  cylinder,  of  non-conducting  materials,  so  that  heat  is  neither 
received  nor  given  out  by  the  gas,  the  curve  of  compression  or  of  expansion 
is  called  an  adiabatic-curve.  The  work  of  compression,  as  internal  work,  is 
converted  into  heat,  which  pervades  the  gas  and  raises  its  temperature; 
and,  reversely,  a  portion  of  the  heat  of  the  gas  is  converted  into  the  work 
of  expansion,  as  internal  work,  and  the  temperature  is  lowered. 

The  general  equation  for  air,  with  the  values  of  the  constants  a  and  a', 
are: — 

(3) 

(4) 


WORK   OF   DRY   AIR   AT   CONSTANT   TEMPERATURE, 
OR   ISOTHERMALLY. 

When  the  temperature  is  constant,  say,  at  62°  F.,  the  absolute  tempera- 
ture is  461°  +  62°  =  523°,  and  the  value  of  the  constant  products  in  formulas 
(3)  and  (4),  are,— 


V  /  =  .  36935  T=  193-2  ......................  (6 

Since  V  P  =  V  P',  in  which  V  and  P'  are  any  other  corresponding  volume 
and  pressure,  at  the  same  temperature,  the  relation  stands  as  follows  :  — 

Isothermal  Compression  or  Expansion  of  Air. 

?-*  ..................................  <» 

The  hyperbolic  ratio  of  expansion,  or  of  compression,  followed  by  air  at  a 
constant  temperature,  has  already,  page  822,  been  taken  as,  practically,  the 
ratio  according  to  which  steam  is  expanded  in  the  cylinder;  and  the  nature 
of  the  relation,  and  the  deductions  from  it,  which  have  there  been  con- 
sidered, apply  also  to  the  case  of  air. 

WORK  OF  COMPRESSION  OF  AIR  AT  CONSTANT  TEMPERATURE, 
WITHOUT  CLEARANCE. 

If  there  be  no  clearance,  and  if  there  were  no  back-pressure,  the  work  of 
simple  compression  in  one  stroke,  calculated  in  terms  of  the  initial  work 
P'V,  or  the  equivalent  PV:— 

yplogV     .....................   (8) 


900  WORK  OF  GASES   COMPRESSED   OR  EXPANDED. 

represented  by  the  area  E  F  B  D,  Fig.  348.     But  there  is  an  assisting  pres- 
E  _  c    sure  P',  —  the  initial  atmospheric  pressure,  —  the  work 

-  of  which   is    (P'  x  V),  represented   by   the   rectangle 
~   DF,  which  is  equal  to  (P  -  P')  V,  the  rectangle  CG; 

-  and  the  net  work  for  simple  compression,  is 

(9) 


Tota(  Work  for  One  Stroke  °f  an  Air-compressor.—- 
ature,  or  isothermaiiy.    There  is  to  be  added  the  work  of  expelling  the  air 
from  the  cylinder  into  a  reservoir,  equal  to, 

(P'-P)V',   .........................  (10) 

and  the  sum  of  (  9  )  and  (  10  ),  reduced,  is  :  — 

Total  Net  Work  for  One  Stroke  of  the  Air-compressor,  isothermally, 
without  Clearance. 

W  =  PVhyplogX.;   ....................   (n) 


(12) 


WORK  OF  COMPRESSION  OF  AIR  AT  CONSTANT  TEMPERATURE, 
WITH  CLEARANCE. 

The  work  of  compression  in  one  stroke,  leaving  back-pressure  out  of 
calculation,  is, 


hyplog  .   ...  ...........  (13) 

Deduct  the  work  of  atmospheric  back-pressure,  or  (P'  -  P)  (V  +  v)  ;  and 
add  the  work  expended  in  expelling  the  air  from  the  cylinder  into  a 
reservoir,  equal  to, 

(P'-P)V'  ............................  (14) 

The  resulting  total  net  work  is, 

Total  Net  Work  for  One  Stroke  of  the  Air-compressor,  isothermaiiy, 
with  Clearance. 

-(P'-P)*  ...........  (15) 


WORK  OF  EXPANSION  OF  AIR,  AT  A  CONSTANT  TEMPERATURE. 

When  the  pressure  of  the  air  at  the  end  of  the  stroke  is  reduced  by 
expansion  to  an  equality  with  the  back-pressure,  the  equations  (8)  to  (15  ), 
for  compression,  express  also  the  relations  of  pressure,  volume,  and  work, 
for  expansion  —  represented  also  by  Fig.  348. 


ADIABATIC   COMPRESSION.  901 

Total  Net  Work  for  One  Stroke  of  a  Compressed-air  Engine,  when  the 
air  is  expanded  down  to  the  back-pressure: — 

Without  clearance...  W  =  P  V  hyp  log— (  16  ) 

With  clearance W  =  P  (V  +  z>)  hyp  log  Yjt^_  (p  -P')z/...  (17) 

When  the  back-pressure  P"  is  less  than  P',  the  lower  limit  of  the  positive 
pressure,  there  is  a  sudden  fall  of  pressure  at  the  end  of  the  stroke,  from 
P'  to  P",  and  the  expression  for  the  total  net  work  is : — 

PV (18) 


WORK   OF   DRY  AIR   IN   A   NON-CONDUCTING    CYLINDER, 
ADIABATICALLY. 

The  specific  heat  of  i  Ib.  of  air  in  foot-pounds  of  work,  is 

Unit.  Foot-pounds. 

At  constant  pressure, 2377x772  =  1 83.45  =  K. 

At  constant  volume, 1688x772=130.3  =K'. 

Difference, 53.15  =a=  K-K'. 

The  difference,  53.15  foot-pounds,  is  equal  to  the  value  of  the  constant  a, 
formula  (  i  ),  for  air,  as  given  in  formula  ( 3 ),  page  899. 

The  ratio  of  the  specific  heats  at  constant  pressure  and  constant  volume 
is  as  1.408  to  i,  or  1.408,  whether  they  are  expressed  in  heat-units  or  in 
foot-pounds. 

ADIABATIC  COMPRESSION  OF  A  GAS. 

Suppose  that  the  gas,  having  the  initial  pressure  P,  volume  V,  and 
temperature  T,  is  compressed  adiabatically,  and  attains  the  pressure  P', 
volume  V,  and  temperature  T';  the  relations  of  the  pressure,  volume,  and 
temperature  are  as  follows: — 

Adiabatic  Compression. 


Showing  that,  for  air,  the  pressure  according  to  the  adiabatic  curve,  varies 
inversely  as  the  1.408  power  of  the  volume: — that,  in  fact,  the  product 
P  Vy  is  a  constant  quantity ;  and  that  the  absolute  temperature  varies  as 


902 


WORK  OF   GASES   COMPRESSED   OR   EXPANDED. 


the  .29  power  of  the  pressure,  and  inversely  as  the  .408  power  of  the 
volume.  For  instance, 

when  the  pressure  is  doubled,  or  as i  to  2 

the  volumes  are  inversely  as  i  to  1.636,  or  directly  as,  i  to    .611 
the  absolute  temperatures  are  as i  to  1.222 

Table  No.  308  contains  the  relative  values  of  the  ratios  of  the  initial 

T"  V  P' 

and  final  temperatures  — ,  and  volumes  — ,  for  given  ratios  — ,  of  initial 

and  final  pressures,  1.2  to  10  times,  calculated  by  means  of  formulas 
(  21  )  and  (  19 ),  with  columns  of  differences  to  facilitate  the  calculation  of 
interpolations  when  required.1 

Table  No.  308. — COMPRESSION  OR  EXPANSION  OF  AIR  WITHOUT 
RECEIVING  OR  GIVING  OUT  HEAT. 

Corresponding  ratios  of  pressure,  temperature,  and  volume,  according 
to  equations  (  19  )  and  (  21 ). 


Ratio  of 
Greater  to 
Less 
Pressures. 

Ratio  of  Greater  to 
Less  Absolute 
Temperatures. 

Inverse  of  these 
Ratios. 

Ratio  of  Greater 
to  Less  Volume. 

Inverse  of  these 
Ratios. 

Ratio  of  Less  to 
Greater  Absolute 
Temperatures. 

Ratio  of  Less  to 
Greater  Volumes. 

numbers. 

differ. 

numbers. 

differ. 

numbers. 

differ. 

numbers. 

differ. 

1.2 

1.054 

48 

-948 

41 

1.138 

132 

.879 

91 

i-4 

I.IO2 

44 

.907 

34 

I.27O 

126 

.788 

72 

1.6 

I.I46 

40 

•873 

30 

1.396 

122 

.716 

57 

1.8 

I.I86 

36 

-843 

25 

1.518 

118 

.659 

48 

2 

1.222 

35 

.8l8 

22 

1.636 

114 

.611 

40 

2.2 

1.257 

32 

.796 

20 

1.750 

112 

-571 

34 

2.4 

1.289 

30 

.776 

18 

1.862 

109 

•537 

30 

2.6 
2.8 

I-3I9 
1.348 

29 

27 

-758 
-742 

16 
15 

I.97I 

2.077 

106 
105 

.^07 
.481 

26 
23 

3 

1-375 

26 

.727 

13 

2.182 

102 

-458 

20 

3-2 

1.401 

25 

.714 

2.284 

100 

.438 

19 

3-4 

1.426 

24 

.701 

ii 

2.384 

99 

.419 

16 

3-6 

1.450 

23 

.690 

ii 

2.483 

97 

-403 

15 

3-8 

1-473 

22 

.679 

10 

2.580 

96 

.388 

14 

4 
4.2 

1.495 
1.516 

21 
21 

.669 
.660 

9 
9 

2.676 
2.770 

94 
93 

-374 
.361 

13 

12 

4-4 

1-537 

2O 

.651 

9 

2.863 

93 

•349 

II 

4.6 

1-557 

ig 

.642 

7 

2.955 

•338 

10 

4.8 

1.576 

19 

-635 

8 

3.046 

89 

.328 

9 

5 

1-595 

86 

.627 

32 

3.135 

434 

•319 

39 

6 

1.681 

77 

•595 

26 

412 

.280 

29 

7 

1.758 

70 

-569 

22 

3^981 

396 

.251 

23 

8 

1.828 

63 

-547 

18 

4-377 

382 

.228 

18 

9 

1.891 

59 

.529 

16 

4-759 

370 

.210 

15 

10 

1.950 

.513 

5.129 

.195 

i 

2 

3 

4 

5 

1  This  table  is  abstracted  from  a  masterly  paper,  "  Etude  Theorique  sur  les  Machines  a 
Air  Comprime;"  by  M.  Mallard  (Bulletin  de  la  Societe  de  V Industrie  Minerals,  1866-67). 


ADIABATIC   COMPRESSION. 


903 


Let  mn  be  the  length  of  a  cylinder  filled  with  i  Ib.  of  gas,  having  the 
initial  pressure  P,  volume  V,  and  temperature  T.  Let  the  gas  be  com- 
pressed adiabatically  by  a  piston  into  the 
volume  V,  to  the  pressure  P',  and  the  tem- 
perature T'.  The  work  of  compression  is 
measured  by  the  area  dgnd',  and  as  shown 
by  Mr.  J.  H.  Cotterill,1  it  is  expressed  by 


for  air,  130.3  (T'-T) 


. 
22  > 


Add  the  work  of  driving  the  compressed  air 

out  of  the  cylinder  into  the  reservoir,  mea- 

sured by  the  rectangle  dm,  equal  to  V'P'; 

and  subtract  the  work  contributed  by  the  air 

from  the  initial  source,—  the  atmosphere,  for 

instance,  —  which  presses  on  the  other  face  of  the  piston,  measured  by  the 

rectangle  gm,  equal  to  V  P.     The  net  work  expended  is,  — 


Fig.  349.—  Compression  of  Air 
adiabatically. 


W  =  K'  (T'-T)  +  V'P'-VP. 
ForairW=i3o.3(T'-T)  +  V'P'-V 


(23) 


By  formula  (  i  ),  V  P'  -  (K  -  K')  T',  and  V  P  -  (K  -  K')  T,  (a  being  =  K 

V'P'-VP  =  (K-K')  (T'-T);) 
for  air,  V'P'- VP  =  53.15  (T'-T)        /  " 

Substituting  the  value  of  V'P'-VP,  in  equation  (23),  and  reducing,  it 
becomes, — 

Work  expended  in  Compressing  i  pound  of  Dry  Gas,  in  terms  of  the 
temperatures, 

W=K(T'-T);     for  air,  W=  183.45  (T'-T) (24) 

That  is,  the  net  work  expended  in  compressing  i  pound  of  gas,  is  equal  to 
the  increase  of  temperature,  or  the  difference  of  the  initial  and  final  temper- 
atures, in  Fahrenheit  degrees,  multiplied  by  the  specific  heat  in  foot-pounds 
at  constant  pressure.  ™f 

When  the  initial  temperature  only  is  given,  T'-T  =  T  (—  -i),  and  by 

substitution  in  formula  (  24 ),  the  final  temperature  may  be  found  when  the 
pressures  are  given : — 

Work  of  Compressing  i  pound  of  Dry  Gas  {formula  to  aid  in  finding  the 
final  temperature). 


for  air,  W=  183.45 


(25) 


P  T' 

Corresponding  to  the  ratio  of  the  pressures  —  r,  the  value  of  —  is  found 

for  air,  in  the  table  No.  308;  thence  the  work,  and  also  the  final  temperature. 

1  Notes  on  the  Theory  of  the  Steam-  Engine.     1871. 


904  WORK   OF   GASES  COMPRESSED   OR   EXPANDED. 

To  express  the  work  in  terms  of  the  pressures,  P  V  =  (K  -  K')  T,  by  for- 

P  V 

mula  ( i ),  and  T  =  — — — 7 .    Substitute  this  value  of  T  in  formula  (24)  -9  and 
Jv  —  Js. 


also  (— ) '29  for  — -;  then,  by  reduction, — 


Work  of  Compressing  i  pound  of  Dry  Gas,  in  terms  of  pressures  and 

initial  volume. 


forair,W  =  3.45PV((.^r-i) 

The  value  of  V,  the  volume  of  a  pound  of  air,  may  be  found  for  various 
pressures  and  temperatures  by  the  formulas  (  i  ),  (  2  ),  page  898. 

P  V  P'  V 

Again,  substitute  the  value  of  T  =  — — — „  and  T'  =  — — — ,  in  equation 

/      \        j     j  "-  —  &-  &-  ~~  •"• 

(24);  and  reduce: — 

Work  of  Compressing  i  pound  of  Dry  Gas,  in  terms  of  pressures  and  volumes. 
W=-A-(P'V'-PV);     for  air,  W  =  3-45  (P'V'-PV) (27) 

JV  —  JV 

To  exemplify  the  rise  of  temperature  by  adiabatic  compression,  take 
atmospheric  air  at  62°  F.,  or  (461+62  =  )  523°  F.  absolute  temperature. 

P' 

In  doubling  the  pressure,  the  ratio  —  =  2,  and  by  the  table  No.  308,  the 

corresponding  ratio  of  the  absolute  temperatures  is  1.222;  whence,  523°  x 
1.222  =  639°,  the  increased  absolute  temperature,  and  639-461  =  178°  F., 
the  final  temperature. 

For  ratios  of  pressure,     2,  3,  4,  5,  10, 

the  ratios  of  the  initial  and  final  absolute  temperatures  are, — 

1.222,     1.375,     1.495,     i-595>     i-95°» 

and  when  the  initial  temperature  is  62°  F.,  the  final  temperatures  are, — 
178°       258°        321°        373°       559°. 

It  may  be  noted  that,  in  this  example,  for  the  ratios  of  pressure,  2,  3,  4,  5, 
and  10,  the  final  temperatures  are,  very  roughly,  3,  4,  5,  6,  and  9  times  the 
initial  temperature  62°. 

ADIABATIC  EXPANSION  OF  GASES. 

Adiabatic  expansion  is  a  duplicate,  in  reverse,  of  the  adiabatic  compres- 
sion of  a  gas  against  a  piston,  and  the  primary  formula  (  i  ),  page  898, — 


(K-K')T, 


with  its  derivatives  (19)  to  (27),  are  applicable,  by  reversing  the  order  of  the 
symbols  of  initial  and  final  pressures,  volumes,  and  temperatures,  defined  at 
page  898. 


ADIABATIC   EXPANSION.  905 

The  compressed-air  engine  differs  from  the  compressing  engine,  in  being 
controlled  by  a  valve  by  which  the  supply  of  air  to  the  cylinder  is  cut  off  at 
any  point  of  the  stroke,  and  any  degree  of  expansion  is  effected.  The  air 
may  thus  be  worked  in  three  ways:  —  ist,  when  it  is  completely  expanded 
down  to  atmospheric  pressure  before  it  is  exhausted;  2d,  when  it  is  admitted 
for  the  whole  of  the  stroke,  and  exhausted  at  full  pressure;  $d,  when  it  is 
only  partially  expanded,  and  exhausted  at  a  pressure  above  atmospheric 
pressure. 

Referring  for  explanations  to  the  discussion  of  adiabatic  compression,  it 
is  sufficient  now  to  repeat  the  formulas  for  compression,  as  adapted  for 
adiabatic  expansive-working. 

When  a  gas  is  completely  expanded  behind  a  piston  from  the  pressure  P, 
volume  V,  and  temperature  T,  to  P',  V,  and  T',  the  relations  are  as 
follows  :  — 

Adiabatic  Expansion  of  a  Gas. 

c       •      P      /V'V4°8  /       x 

'>     for  air>  ^  =        ............  (29) 


The  table,  No.  308,  contains  corresponding  values  of  ratios  of  pressures, 
volumes,  and  temperatures,  to  save  calculation. 

IST.  WHEN  THE  GAS  is  COMPLETELY  EXPANDED  DOWN  TO  AN  EQUALITY 
WITH  THE  BACK-PRESSURE. 

In  the  diagram,  Fig.  349,  let  m  n  be  the  length  of  the  stroke  of  a  cylinder, 
into  which  i  pound  of  a  gas  of  the  pressure  P  is  admitted,  occupying  the 
portion  of  the  stroke  cd,  or  the  volume  V;  and  let  the  gas  be  expanded  to 
the  end  of  the  stroke,  and  the  volume  V,  and  the  pressure  P',  equal  to  the 
pressure  of  the  surrounding  medium,  constituting  back-pressure.  The  initial 
work,  during  admission,  is  measured  by  the  rectangle  dm,  equal  to  VP,  and 
the  back-pressure  by  the  rectangle  gm,  equal  to  V  P'.  The  work  of  ex- 
pansion between  the  initial  and  final  temperatures  T  and  T',  is  measured  by 
the  area  dgnd',  and  is  expressed  by, 

J>4'(T-T')  =  K'(T-T')j     for  air,  130.3  (T-T)  ......  (32) 

That  is,  the  work  by  simple  expansion  is  equal  to  the  fall  of  temperature, 
or  the  difference  of  the  initial  and  final  temperatures  in  Fahrenheit  degrees, 
multiplied  by  the  specific  heat  in  foot-pounds  at  constant  volume. 

Add  the  initial  work,  and  deduct  the  work  of  back-pressure,  and  the  net 
total  work  expended  is, 

W  =  K'(T'-T)  +  VP-V'P',      )  y-\ 

for  air,  W=  130.3  (T  -T)  +  VP  -  V  F  j  V 

By  substitution  and  reduction,  as  was  done  for  compression,  page  903:  — 


906  WORK  OF  GASES  COMPRESSED  OR  EXPANDED. 

Work  performed  by  One  Pound  of  Dry  Gas  expanded  down  to  the  back- 
pressure, in  terms  of  the  temperatures. 

W  =  K(T-T);     for  air,  W=  183.45  (T-T)  (34) 

That  is,  the  net  work  performed  is  equal  to  the  fall  of  temperature,  or  the 
difference  of  the  initial  and  final  temperatures,  in  Fahrenheit  degrees,  mul- 
tiplied by  the  specific  heat  in  foot-pounds  at  constant  pressure.  T, 

When  the  initial  temperature  only  is  given,  T-T  =  T  ( i  -  J_  )-}  and 
by  substitution  in  formula  (  34  ): — 

Work  performed  by  One  Pound  of  Dry  Gas  expanded  down  to  the  back- 
pressure {formula  to  aid  in  finding  the  final  temperature). 

W  =  KT(i-^);    for  air,  W=  183.45  T(i- -II) (35) 

T' 

The  value  of  —  corresponds  in  table  No.  308,  column  3,  to  the  ratio  of 

p 

the  initial  and  final  pressures,  — .  Thence  the  work  may  be  found;  also  the 
final  temperature. 

To  express  the  work  in  terms  of  the  pressures,  P  V  =  (K  -  K')  T,  by 

P  V 

formula  (  28  ),  and  T  =  — ,.     Substitute  this  value  for  T  in  formula 

K  —  K 

(35  );  and  also  (-p-)'29  for  — ;  then,  by  reduction,— 

Work  performed  by  One  Pound  of  Dry  Gas,  in  terms  of  pressure  and 

initial  volume. 


.          _    _         .  (36) 

for  air,  W  =  3.45 

P  V  P'  V 

Again,  substitute  the  value  of  T  =  — — — 7,  and  T'  =  — — —  in  equation  (34)- 

and  reduce,— 

Work  performed  by  One  Pound  of  Dry  Gas,  in  terms  of  pressures 
and  volumes. 

.(PV-FV);    for  air,  W  =  3.45  (PV-FV)....  (38) 


Jv  —  Js. 

To  exemplify  the  fall  of  temperature  by  adiabatic  expansion,  take  atmo- 
spheric air  at  62°  R,  or  (461  +  62=)  523°  F.  absolute  temperature.     In 

p 

reducing  the  pressure  to  a  half,  the  inverse  ratio  —  =  2,  and  the  correspond- 
ing ratio  of  temperatures,  column  3,  table  No.  308,  is  .818;  whence  523°  x 


ADIABATIC  EXPANSION.  9O/ 

.818  =  428°,  the    final   absolute   temperature,  and  461-428=    -33°  F., 
the  final  temperature.     Similarly, 

for  inverse  ratios  of  pressure,  2,  3,  4,  5,  10, 

the  ratios  of  the  initial  and  final  absolute  temperatures  are, 

.818,  .727,  .669,  .627,  .513, 
and  when  the  initial  temperature  is  62°  F.,  the  final  temperatures  are, 

-33°,     -81°,     -in0,     -133°,     -i93°F. 

These  instances  illustrate  the  limitless  possibilities  of  producing  cold  by 
the  expansion  of  air.  It  is  clearly  as  impracticable  to  work  a  compressed- 
air  engine  in  such  low  temperatures,  when  every  particle  of  moisture  and 
lubricant  would  be  frozen,  as  amongst  the  high  temperatures  previously 
noticed. 

2D.  WHEN  THE  GAS  is  ADMITTED  TO  THE  CYLINDER  FOR  THE  WHOLE 

OF  THE  STROKE. 

In  this  case,  there  is  no  expansive  working,  and  the  gas  is  exhausted  at 
full  pressure.  The  work  done  by  i  pound  of  dry  gas  is  — 

W  =  V(P-P'),   ........................   (39) 

in  which  P  and  P'  are   the  initial  and  the  exhaust   pressures.     P  V  = 
(K  -  K')  T,  by  formula  (  28  ),  page  904,  and,  by  inversion, 


and,  by  substitution  and  reduction, 

W  =  (K-K')T(i-^);     for  air,  W  =  53.15  T(i-J').  ...  (41) 

A.gain,  the  general  equation  for  the  work  done  by  i  pound  of  dry  gas  is 
(formula  (  34  ),  page  906), 


W  =  K(T-T);     andW  =  KT(i-),  .......  (42) 

in  which  T  and  T'  are  the  initial  and  the  final  temperatures. 

Equating  these  expressions  for  W,  (  41  )  and  (  42  ),  and,  reducing,1 


for  air,      =  .  71  +.  29      ;  .....  (  43  ) 


By  either  of  these  formulas,  (43,  44),  the  final  temperature  T'  is  found, 

P' 

when  the  initial  temperature  T  is  given.     For  a  ratio,  for  air,  —  -=  ^,  or 

p 

—  -,  =  2,  for  instance,  with  the  initial  temperature  62°  F.,  or  absolute  tempera- 

1  This  method  of  finding  the  final  temperature,  by  equating  the  two  expressions  for  W, 
is  borrowed  from  M.  Mallard.     See  the  preceding  note,  page  902. 


908 


WORK  OF  GASES  COMPRESSED   OR   EXPANDED. 


ture  523°, the  final  temperature  T'  =  5 23  (.7i+.29x  %)  =  523 x. 855  =  447°; 
and  461  -  447  =  -  14°  F. 

To  facilitate  calculation,  by  means  of  formula  ( 43 ),  the  values  of  the 
ratios  of  the  absolute  temperatures  corresponding  to  given  ratios  of  the 
pressures,  are  given  in  table  No.  309. 

Table  No.  309. — COMPRESSED-AIR  ENGINE: — AIR  ADMITTED  FOR  THE 
WHOLE  OF  THE  STROKE. — CORRESPONDING  RATIOS  OF  PRESSURES 
AND  TEMPERATURES. 


Ratio  of  the 
Final  to  the 
Initial  Pressure. 

Ratio  of  the 
Initial  to  the 
Final  Pressure. 

Ratio  of  the 
Final  to  the 
Initial  Absolute 
Temperatures. 

Ratio  of  the 
Final  to  the 
Initial  Pressure. 

Ratio  of  the 
Initial  to  the 
Final  Pressure. 

Ratio  of  the 
Final  to  the 
Initial  Absolute 
Temperatures. 

I 

I 

I 

'/6 

6 

.758 

X 

2 

.855 

«// 

7 

•751 

'/s 

3 

.806 

# 

8 

.746 

1A 

4 

.782 

'/9 

9 

.742 

'/s 

5 

.768 

x/xo 

10 

•739 

The  final  temperatures  of  air  under  adiabatic  expansion,  and  also  when 
exhausted  at  full  pressure,  without  expansion,  due  to  given  ratios  of  pres- 
sure, are  detailed,  for  comparison,  in  table  No.  310,  in  the  second  and  third 
columns.  The  reduced  efficiency  by  adiabatic  expansion,  supposing  the  initial 
temperature  to  fall  to  62°  F.,  given  at  page  910,  is  here  given  in  column  4. 
The  same,  for  full  pressure,  without  expansion,  is  given  in  column  5.  It  is 
calculated  thus,  in  the  first  instance,  for  example : — The  final  temperature, 
column  3,  is  -  14°  F.,  and  is  (62  +  14  =  )  76°  below  62°, — being  the  range  of 
the  temperature  in  doing  work.  But  the  range  of  temperature  in  compress- 
ing the  air  adiabatically  to  twice  the  initial  pressure  is  (178°  (as  at  page  904) 

-62  =  )  116°;  and  (T~?X  100  =  )  66  per  cent,  is  the  reduced  efficiency 
without  expansion,  as  in  column  5.    The  ratios  of  these  reduced  efficiencies, 


Table  No.  310. — COMPRESSED-AIR  ENGINE: — AIR  EXPANDED  ADIABATI- 
CALLY, AND  AIR  ADMITTED  FOR  THE  WHOLE  STROKE. — COMPARA- 
TIVE FINAL  TEMPERATURES,  AND  REDUCED  EFFICIENCIES. 

Initial  temperature  =  62°  F. 


Ratio  of  the 

Final  Temperature. 

Reduced  Efficiency. 

Ratio  of  Reduced 
Efficiencies  :  — 

Initial  to  the 
Final  Pressure. 

With  Adiabatic 
Expansion. 

Without 
Expansion. 

With  Adiabatic 
Expansion. 

Without 
Expansion. 

sion  and  with 
Complete 
Expansion. 

Fahr. 

Fahr. 

per  cent. 

per  cent. 

per  cent. 

2 

-33° 

-14° 

82 

66 

80 

3 

-81 

-40 

73 

52 

71 

4 

-  in 

-52 

67 

44 

66 

5 

-133 

-60 

63 

39 

62 

10 

-193 

-75 

5i 

27.5 

54 

EFFICIENCY  OF   COMPRESSED-AIR   ENGINES.  909 

in  columns  4  and  5,  are  given  in  the  last  column;  found  thus,  in  the  first 

example,  for  instance:  —  (—  x  100  =  )  80  per  cent.     These  ratios  may  also 
82 

be  calculated  as  the  ratios  of  the  ranges  of  temperature  in  the  two  cases.  In 
the  first  instance,  for  example,  (  -  33  +  62  =  )  95°,  and  (  -  14  +  62  =  )  76°,  are 

the  ranges  for  adiabatic  expansion,  and  without  expansion;  and  (L-  x  100  =  ) 

95 

80  per  cent,  is  the  ratio  of  the  reduced  efficiencies.  The  table  indicates, 
generally,  the  economical  disadvantage  of  working  compressed  air  without 
expansion. 

30.  WHEN  THE  GAS  is  BUT  PARTIALLY  EXPANDED. 

The  absolute  temperature  of  the  gas,  when  expanded,  falls  from  T  to  T'' 
at  the  end  of  the  stroke.  Here,  it  is  suddenly  exhausted  into  the  surround- 
ing medium,  and  the  temperature  falls  still  further,  to  T".  The  work  done 
by  i  pound  of  gas,  in  terms  of  the  extreme  temperatures,  is,  by  the  general 
formula  (  34  ),  page  906, 

W  =  K(T-T");  for  air,  W=  183.45  (T-T)  ............  (45) 

whence,  as  in  (  35  ), 

W=KT  (!-);•  for  air,  W=  183.45  T(i-l^)  ........  (46) 

(47) 

When  the  successive  pressures,  P,  P',  P",  are  known,  the  ratios  of  the 

temperatures  in  these  last  two  formulas  are  easily  found  in  the  table  No.  308, 

p"         p"  p' 

page  902,  from  the  ratios  of  the  pressures  —  ,  or  —7,  and  —  ;   when  the 

calculation  for  the  work  may  be  completed. 

The  temperatures  T'  and  T'  may  be  found  from  T;  first,  for  T',  by 
inverting  equation  (31  ),  page  905, 


for  air,T  =  T    - 


Thence,  the  value  of  T',  the  ultimate  temperature,  is  found  according  to 
formula  (  44  ),  page  907,  to  be, 

''=T'(.7i  +  .29        .....  (49) 


EFFICIENCY   OF   COMPRESSED-AIR   ENGINES. 

The  work  by  expansion  would  be  an  exact  duplicate,  in  reverse,  of  the 
work  expended  for  compression,  and  the  two  works  would  be  equal  to  each 
other,  if  the  reverse  actions  took  place  between  the  same  temperatures, 
pressures,  and  volumes.  The  efficiency  of  the  combined  compressor  and 
motor  would  be  equal  to  100  per  cent.,  irrespective  of  losses  by  friction  and 
clearance.  But,  under  practical  conditions,  the  initial  temperature  for 


9IO  WORK  OF  GASES   COMPRESSED   OR   EXPANDED. 

expansion  is  not  more  than  that  of  the  surrounding  atmosphere;  and,  in 
working,  by  expansion,  back  to  atmospheric  pressure,  even  between  the 
same  extremes  of  pressure,  the  volumes  are  smaller,  since  the  temperatures 
are  lower;  and  the  efficiency  must,  of  course,  be  less  than  100  per  cent. 

In  working  air,  under  these  conditions,  between  two  given  pressures, 
first  compressively,  and,  second,  expansively,  let  the  ratios  of  the  pressures, 

P 

which  are  the  same  in  both  actions,  be  — ,  P  being  the  higher  pressure,  and 

P'  the  lower,  or  atmospheric  pressure.  Put  T"  for  the  higher  temperature 
by  adiabatic  compression,  whilst  T  is,  as  before,  the  atmospheric  tempera- 
ture, and  T'  the  final  temperature  by  expansion.  Then,  according  to  for- 

T"     T      /PY29 

mula(3i),  —  =  — -  =  (— - )     ;  that  is  to  say,  the  ratios  of  the  absolute  tempera- 

/P\.29 

tures  are  equal  to  each  other,  since  they  are  each  equal  to  \p,J    .  It  follows 

that, 

T"  :T  :  :T  :  T';  and  that  T"-T  :  T-T'  :  :  T"  :  T; 

that  is  to  say,  the  range  or  difference  of  the  temperatures  for  compression, 
<T"-T),  is  to  the  range  for  expansion  (T-T'),  in  the  ratio  of  the  higher 
absolute  temperatures,  T"  and  T,  for  compression  and  for  expansion  respec- 
tively; and  the  loss  of  efficiency  by  the  intermediate  fall  of  the  temperature 
of  the  compressed  air  from  that,  T",  due  to  the  compression,  to  T,  the 
atmospheric  temperature,  is  simply  the  proportion  which  this  fall,  T"  -  T, 
bears  to  the  maximum  temperature  T". 

It  is  so,  because  the  volume  is  as  the  absolute  temperature  T",  and  the 
loss  of  temperature  T"-T,  indicates  the  loss  of  volume  by  contraction, 
under  the  same  pressure.  For  instance,  in  compressing  dry  air  at  62°  F., 
to  two  atmospheres  of  pressure,  in  a  non-conducting  vessel,  the  temperature 
is  raised  to  1 78°,  and  the  fall  in  reverting  to  62°  is  (i  78  -  62  =  )  1 16°.  The 
loss  of  efficiency  is  the  proportion  of  116°  to  (461  +  178  = )  639°,  the  maxi- 
mum absolute  temperature,  thus : — 

(461 +  178  =  )  639° 
(461+  62  =  )  523 

Difference,  or  loss,....  116°=  18  per  cent,  of  the  maximum  absolute  temperature. 
Leaving 523  =82 

For  ratios  of  pressure,  or  atmospheres 

2,  3,  4,  5,  10, 

the  final  temperatures  for  compression  are, 

178°,          258°,          321°,          373°,  559°Fahr.; 

and  the.  reduced  efficiency,  supposing  the  initial  temperature  for  expansion 
becomes  62°  F.,  is 

82,  73,  67,  63,  5 1  per  cent, 

whilst  the  loss  of  efficiency  is 

1 8,  27,  33,  37,  49 

Here  it  is  obvious  that  the  lower  the  degree  of  compression  applied  to  the 
air,  the  less  is  the  rise  of  temperature,  the  less  is  the  loss  of  heat  by 
dissipation,  and  the  greater  is  the  efficiency  of  the  machine.  When  an 
initial  temperature  can  be  maintained  for  the  expansion-engine,  higher  than 
that  of  the  surrounding  atmosphere,  the  range  of  temperature  within  which 


EFFICIENCY  OF  COMPRESSED-AIR  ENGINES. 


911 


the  air  may  be  expanded  before  it  arrives  at  the  freezing-point,  as  a  lower 
limit,  is  greater  than  if  it  commence  at  atmospheric  temperature;  and  the 
performance  is  also  greater  in  the  same  proportion. 

When  the  compression  is  carried  to  10  atmospheres,  the  efficiency  for 
working  in  a  compressed-air  engine,  above  indicated,  is  only  5 1  per  cent. 

Add,  that  the  efficiencies  of  the  machines  themselves, — the  compressor 
and  the  power-engine, — are  factors  for  the  calculation  of  their  resul- 
tant efficiency;  and  if  the  efficiency  of  each  machine  be  taken  at 

80  per  cent.,  the  combined  percentage  of  the  two  machines  is  (          °  = ) 

100 

64  per  cent.,  or  two-thirds;  and  64  per  cent  of  51  per  cent,  is  33  per  cent, 
the  resultant  efficiency  of  the  combined  compressor  and  engine,  working  to 
i  o  atmospheres.  Similarly,  it  is  found  that  the  resultant  efficiency,  working 
to  2  atmospheres,  is  52  per  cent.  The  less  the  degree  of  compression,  the 
greater  is  the  efficiency;  because  the  less  is  the  proportional  loss  from  the 
intermediate  reduction  of  temperature.  In  general  practice,  the  resultant 
efficiency  rarely  exceeds  30  per  cent. 

M.  Piccard's  illustration. — M.  Piccard1  happily  illustrates  by  examples 
the  difference  of  the  conditions  and  the  efficiency  of  the  work  of  compressed 
air,  in  three  cases,  for  which  he  adopts  the 
initial  temperature  32°  F.  He  supposes  that, 
in  the  ist  and  2d  cases,  the  pressure  and 
temperature  are  raised  adiabatically  during 
compression;  and  that  the  temperature  re- 
lapses to  the  normal  point,  32°  F.,  before 
the  air  is  applied  to  work,  illustrated  by 
Figs.  350  and  351;  and,  in  the  3d  case, 
that  the  temperature  is  constant  at  32°  F., 
whilst  the  air  undergoes  compression  iso- 
thermally;  Fig.  352. 


volumes 


Fig.  350. — ist  case — 
Compression  Adiabatically. 


Fig.  351.— 2d  case- 
Compression  Adiabatically. 


Volumes 

Fig.  352. — 3d  case — 
Compression  Isothermally. 


Illustrations  of  Adiabatic  and  Isothermal  Compression  of  Air ;  with  Adiabatic  Expansion. 

ist  case: — Air  compressed,  cooled,  and  expanded  within  the  same 
cylinder,  without  any  reservoir.  2d  case,  Compressed  air  cooled  in  a 
reservoir.  3d  case,  Air  cooled  during  compression.  The  pressure  to  which 
the  air  is  compressed  is,  in  each  case,  6  atmospheres;  whilst  the  final 
volumes  to  which  it  is  compressed,  taking  the  initial  volume  as  i,  are, — 

1  "Du  Rendement  de  1'Air  Comprime,"  Bulletin  de  la  Societe  Vaudoise  des  Ingenimrs  et 
des  Architectes,  June,  1876,  page  10;  abstracted  in  the  Proceedings  of  the  Institution  of 
Civil  Engineers,  vol.  xlv.,  1875-76,  page  273. 


912  WORK  OF   GASES   COMPRESSED   OR   EXPANDED. 


Final  Volume.  Vohime  During  the  Interval 

ist  case,...  0.281  or  I/3.5  pressure  falls  to  3.56  atmospheres. 
2d  case,...  0.281  or  i/3.5  final  volume  reduced  to  x/e  initial. 
3d  case,...  0.167  or  J/6  pressure  and  volume  stationary. 

The  final  pressures,  volumes,  and  temperatures  are  subjoined;  and  to  these 
are  added  the  efficiency  for  each  case,  or  the  ratio  of  the  useful  work  done 
to  the  work  expended  in  producing  the  supply  of  compressed  air  :  — 

Volume  Final  Temperature.  Efficiency. 


Atneres.      T,  Initial  =  ^  F'  Compression  =  ,. 

ist  case,  .........  i     .......  69       ......     -ii9°F  .......     36.4  per  cent. 

2d  case,  .........  i     .......  595     ......     -  168  ......     59.2      „ 

3d  case,  .........  i     .......  595     ......     -  168  ......     78.0      „ 

Ordinary  practical  conditions  oscillate  between  cases  2  and  3;  and  it 
is  clear  that,  the  more  the  air  is  cooled  during  the  process  of  compression, 
the  less  is  the  expenditure  of  work  on  compression,  and  the  greater  is  the 
resultant  efficiency.  M.  Piccard  gives  the  following  for  the  respective 
efficiencies  for  various  pressures:  —  in  the  3d  case,  and  in  the  case  when 
the  air  is  admitted  for  the  whole  of  the  stroke,  without  expansion  :  — 

Pressures.  Efficiency  in  the  sd  Case.  Efficiency,  without  Expansion. 

atmospheres.  per  cent.  per  cent. 

1  ..................  100  ..................  100 

2  ..................  90.6  ..................  72.1 

4  ..................  82.4  ..................  54.1 

6  ..................  78.0  ..................  46.0 

8  ..................  75-2  ..................  42.1 

10  ..................  72.9  ..................  39-1 

It  may  be  inferred  that,  under  every  condition,  the  efficiency  is  reduced 
as  the  pressure  is  multiplied. 

COMPRESSION  AND  EXPANSION  OF  MOIST  AIR. 

M.  Mallard  has  investigated  the  influence  of  moisture  in  air  upon  the 
variations  of  temperature,  and  on  the  work  of  compression  or  expansion. 
The  principal  results  of  the  investigation  are  here  given.  It  is  assumed 
that  the  vapour  generated  from  the  moisture  is  always  in  the  condition 
of  saturation. 

Temperature  in  Compression.  —  The  rise  of  temperature  is  much  less  when 
moisture  is  present  in  the  air,  than  when  the  air  is  dry,  and  is  compressed 
adiabatically.  Atmospheric  air  at  68°  F.  initial  temperature,  when  com- 
pressed to  7^  atmospheres,  rises,  if  dry,  to  490°  F.;  and,  if  sufficiently  moist, 
to  194°  F.  only. 

Work  for  Compression.  —  The  work  is  the  same  for  dry  air  and  moist  air 
at  68°  F.  when  compressed  to  i^  atmospheres.  For  a  less  degree  of 
compression,  it  is  rather  less  for  dry  air;  but  for  higher  compressions,  it  is 
less  for  moist  air.  For  7^  atmospheres,  it  is  14  per  cent.  less. 

Proportion  of  Moisture  in  Saturation.  —  The  weight  of  saturated  vapour 
in  moist  air  at  68°  F.,  compressed  to  from  i^  to  7%  atmospheres,  is  from 
2>£  to  6^  per  cent,  of  the  weight  of  the  air. 

Particulars  of  the  compression  of  air,  dry  and  moist,  are  given  in 
table  No.  311:  — 


COMPRESSION   AND   EXPANSION   OF   MOIST  AIR. 


913 


Table  No.  311. — COMPRESSION  OF  AIR,  DRY  AND  MOIST. — TEMPERATURE 

AND  WORK. 

(Deduced  from  M.  Mallard's  data.) 


Final 
Pressures. 

Final  Temperatures  for 
Compression. 

Work  Expended  in  Com- 
pressing i  pound  of  Air. 

Moisture 
Required  to 
Produce 
Saturation  in 

Initial  Temperature=68°  Fahr. 

Imtia.1  Pres- 
sure =  i 

Air  with 

Air  with 

parts  of  the 
Weight  of 

Atmosphere. 

Dry  Air. 

Sufficient 
Moisture. 

Dry  Air. 

Sufficient 
Moisture. 

the  Air  Com- 
pressed. 

atmospheres. 

Fahr. 

Fahr. 

foot-pounds. 

foot-pounds. 

per  cent. 

i# 

133° 

94° 

13,300 

13,200 

2.4 

2 

I85 

III 

23,500 

22,500 

3-0 

2^ 

229 

124.5 

30,500 

29,OOO 

3-6 

3 

266 

135-5 

37,000 

35,000 

4.0 

^ 

300 

145-4 

43,200 

40,500 

44 

4 

330 

153-5 

48,500 

45,000 

4-8 

4/2 

357 

161.6 

53,600 

49,000 

5-1 

5 

383 

167 

58,500 

52,500 

54 

slA 

407 

173 

63,200 

56,500 

5-7 

6 

428 

179 

67,000 

60,000 

6.0 

6/2 

440 

184 

71,000 

63,000 

6.2 

7 

470 

190 

75,000 

66,000 

6.4 

7/2 

49° 

194 

78,300 

68,300 

6.6 

Work  in  Expansion. — There  is  a  slight  gain  in  work  done,  by  the 
presence  of  vapour  in  the  air,  in  a  state  of  saturation;  but  it  may  be 
neglected  in  ordinary  calculations. 

Table  No.  312. — EXPANSION  OF  AIR,  DRY  AND  MOIST. — 
TEMPERATURES. 

(Reduced  from  M.  Mallard's  data.) 


Temperatures. 

Ratio  of  Expansion. 

Final. 

Initial. 

Dry  Air. 

Air  with  Suffi- 
cient Moisture. 

Fahr. 

Fahr. 

ratio. 

ratio. 

32° 

40° 

1.05 

1.  10 

32 

50 

1.13 

1.24 

32 

60 

1.22 

1.38 

32 

62 

1.23 

I.4I 

32 

68 

I..28 

1.50 

32 

70 

1.30 

1.56 

32 

80 

1-37 

1-75 

32 

90 

1.47 

2.00 

32 

100 

2.28 

32 

no 

1.67 

2.63 

32 

120 

1.76 

3.00 

32 

130 

1.88 

345 

32 

140 

2.0O 

4.00 

914  WORK   OF  GASES   COMPRESSED   OR  EXPANDED. 

Temperature  in  Expansion. — When  moisture  is  present  in  air  in  the 
condition  of  saturation,  the  fall  of  temperature  during  expansion,  is  greatly 
less  than  what  takes  place  when  dry  air  is  expanded.  That  a  compressed- 
air  engine  may  work  without  the  freezing  of  any  moisture  or  vapour  in  the 
air,  it  should  not  exhaust  at  a  temperature  lower  than  the  freezing-point. 
Table  No.  312,  page  913,  shows  a  few  examples  of  the  maximum  ratio  of 
expansion  that  may  be  practised,  with  given  initial  temperatures,  when  the 
final  temperature  is  to  be  32°  F. : — 

The  table  shows  that  air  at  120°  F.  may  be  introduced  into  the  cylinder 
at  a  pressure  of  3  atmospheres,  and  expanded  to  atmospheric  pressure, 
without  risk  of  interference  from  the  freezing  of  moisture;  whilst  with  dry 
air,  the  maximum  pressure,  under  the  same  condition,  is  only  1.76  atmo- 
spheres. 


AIR    MACHINERY. 


MACHINERY  FOR  COMPRESSING  AIR,  AND   FOR  WORKING 
BY  COMPRESSED  AIR. 

COMPRESSION  OF  AIR  BY  WATER  AT  MONT  CENIS  TUNNEL  WORKS 

(COMPRESSEURS   A    COLONNE    D'EAU).1 

The  motive  power  was  derived  from  the  fall  of  a  column  of  water,  having  a 
head  of  85^  feet,  acting  on  the  principle  of  a  hydraulic  ram,  —  the  water,  by 
the  power  of  its  fall,  compressing  a  given  quantity  of  air  at  each  stroke. 
There  were  1  1  rams,  to  each  of  which  the  water  was  conducted  from  the 
reservoir  by  a  24-inch  pipe.  Each  ram  made  from  2^  to  3  strokes  per 
minute,  and  the  air  was  compressed  to  6  atmospheres  of  total  pressure. 
The  volume  of  air  at  atmospheric  pressure,  shut  in  and  compressed  for 
service  at  each  stroke  of  the  ram,  was  measured  by  a  column  in  the  air-limb 
of  the  pipe,  2.04  feet  in  diameter  and  14.1  feet  high,  making  a  volume  of 

46.1  cubic  feet  of  atmospheric  air,  or  (46.1^6  =  )  7.68  cubic  feet  of  com- 
pressed air.     The  volume  of  compressed  air  for  2  ^  strokes  per  minute  was 

19.2  cubic  feet  per  minute.     The  net  horse-power  is 

[  (6  x  15)  x  144  x  19.2  xhyp  log  6]-^33ooo  =  13.51  horse-power. 

The  total  expenditure  of  power  in  the  water  for  generating  compressed 
air  was,  — 


2.o42x.7854x  14.1  x62^  Ibs.  x85^  feet  x  2^  strokes  _  jg     H  p 

33000 
The  efficiency  was,  thus,  equal  to  73  per  cent. 

COMPRESSION  OF  AIR  BY  DIRECT-ACTION  STEAM-PUMPS. 

In  the  temporary  machines  used  at  the  works  for  the  St.  Gothard  tunnel, 
the  steam-piston,  19.7  inches  in  diameter,  was  fixed  to  the  same  rod  with 
the  air-piston  of  17.73  inches,  with  a  stroke  of  4  feet.  The  air-pumps 
worked  in  water.  The  minimum  number  of  double  strokes  per  minute 
was  5,  but  the  machine  could  make  20  per  minute.  In  compressing  air 
to  3  atmospheres,  the  efficiency,  according  to  the  indicator-diagrams,  was 
84  per  cent. 

These  pumps  have  been  replaced  by  others  on  Colladon's  system,  in 
which  the  air-cylinder  is  kept  cool  by  exposing  every  piece  that  is  in  contact 
with  the  air  when  undergoing  compression,  to  currents  of  cold  water.  The 
pump  makes  90  revolutions  per  minute,  and  is  maintained  sufficiently  cool 
in  compressing  air  to  8  atmospheres  of  pressure. 

1  Simms1  Practical  Tunnelling,  3d  edition,  1877,  page  261. 


916 


AIR  MACHINERY. 


COMPRESSED-AlR   MACHINERY   AT   POWELL   DUFFRYN   COLLIERIES.1 

This  machinery  was  constructed  by  Messrs.  J.  Fowler  &  Co.,  for  Sir  George 
Elliott.  There  is  a  pair  of  horizontal  air-compressing  engines  connected  to 
one  shaft,  the  steam-cylinder  and  the  air-cylinder  being  in  one  line,  on  the 
same  rod.  The  steam-cylinders  are  34  inches,  and  the  air-cylinders  40  inches 
in  diameter,  with  a  stroke  of  6  feet.  The  engine  is  worked  with  steam  of 
70  Ibs.  effective  pressure,  cut  off  at  one-fourth,  and  is  fitted  with  Cornish 
steam-  and  exhaust-valves,  8  inches  and  9  inches  in  diameter.  The  engines 
make  20  turns  per  minute,  giving  240  feet  of  piston  per  minute,  to  indicate 
482  horse-power,  against  a  pressure  of  air  of  40  Ibs.  per  square  inch  above 
the  atmosphere.  The  air-cylinders  are  immersed  each  in  a  cold-water 
bath,  open  at  the  upper  side. 

Experiments  were  made  with  a  double-cylinder  air-compressing  engine, 
similar  in  arrangement  to  the  above,  having  1 6-inch  cylinders  for  steam  and 
for  air,  of  30  inches  stroke,  with  an  air-receiver  5  feet  in  diameter  and  24 
feet  long.  The  steam  was  cut  off  at  80  per  cent.  The  air-engine  was  an 
ordinary  semi-portable  engine,  having  two  lo-inch  cylinders  of  12  inches 
stroke,  cutting  off  at  three-fourths.  The  air  from  the  receiver  was  led 
into  and  passed  through  the  boiler  of  the  portable  engine,  and  was  thereby 
cooled  down  to  within  5°  of  the  atmospheric  temperature  before  it  passed 
into  the  cylinder.  The  principal  results  of  the  trials  are  quoted  from  the 
paper  and  given  in  table  No.  313;  in  which  the  two  lines,  7  and  12,  have 
been  calculated  and  added  by  the  author. 

Table  No.  313. — AIR -COMPRESSING  ENGINES,  AND  COMPRESSED-AIR 
ENGINES,  AT  POWELL  DUFFRYN  COLLIERY — RESULTS  OF  TRIALS. 


Pressure  of  Air  in  Receiver,  Effective,  Ibs. 

40.0 

34-0 

28.5 

24 

19 

i.   Effective  mean  pressure  in  steam-cylinders,  Ibs. 
2.             Do.             do.           air-cylinders,  —  Ibs. 
3.   Speed  of  piston,  per  minute,  feet 
4.   Effective  mean  pressure  in  air-engine,  Ibs. 
5    Speed  of  piston   per  minute      feet 

26.3 
24.0 
190 
35-6 
1  08 

25.1 
22.7 

155 
29.8 
IO4 

21.5 
19-5 
140 
24.7 
104 

19.7 
16.5 
no 

21.0 
1  08 

16.6 

'4-5 
60 
17.0 

88 

Air-compressing  engine  — 
6          In  steam-cylinder  (A),                   .    I.H.P. 

Co.  4 

46.2 

35.8 

25.8 

11.  8 

7          In  air-cylinder  (B),  I.H.P. 

52.6 

4O.7 

32.2 

21.7 

10.  1 

8    Air-engine   cylinder  (C),                   .     I.H.P. 

18.3 

14.7 

12.2 

10.8 

7.1 

9          Do          brake  (D),    H.P. 

1C.  7 

12.  C, 

IO.2 

9.O 

5.4 

10.   Efficiency  of  D  in  parts  of  A,  per  cent. 
u.           Do.         C          ,,          A,  percent. 
12.           Do.         B         ,,          A,  percent. 

25.8 
30.8 
87.7 

27.1 
31-8 

88.0 

28.5 

34-1 
£9.8 

34-9 
41.9 

84-3 

45-8 
60.2 

85-4 

13.  Total  Pressure  in  receiver,  atmospheres 
14.   Actual  final  volume  in  air  —  cylinder  of  com- 
pressing engine              initial  vol.  —  I 

3-72 
.380 

3-31 

•425 

2.94 
.470 

2.63 
.518 

2.29 

•575 

15.  Final   volume  according    to   the    adiabatic 
curve                                  .  .    initial  vol.  ~~  I 

•393 

.427 

•465 

•503 

•555 

1  6.   Final  volume  according  to  the   hyperbolic 
curve                                       initial  vol.  —  I 

.269 

.302 

•340 

.380 

•437 

Actual  mean  pressure  :  — 
17          From  indicator-diagram      Ibs. 

24.0 

22.7 

19.5 

16.5 

14.5 

18.         By  the  adiabatic-  curve,  Ibs. 
IQ          By  the  hyperbolic-curve       .         .         Ibs. 

23-5 

10.7 

21.  1 
17.6 

18.6 
ic..  9 

164 
14.2 

13-8 

12.2 

Proceedings  of  the  Institution  of  Mechanical  Engineers,  1874. 


HOT-AIR  ENGINES.  QI/ 


HOT-AIR  ENGINES. 

Engines  worked  by  heated  air  are  of  two  classes: — ist.  Those  in  which 
the  air  is  heated  and  cooled  alternately  by  contact  with  hot  and  cold 
surfaces;  and,  2d,  those  in  which  the  air  is  mixed  with  the  hot  products  of 
combustion  when  heating  surface  is  not  used. 

IST  CLASS. — RIDER'S  Hox-Am  ENGINE. 

In  this  engine,  which  is  called  a  compression-engine,  two  single-acting 
cylinders  are  placed  vertically,  a  little  apart,  connected  at  the  upper  part 
by  a  regenerator  composed  of  thin  plates.  One  of  these  is  the  working 
or  hot  cylinder,  under  which  a  fire  is  maintained,  the  other  is  the  air-pump, 
or  cold  cylinder,  surrounded  by  water  to  cool  the  air  which  is  drawn  into 
it,  and  which  is  pumped  back  into  the  hot  cylinder.  The  plungers  of 
these  cylinders  are  worked  by  cranks  placed  at  an  angle  of  95°  on  a  shaft 
overhead.  The  working  plunger  of  the  i  horse-power  engine  has  a  dia- 
meter of  6^  inches,  with  a  stroke  of  9^  inches;  the  pump-plunger  is 
6^  inches  in  diameter,  with  a  stroke  of  8.6  inches. 

"  The  compression  (pump)  piston  first  compresses  the  cold  air  in  the 
lower  part  of  the  compression-cylinder  into  about  one-third  of  its  normal 
volume,  when,  by  the  advancing  or  upward  motion  of  the  power  (working) 
piston,  and  the  completion  of  the  down-stroke  of  the  compression-piston, 
the  air  is  transferred  from  the  compression-cylinder,  through  the  regenerator, 
and  into  the  heater,  without  any  appreciable  change  of  volume.  The  result 
is  a  greater  increase  of  pressure,  corresponding  to  the  increase  of  temperature, 
and  this  impels  the  power-piston  up  to  the  end  of  its  stroke.  The  pressure 
still  remaining  in  the  power-cylinder,  and  reacting  on  the  compression- 
piston,  forces  the  latter  upward  till  it  reaches  nearly  to  the  top  of  its  stroke, 
when,  by  the  cooling  of  the  charge  of  air,  the  pressure  falls  to  its  minimum 
[about  atmospheric  pressure],  the  power-piston  descends,  and  the  compres- 
sion again  begins.  In  the  meantime  the  heated  air,  in  passing  through  the 
regenerator,  has  left  the  greater  portion  of  its  heat  in  the  regenerator-plates, 
to  be  picked  up  and  utilized  on  the  return  of  the  air  towards  the  heater." 

From  indicator  diagrams,  taken  at  120  turns  per  minute,  it  appears  that 
the  effective  mean  pressure  in  the  working  cylinder  was  16.8  Ibs.,  and  that 
in  the  pump  was  7.15  Ibs.  per  square  inch.  Reducing  the  pump-pressure 

o  {• 

in  the  ratio  of  the  strokes,  it  becomes  7.15  x  —  =  6.47  Ibs.;  then  (i 6. 8 

9-5 

-  6.47  = )  10.33  Ibs.  per  square  inch  is  the  net  effective  pressure  on  the 
working  plunger,  from  which  the  power  is  to  be  calculated.  The  area  of 
the  plunger  is  35.78  square  inches,  and  the  net  indicator  horse-power  is — 

35.78  Ibs.  x  I0#  Ibs.  x  .80  foot  x  120  =        6  hors          er 
33,000 

It  is  stated  that  the  quantity  of  coal  consumed  is  from  2  to  3  Ibs.  per 
net  indicator  horse-power. 

An  engine  of  ^  horse-power  was  tested  to  deliver  from  650  to  700  gallons 


91 8  AIR   MACHINERY. 

of  water  per  hour,  90  feet  high,  with  a  consumption  of  4  Ibs.  of  coal  per 
hour.1  Taking  a  mean  of  675  gallons,  the  performance  is  equivalent  to 
(675x10  Ibs.  x  90  feet  -•-  60=)  10,125  foot-pounds  per  minute,  or  to 
(10,125  •*•  33>ooo  =  )  -3°7  horse-power  of  net  duty,  for  which  (4  Ibs.  -^.307  =  ) 
13  Ibs.  of  coal  was  consumed  per  horse-power. 

2D  CLASS. — BELOU'S  Hox-AiR  ENGINE  AT  CUSSET. 

The  air  is  supplied  by  a  feeding  cylinder,  i  metre  in  diameter,  with  i  y% 
metres  of  stroke,  in  which  it  is  compressed,  and  from  which  it  is  discharged 
into  a  close  furnace,  where  it  is  heated  by  the  combustion  produced  by  it. 
Thence,  it  is  passed  to  the  working  cylinder,  1.4  metres  in  diameter,  with 
i^  metres  of  stroke,  where  it  acts  with  full  pressure  and  expansively,  after 
which  it  is  exhausted  into  the  atmosphere.  These  cylinders  are  double- 
acting.  The  feeding  cylinder  draws  i  cubic  metre  of  air  at  each  stroke. 
The  furnace  is  inclosed  in  a  horizontal  cast-iron  cylinder;  the  grate  is 
inclined,  and  has  an  area  of  .80  square  metre,  or  8.6  square  feet.  The 
greater  portion  of  the  air  passes  through  the  grate.  The  engine  makes 
23  turns  per  minute,  giving  a  speed  of  pistons  of  225  feet  per  minute.  The 
temperature  in  the  chimney  is  480°  F. 

The  absolute  pressure  in  the  feeding  cylinder,  is  raised  to  1.94  atmo- 
spheres, for  which  the  period  of  compression  is  51.5  per  cent,  of  the  stroke. 
In  the  working  cylinder,  the  initial  pressure  is  1.68  atmospheres;  the  air  is 
cut  off  on  the  upper  side  at  39  per  cent,  of  the  stroke,  and  expanded 
exactly  to  atmospheric  pressure  at  the  end  of  the  stroke;  on  the  lower  side, 
the  admission  is  longer,  to  compensate  for  the  weight  of  the  piston — about 
2  tons.  The  difference  of  the  pressures,  (1.94-  1.68  =  )  .26  atmosphere,  or 
3.8  Ibs.  per  square  inch,  represents  the  resistance  in  the  furnace  and  the 
passages.  The  average  effective  pressures  are,  in  the  feeding  cylinder, 
9.4  Ibs.,  and  in  the  working  cylinder  7.13  Ibs.  per  square  inch;  yielding 
respectively  80.62  and  119.74  indicator  horse-power.  Thus,  it  is  seen  that 
two-thirds  of  the  working  indicator  power  is  expended  in  supplying  air 
to  the  working  cylinder.  Allowing  only  10  per  cent,  of  the  indicator  power 
for  general  resistances,  and  so  reducing  it  to  107.77  horse-power,  the  net 
useful  work  is  107.77  -  80.62  =  27.15  horse-power,  which  is  22.67  per  cent, 
of  the  indicator  power. 

The  quantity  of  coal  consumed  is  88  Ibs.  per  hour,  being  at  the  rate  of 
.735  Ibs.  per  indicator  horse-power,  or  3.24  Ibs.  per  net  horse-power,  as  at 
the  brake.2 


GAS-ENGINES. 

Gas-engines  are  worked  by  the  explosion  of  a  mixture  of  coal-gas  and  air, 
which  acts  on  a  piston  within  a  cylinder.  They  may  be  double-acting  or 
single-acting,  and  the  explosion  may  be  effected  by  means  of  an  electric 
battery,  or  of  lighted  jets  of  gas  placed  in  communication  with  the  mixture. 

1  At  the  meeting  of  the  Royal  Agricultural  Society  at  Birmingham ;  Messrs.  Eastons 
and  Anderson,  Engineers. 

2  See  the  Annales  du  Conservatoire  des  Arts  et  Metiers,  vol.  vii.,  for  full  particulars  of 
Belou's  engines.     The  data  above  given  are  drawn  from  this  source. 


GAS-ENGINES.  919 

LENOIR'S  DOUBLE-ACTING  GAS-ENGINE. 

Two  horizontal  engines  of  this  kind,  fired  by  electricity,  were  tested  by 
M.  Tresca.1  During  a  part  of  the  stroke,  the  gas  and  air,  in  fixed  propor- 
tions, are  admitted  into  the  cylinder,  and  then  exploded  by  an  electric 
spark.  By  the  explosion,  heat  and  pressure  are  generated,  and  the  pressure 
acts  on  the  piston  during  the  remainder  of  the  stroke.  During  the  return- 
stroke,  the  gaseous  products  are  exhausted  into  the  atmosphere;  whilst  the 
explosive  action  takes  place  on  the  other  face  of  the  piston.  The  heat  of 
the  cylinder  is  reduced  by  a  continuous  current  of  cold  water  applied  on 
the  outside. 

In  the  first  engine,  the  cylinder  was  7.1  inches  in  diameter,  with  a  stroke 
of  4  inches.  The  mixture  of  gas  and  air  was  cut  off  at  half-stroke,  and  the 
maximum  absolute  pressure  in  the  cylinder  was  a  little  less  than  6  atmo- 
spheres. The  average  speed  of  the  engine  was  129  turns  per  minute, 
giving  a  speed  of  piston  of  153  feet  per  minute.  The  power  measured  by 
the  brake  was  .57  horse-power,  and  the  quantity  of  gas  consumed  amounted 
to  1 1 2  cubic  feet  per  brake  horse-power  per  hour.  The  gas  and  air  were 
mixed  in  proportions  of  i  to  10.  Fifty-three  per  cent,  of  the  heat  gener- 
ated in  the  cylinder  was  carried  off  by  the  water  outside.  The  combustion 
of  the  gases  was  very  nearly  complete. 

For  the  second  trial,  the  engine  had  a  cylinder  9^  inches  in  diameter, 
with  a  stroke  of  4^  inches.  The  weight  of  the  engine  complete  was 
14  cwts.  The  speed  of  the  engine  was  100  turns  per  minute,  giving  158  feet 
of  piston  per  minute.  The  period  of  admission  was  a  little  more  than  half- 
stroke,  and  the  maximum  absolute  pressure  was  5.36  atmospheres.  The 
quantity  of  gas  consumed  amounted  to  97  cubic  feet  per  brake  horse-power 
per  hour;  the  power  developed  at  the  brake  being  about  i  horse-power. 
The  gas  and  air  were  mixed  in  the  proportion  of  i  to  1 1  yz ;  and  the  volume 
of  gas  admitted  for  each  stroke  was  24^  cubic  inches,  the  heat  of  combus- 
tion of  which  is,  according  to  M.  Tresca,  96  English  units.  It  is  not  sur- 
prising that  the  temperature  and  the  pressure  after  explosion,  are  lowered, 
as  is  shown  by  diagrams,  almost  instantaneously  by  contact  with  the  metal; 
and  it  is  for  this  reason,  probably,  that  the  stroke  is  made  so  short  in  pro- 
portion to  the  diameter.  The  quantity  of  water  consumed  for  cooling  the 
cylinder  amounted  to  4^  cubic  feet  per  horse-power  per  hour,  the  temper- 
ature being  raised  140°  F. 

M.  Tresca  has  estimated  the  distribution  of  the  heat  generated  in  the 
cylinder  as  follows : — 

Heat  carried  off  by  the  water  and  the  products  of  combustion,  69  per  cent. 

Heat  converted  into  work  at  the  brake, 4        „ 

Losses,  not  estimated, 27        „ 

IOO 

The  net  efficiency  at  the  brake  is  thus  taken  as  4  per  cent. 

OTTO  AND  LANGEN'S  ATMOSPHERIC  GAS-ENGINE — SINGLE-ACTING. 

In  this  engine,  the  cylinder  is  vertical,  open  to  the  atmosphere  at  the  upper 
end;  it  has  a  "  free  piston,"  its  principal  feature,  which  is  impelled  upwards 

1  Annales  du  Conservatoire  des  Arts  et  Metiers,  vol.  i.,  1861,  page  894. 


Q20  AIR   MACHINERY. 

against  the  atmosphere,  by  the  explosion  of  gas  below  it.  The  stress  of  the 
explosion  of  gas  is  intense,  but  momentary,  and  the  free  piston  mounts 
instantly  and  quickly  against  the  atmospheric  resistance,  whilst  yet  the 
explosive  force  continues.  Thus  the  explosive  force  is  utilized  efficiently, 
and  the  power  is  derived  from  the  pressure  of  the  atmosphere,  by 
which  the  piston  is  driven  downwards  against  the  partial  vacuum  formed 
under  the  piston  by  the  collapse  of  the  gaseous  products.  To  cool  and 
contract  the  gases,  the  lower  half  and  the  bottom  of  the  cylinder  are 
jacketed  with  cold  water.  The  piston-rod  is  formed  as  a  rack,  and  gears 
into  a  pinion  loose  on  the  fly-wheel  shaft.  The  pinion  turns  loose  with  the 
rack  during  the  ascent,  but,  during  the  descent  it  engages  with  and  turns 
the  shaft  by  means  of  a  friction-clutch,  making  two  revolutions  during  one 
descent.  The  routine  of  the  engine  is  as  follows: — i.  The  piston  is  lifted 
through  '/nth  of  the  stroke  to  receive  the  charge  of  gas  and  air.  2.  The 
mixture  is  fired  by  a  gas-light.  3.  The  piston  makes  the  up-stroke.  4.  The 
plenum  under  the  piston  becomes  a  vacuum  of  22  inches  of  mercury,  at 
the  beginning  of  the  down-stroke.  5.  The  down-stroke  is  made  under  an 
effective  pressure  of  1 1  Ibs.  per  square  inch,  and  the  force  is  transmitted 
to  the  shaft.  6.  When  the  piston  arrives  near  to  the  bottom,  the  vacuum 
becomes  a  plenum,  by  compression  of  the  gases;  and,  by  the  weight  of  the 
piston  and  rack^  the  gaseous  products  are  expelled  from  the  cylinder.  The 
intermittent  motion  is  worked  by  a  tappet  on  the  rack  to  raise  the  piston 
for  the  next  charge. 

According  to  Mr.  Crossley,1  for  a  ^  horse-power  engine,  the  cylinder  is 
6  inches  in  diameter,  with  a  stroke  of  40  inches;  and  the  explosions  are 
made  at  various  rates  up  to  that  of  30  per  minute.  The  mixture  consists  of 
6^  volumes  of  air  to  i  volume  of  coal-gas.  He  takes  the  heating  power 
of  i  Ib.  of  coal-gas,  of  density  .40,  at  24,000  units  of  heat;  and,  for  a  con- 
sumption of  1.05  cubic  feet  of  gas  per  minute,  the  heat  supplied  to  the 
engine  is  equivalent  to  584,000  foot-pounds,  of  which  70,000  foot-pounds, 
or  12  per  cent.,  is  yielded  at  the  brake.  The  power  at  the  brake  is 
(70,000  -=-33,000  =  )  2.12  horse-power,  and  the  consumption  of  gas  is  at  the 
rate  of  (1.05  x  60  -f  2.12  = )  30  cubic  feet  per  horse-power  at  the  brake  per 
hour.  From  indicator-diagrams,  it  appears  that,  in  the  down-stroke,  the 
effective  pressure  varies  from  1 1  Ibs.  per  square  inch  to  zero  at  four-fifths  of 
the  stroke,  averaging  9  Ibs.  for  four-fifths,  or  about  7  Ibs.  for  the  whole  of 
the  stroke. 

M.  Tresca2  tested  a  6-inch  single-acting  gas-engine,  in  which  the  power 
at  the  brake,  making  81  turns  per  minute,  was  .456  horse-power.  The  gas 
consumed  per  hour  was — 

Per  Minute.  fc.lhj.g-.*-. 

For  work  in  cylinder, 20.09  cubic  feet.  44.06  cubic  feet. 

For  inflaming, 2.08          „  4.57         „ 

22.17          „  48.63         „ 

The  water-jacket   absorbed   only  800    English   units   of  heat  per   hour. 
M.  Tresca  allows  a  heating  power  of  only  6000  French  units  per  cubic 

1  See  a  paper  by  Mr.  F.  W.  Crossley,  on  "Otto  and  Langen's  Gas-Engine,"  in  the 
Proceedings  of  the  Institution  of  Mechanical  Engineers,  1875*  Page  IQ*« 
"*  Annales  du  Conservatoire  des  Arts  et  Metiers,  vol.  vii.,  page  628. 


GAS-ENGINES.  Q2I 

metre  of  coal-gas,  equivalent  to  (6000  x  4  x  —  =  )  21,000  English  units  per 

O  J 

pound  of  30  cubic  feet,  in  round  numbers.  The  quantity  of  heat  generated, 
according  to  this  allowance,  was,  then,  (2i,ooox— =  )  14,000  units  per 

hour.  The  work  at  the  brake  was  (.456  x  33,000  x  60  -r  772  = )  1 1 70  units 
per  hour,  which  represents  an  efficiency  of  (1170  x  IOO-T-  14,000  =  )  8.4  per 
cent. 

THE  OTTO  GAS-ENGINE. 

Whilst  the  Otto  and  Langen  atmospheric  gas-engine  superseded  the 
Lenoir  gas-engine,  it  was  in  its  turn  superseded  by  the  "  Otto  Silent 
Gas-engine,"  the  invention  of  Dr.  Otto,  of  Deutz:  called  silent  in  com- 
parison with  other  gas-engines,  but  now  known  as  the  Otto  Gas-engine. 
A  novel  and  important  feature  was  the  compression  of  the  explosive 
mixture  before  being  fired:  effecting  economy  of  gas  by  increase  of 
efficiency,  and  facilitating  the  use  of  engines  of  greater  power  than 
before.  Constructed  by  Crossley  Brothers,  Manchester,  the  Otto  engine 
is  horizontal  in  its  disposition,  resembling  generally  a  steam-engine.  It  is 
single-acting,  having  the  cylinder  open  at  one  end,  with  a  trunk-piston. 
The  cycle  of  operations  is  fourfold:  in  the  first  out-stroke  a  charge  of  gas 
and  air  in  mixture,  in  the  ratio  of  i  to  16,  including  the  burnt  gases,  is 
drawn  in;  and  in  the  first  in-stroke,  following,  the  charge  is  compressed 
until  it  reaches  a  pressure  of  35  Ibs.  per  square  inch;  at  the  beginning  of 
the  second  out-stroke  the  compressed  charge  is  ignited  and  exploded,  and 
acts  on  the  piston  for  propulsion,  during  this,  the  working  stroke;  by  the 
second  in-stroke  the  gaseous  products  of  combustion  are  expelled  from  the 
cylinder.  Thus  there  are  one  charge  and  one  explosion  for  every  four 
single-strokes,  or  two  double-strokes  or  revolutions.  Strictly,  therefore,  the 
engine  is  half  single-acting;  and  a  heavy  fly-wheel  is  necessary,  to  work  the 
piston  through  the  negative  part  of  the  cycle.  The  cylinder  serves 
alternately  as  a  compression-pump  and  as  a  motor-cylinder.  It  is  jacketed 
with  cold  water  to  prevent  overheating,  although  it  is  estimated  that  a 
loss  of  42  per  cent  is  thus  incurred. 

The  ignition  of  the  charge  has  been  effected  by  means  of  a  slide-valve, 
carrying  a  gas-jet,  kept  constantly  burning.  This  form  has  recently  been 
superseded  by  .a  system  of  ignition  within  a  tube  opening  to  the  cylinder, 
charged  with  a  strong  igniting  mixture.  Space  is  provided  for  the  com- 
pressed charge  by  a  prolongation  of  the  closed  end  of  the  cylinder.  The 
initial  pressure  in  the  cylinder  varies  from  120  Ibs.  to  190  Ibs.  per  square 
inch  above  the  atmosphere. 

The  Otto  engine  is  constructed  with  a  single  cylinder,  of  various  nominal 
power,  of  from  ^  H.P.  to  20  H.P.,  indicating  from  2  H.P.  to  50  H.P.  ;  and 
with  double  cylinders,  indicating  from  16  H.P.  to  100  H.P.  The  12  N.H.P. 
engine  has  been  proved  to  develop  28  I.H.P.,  and  23  H.P.  at  the  break,  or 
82  per  cent  of  the  indicator  power;  with  a  consumption  of  20  cubic  feet  of 
gas  per  indicator  horse-power  per  hour,  or  24.3  cubic  feet  per  brake  horse- 
power. The  total  consumption  of  gas  was  at  the  rate  of  560  cubic  feet  per 
hour;  when  running  without  load,  100  cubic  feet  per  hour.  In  a  4  H.P. 
engine,  23.3  cubic  feet  was  consumed  per  brake  horse-power.  In  the  use 


922  AIR   MACHINERY. 

of  Dowson  gas  instead  of  ordinary  coal-gas,  i  y*>  pounds  of  anthracite  coal 
is  consumed  per  indicator  horse-power  per  hour.  The  consumption  has 
occasionally  been  only  i.i  pounds. 

CLERK'S  GAS-ENGINE. 

The  gas-engine  of  Mr.  Dugald  Clerk  is  single-acting.  A  charge  is 
exploded  at  every  out-stroke,  the  mixture  of  gas  and  air  being  as  i  to  9, 
admitted  for  the  first  half  of  the  stroke.  During  the  second  half-stroke 
pure  air  is  admitted.  In  order  to  effect  the  explosion  at  every  out-stroke 
a  displacer  cylinder  is  employed,  and  the  charged  air  is  compressed  till  it 
attains  a  pressure  of  38  Ibs.  per  square  inch,  when  it  is  exploded  and  burns 
during  the  out-stroke.  The  exhaust  takes  place  near  the  end  of  the  stroke, 
and  as  the  piston  returns,  the  pure  air  of  the  charge  is  exhausted  through 
the  pipe,  which  is  cooled. 

GASEOUS    FUEL. 
THE  WILSON  GAS-PRODUCER. 

The  Wilson  gas-producer,  introduced  by  Mr.  Bernard  Dawson,  is  an 
upright  cylindrical  chamber  of  firebrick,  having  a  solid  hearth,  kept  nearly 
full  of  small  bituminous  coal.  A  mixture  of  air  and  steam,  comprising 
about  20  parts  of  air  to  i  part  of  steam  by  weight,  is  delivered  into  the  lower 
part  of  the  chamber,  the  air  being  induced  by  two  small  jets  of  steam  from 
a  steam  boiler.  The  fuel  is  resolved  into  combustible  gases, — hydrogen, 
carbonic  oxide,  and  hydrocarbons, — in  the  manner  of  the  ordinary  gas- 
furnace;  and  the  gases  pass  through  a  number  of  openings  above  the  level 
of  the  fuel,  into  an  annular  flue,  whence  they  are  conducted  by  an  under- 
ground conduit  to  the  place  of  consumption.  The  fuel  is  charged  in  at  the 
top,  which  is  closed  by  a  pendulous  conical  valve  or  plug. 

As  applicable  for  supplying  heat  to  steam  boilers,  the  results  of  a  test- 
trial,  in  November,  1886,  at  Apsley  Paper  Mill,  Hemel-Hempstead,  con- 
ducted by  the  author,  may  be  noticed.  Four  Cornish  boilers  were  fitted 
with  two  4-cwt  Wilson  gas-producers,  for  generating  steam  by  gas-firing. 
The  boilers  are  each  5^  feet  in  diameter,  with  a  3-feet  furnace-tube,  and 
21  feet  long;  having  eight  Galloway  tubes  in  each.  The  producers  stand 
side  by  side  in  an  open  yard  adjacent  to  the  boiler-house.  Each  producer 
is  cylindrical,  8  feet  in  diameter,  9  feet  high,  of  firebrick  cased  in  plate-iron. 
The  internal  hearth  is  5  feet  in  diameter,  having  20  square  feet  of  area,  the 
fuel  space  above  the  hearth  is  4^  feet  deep,  and  the  gases  pass  through 
openings  into  the  annular  flue  surrounding  the  neck  or  upper  part  of  the 
furnace,  whence  they  are  conducted  underground  to  the  boilers.  Here  the 
supply  of  gas  to  each  boiler  is  regulated  by  means  of  a  valve.  The  gas  is 
delivered  through  the  doorway,  together  with  air,  into  the  furnace-tube,  and 
combustion  takes  place. 

The  four  steam  boilers  are  set  in  a  row.  No.  i  boiler  was  separated 
from  Nos.  2,  3,  and  4,  and  was  devoted  to  the  generation  of  steam  for 
supplying  the  blast  injector  attached  to  each  gas-producer.  The  three 
other  boilers  were  connected  for  the  supply  of  steam  to  the  factory.  The 
feedwater  was  measured  separately  into  No.  i  boiler.  The  coal  used  in  the 
producers  was  cobbles  from  Wyken  Colliery,  broken  up  by  hand;  charged 
into  each  hopper  about  four  times  per  hour. 


THE  WILSON  GAS-PRODUCER.  923 

The  leading  results  of  the  test-trial  are  as  follows;  and  for  comparison, 
the  results  of  a  six-days'  test  of  the  same  boilers,  which  had  previously  been 
made  with  hand-firing,  are  prefixed.  In  this  case,  the  fire-grates  were  4  feet 
8  inches  long,  presenting  an  area  of  14  square  feet  for  each  boiler. 

Hand-firing.  Gas-firing. 

Coal  consumed  per  boiler  in  full  steam  per  hour,     163  Ibs.  258.6  Ibs. 

Water  evaporated        do.            do.            do.         !4«93  cub.  ft.      24.94  cub.  ft. 
Water  per  pound  of  coal  from  and  at  212°  F 6.79  Ibs.  7.16  Ibs.  (net) 

It  is  shown  that  the  boilers  in  full  steam  did  two-thirds  more  evaporative 
duty  by  gas-firing  than  by  hand-firing;  and,  with  5^  per  cent,  more  evapo- 
rative efficiency  net,  after  allowance  made  for  steam  consumed  in  blowing 
the  producers. 

It  is  also  shown  that  the  weight  of  steam  consumed  by  the  producers 
was  equal  to  8.29  per  cent,  of  the  total  quantity  generated  in  the  four 
boilers. 

The  total  evaporative  efficiency  of  the  boilers  with  gas-firing,  if  no  deduc- 
tion be  made  for  the  demands  of  the  producers,  is  expressed  by  6.56  pounds 
of  water  per  pound  of  coal,  or  an  equivalent  of  7.81  pounds  from  and  at 
212°,  which  is  (7.81  -6.79  =  .98  pound,  or  14.4  per  cent,  more  efficiency 
than  was  obtained  by  hand-firing.  This  is  an  expression  of  the  absolute 
difference  of  efficiency  in  favour  of  gas-firing.  The  practical  difference, 
after  making  the  needful  allowance,  is,  as  above  stated,  5  */£  per  cent. 

At  intervals  no  smoke  was  visible  with  gas-firing;  and  there  is  no  good 
reason  why,  with  good  draught,  gas-firing  should  not  be  conducted  entirely 
without  smoke. 

Comparative  trials  have  been  made  at  Plas  Power  Colliery,  by  Mr.  John 
H.  Darby,  in  which  it  was  shown  by  the  best  result  that  a  greater  absolute 
evaporative  efficiency  was  attained  of  9.85  per  cent,  in  favour  of  gas-firing. 
The  following  was  the  average  composition  of  the  gases  produced: — 

Carbonic  acid 6.26 

Oxygen o.oo 

Hydrogen 14.68 

Carbonic  oxide 23.98 

Marsh  gas 4.72 

Nitrogen 5°-36 

100.00 
THE  DOWSON  GENERATOR  GAS. 

It  is  known  that  highly-heating  non-luminous  gases  can  be  produced  by 
decomposing  steam  in  the  presence  of  carbon:  passing  steam  and  air 
through  a  fire  of  incandescent  fuel.  Mr.  J.  Emerson  Dowson  practises 
this  system;  and,  in  addition,  he  employs  special  means  of  generating  and 
superheating  the  steam.  The  steam  producer  and  superheater  consists  of 
a  long  coil  of  tube,  of  such  a  form  that  nearly  all  of  it  is  exposed  to  the 
action  of  gas  flame.  Water  is  forced,  under  a  pressure  of  from  20  Ibs. 
to  25  Ibs.  per  square  inch,  into  the  coil,  in  which  it  is  converted  into 
superheated  steam.  The  gas  required  for  heating  the  coil  is  drawn  from 
the  gas-holder.  The  retort  or  generator  is  of  iron,  lined  with  ganister. 
The  fuel,  anthracite,  rests  on  a  grate,  above  an  inclosed  chamber,  into 


924  AIR   MACHINERY. 

which  a  jet  of  superheated  steam  is  directed  through  a  small  opening, 
carrying  with  it,  by  induction,  a  current  of  air  into  the  furnace,  for  combus- 
tion. The  gas  produced  contains  by  volume,  approximately,  20  per  cent, 
of  hydrogen,  30  per  cent,  of  carbonic  oxide,  3  per  cent,  of  carbonic  acid,  and 
47  per  cent,  of  nitrogen.  The  gross  quantity  of  fuel  consumed  in  working 
Otto  Gas-engines  averages,  as  before  stated,  1.3  pounds  per  indicator  horse- 
power. Professor  Witz,  of  Lille,  tested  a  9  horse-power  gas-engine  on  the 
Delamare-Debouteville  system;  and  proved  a  consumption  of  89  cubic  feet 
of  the  Dowson  gas,  or  1.33  pounds  of  coal  per  brake  horse-power  per  hour. 


FANS    OR    VENTILATORS. 

COMMON  CENTRIFUGAL  FAN. 

The  ordinary  fan  consists  of  a  number  of  blades  fixed  to  arms,  re- 
.volving  on  a  shaft  at  high  speeds.  It  appears  from  the  results  of  Mr. 
Buckle's  experiments,2  that  when  the  fan  is  revolved  in  its  case  without 
any  air  being  discharged,  the  pressure  generated  at  the  circumference 
of  the  fan  varies  as  the  square  of  the  velocity  of  the  fan,  and  the 
horse-power  required  to  maintain  the  speed  varies  as  the  cube  of  the 
velocity.  It  further  appears  that  the  pressure  generated  at  the  circum- 
ference is  one-ninth  greater  than  that  which  is  due  to  the  actual  circumfer- 
ential velocity  of  the  fan.  To  express  the  relation  of  the  pressure  and  the 

-.2 

velocity  of  an  air-current,  the  height  due  to  the  velocity  is  h  =  —  •  h  is  also 

64 

equal  to  the  height  of  a  column  of  air  equal  in  weight  to  the  pressure. 
The  velocity  due  to  the  pressure  may  thence  be  deduced  by  means  of  the 
ordinary  relation  v-^^Jh. 

Mr.  Buckle  recommends  the  following  proportions  for  fans  of  from  3  to  6 
feet  in  diameter,  and  for  pressures  ranging  from  3  to  6  oz.  per  square 
inch : — 

The  width  and  length  of  the  vanes  equal  to  one-fourth  of  the  dia- 
meter. 

The  diameter  of  the  inlet  openings  in  the  sides  of  the  fan-chest  equal  to 
half  the  diameter  of  the  fan. 

For  higher  pressures,  of  from  6  to  9  oz.  and  upwards,  Mr.  Buckle 
recommends  that  the  vanes  should  be  narrower  and  longer,  and  the  inlet 
opening  smaller,  than  are  prescribed  by  the  above  proportions.  He  gives 
the  following  table  of  dimensions.  The  number  of  blades  may  be  4  or  6. 
The  case  is  made  of  the  form  of  an  arithmetical  spiral,  widening  the  space 
between  the  case  and  the  revolving  blades,  circumferentially,  from  the 
origin  to  the  opening  for  discharge;  and  it  appears  that  the  upper  edge 
of  the  opening  should  be  level  with  the  lower  side  of  the  sweep  of 
the  fan : — 

1  English  Mechanic,  December  29,  1876,  page  387. 

2  "Experiments  Relative  to  the  Fan-Blast,"  by  Mr.  Buckle;  Proceedings  of  the  Institu- 
tion of  Mechanical  Engineers,  1847. 


FANS   OR  VENTILATORS. 


925 


Table'No.  314. — DIMENSIONS  OF  FANS. 

(Mr.  Buckle.) 
Pressure,  from  3  to  6  oz.  per  square  inch. 


Vanes. 

Diameter  of 

Diameter  of 

Fans. 

Width. 

Length. 

Inlet  Openings. 

feet,      inches. 

feet.      inches. 

feet,      inches. 

feet.      inches. 

3        o 

o        9 

o        9 

i        6 

3        6 

0         10% 

0         10% 

i        9 

4        o 

I             0 

I            0 

2            0 

4        6 

I      IX 

i       iX 

2         3 

5        o 

i         3 

i         3 

2            6 

6        o 

i        6 

i        6 

3        o 

Pressure,  from  6  to  9  oz.  per  square  inch,  and  upwards. 

3        o 

o        7 

0 

I            0 

3        6 

o        8>£ 

i# 

i         3 

4        o 

o        9^ 

3X 

i        6 

4        6 

0         10% 

4X 

i         9 

5        o 

I            0 

6 

2           0 

6        o 

I            2 

10 

2           4 

MlNE   VENTILATORS.1 — GuiBAL's    FAN. 

The  blades  are,  for  the  most  part  of  their  length,  straight;  but  they  curve 
forwards  at  the  outer  ends.  They  are  fixed  on  polygonal  centres,  and  at  a  con- 
siderable backward  inclination — usually  45°, — to  the  radius.  The  wheel  is 
closely  surrounded  for  about  two-thirds  of  the  circumference,  by  a  casing  of 
brickwork;  for  the  remaining  third,  the  casing  gradually  opens  out  into  the 
discharge  vent,  which  expands  upwards  as  an  inverted  cone.  By  so  forming 
the  vent,  the  velocity  of  the  discharged  air  is  reduced,  and  converted  into 
outward  pressure,  by  the  action  of  which  the  velocity  through  the  fan  is  in- 
creased, and  the  efficiency  is  raised.  A  Guibal  fan,  working  at  Staveley 
Colliery,  is  30  feet  in  diameter,  and  10  feet  wide.  It  makes  60  revolutions 
per  minute  in  the  day.  The  following  are  particulars  of  its  performance : — 


Speed,  in 
Turns  per 
Minute. 

Draft  in  Inches 
of  Water. 

Volume  of  Air 
Discharged  per 
Minute. 

Efficiency,  in  parts 
of  the  Gross 
Indicator  Power 
of  the  Engine. 

cubic  feet. 

per  cent. 

32 

.70 

43,852 

40.38 

51 

1.70 

86,283 

43-09 

64 

2.77 

101,773 

53-27 

68 

3.10 

1  10,005 

53.85 

1  These  particulars  of  mine-ventilators  are  derived  from  papers  on  "Ventilation  of  Mines," 
by  Mr.  J.  S.  E.  Swindell,  and  Mr.  W.  Daniel,  in  the  Proceedings  of  the  Institution  of 
Mechanical  Engineers,  1869,  1875. 


926  AIR   MACHINERY. 

The  advantage  of  surrounding  the  fan  by  a  casing,  and  of  adjusting,  by 
means  of  a  slide,  the  size  of  the  opening  into  the  vent,  is  shown  by  the 
following  results  of  trials  at  a  mine  in  Belgium : — 

Gross  Efficiency. 

Without  casing, 22  per  cent. 

With  casing, 31        „ 

With  casing  and  expanding  vent, 57       „ 

With  casing  and  expanding  vent,  and  with  )  , 

slide  adjusted, \  » 

These  are  the  efficiencies  in  parts  of  "  the  gross  power  supplied  from  the 
boiler."  An  efficiency  of  60  per  cent,  is  generally  obtained  by  this  venti- 
lator; equivalent  to  80  per  cent,  of  the  net  power  of  the  engine. 

COOK'S  VENTILATOR. 

This  is  a  positive  ventilator,  making  a  given  discharge  for  each  revolu- 
tion. It  consists  of  a  revolving  eccentric  within  a  circular  case,  against 
which  a  flap-valve  is  maintained  constantly  in  contact,  to  separate  the 
entering  current  from  the  outgoing  current.  Two  ventilators  working  at 
Saltburn  have  casings  of  22  feet  in  diameter,  and  n  feet  6  inches  wide; 
the  eccentric  has  a  diameter  equal  to  two-thirds  of  that  of  the  casing, 
and  the  eccentricity  is  one-fourth  of  its  diameter.  The  period  of  inlet  and 
discharge  of  air  is  235°,  or  about  two-thirds  of  a  revolution.  Making  from 
26  to  29  turns  per  minute,  with  a  draught  of  from  i  to  3.25  inches,  the 
efficiency  was  found  to  be  from  58.5  to  64  per  cent,  of  the  indicator  horse- 
power. 

BLOWING  ENGINES. 

Blowing  engines  of  recent  design  are  direct-acting,  the  steam-piston  and 
the  air-piston  being  fixed  to  one  rod,  and  the  steam-  and  air-cylinders  in 
line.  There  is  a  pair  of  blowing  cylinders,  each  of  which  is  worked  by  a 
steam-cylinder;  and  the  two  steam-cylinders  are  either  a  pair  or  are 
arranged  as  compound  cylinders.  The  clearance  in  the  air-cylinders 
should  be  reduced  to  the  smallest  practicable  limits.  At  Lackenby  Iron- 
works it  is  only  3  per  cent,  at  each  end ;  the  total  area  of  valve-opening  at 
each  end,  for  the  inlet,  is  J/6th  of  the  area  of  the  piston,  and  for  the  outlet, 
^th.  These  proportions  are  unusually  liberal.  The  two  air-cylinders  are 
80  inches  in  diameter,  with  54  inches  of  stroke,  having  each  a  capacity  of 
157  cubic  feet.  They  make  24  double  strokes  per  minute,  giving  a  speed 
of  piston  of  216  feet  per  minute;  190,000  cubic  feet  of  atmospheric  air  are 
supplied  per  ton  of  iron  made,  and  the  supply  is  sufficient  for  the  produc- 
tion of  800  tons  of  iron  per  week.  The  blast-main  is  30  inches  in  diameter, 
and  has  a  capacity  12^  times  the  united  volumes  of  the  cylinders.  The 
pressure  in  the  main  is  4^  Ibs.  per  square  inch  above  the  atmosphere,  and 

it  is  free  from  fluctuations.     The  ratio  of  compression  is  — '- — — —  =  1.3, 

and  the  valves,  therefore,  open  to  the  main  when  the  air-piston  has  passed 
through  20  per  cent,  of  the  stroke,  approximately,  allowing  for  clearance; 
whilst  the  air  is  driven  into  the  compressor  during  80  per  cent,  of  the  stroke. 
The  clearance  is,  proportionally,  3  x  1.3  =  4  per  cent,  of  the  volume  of  com- 
pressed air;  and  thus  the  effective  charge  of  air  is  (100-4  =  )  9 6  per  cent. 


BLOWING  ENGINES.  Q2/ 

of  the  quantity  compressed.  The  steam-cylinders  are  32  and  60  inches  in 
diameter,  and  their  indicator  horse-power  is  290  horse-power;  whilst  that  of 
the  air-cylinders  is  258  horse-power,  representing  an  efficiency  of  89  per 
cent.1 

An  instance  of  very  low  pressure  of  blast  produced  by  a  blowing  engine 
is  given  by  Mr.  Briggs.  A  pair  of  1 2-inch  steam-cylinders  drive  directly  a 
pair  of  48-inch  air-cylinders,  with  a  stroke  of  24  inches.  The  steam-valves 
cut  off  at  ^ths,  and  they  have  "negative"  lead  and  ample  cover  to  the 
exhaust,  for  the  purpose  of  counteracting  the  expansive  force  of  the  com- 
pressed air  left  in  the  clearance.  But  the  clearance  in  the  air-cylinders  is 
very  considerable;  it  is  equivalent  to  JJ/2  inches  of  the  stroke,  or  31^  per 
cent.  At  60  double  strokes  per  minute,  when  the  speed  of  piston  was 
240  feet  per  minute;  the  indicator  diagrams  showed  an  average  effective 
pressure  of  17.8  Ibs.  per  square  inch  for  steam,  and  0.8  Ibs.  for  air,  repre- 
senting an  efficiency  of  72  per  cent.2 

In  French  blowing  engines,  according  to  M.  Claudel,  the  proportion  of  the 
air  discharged  is  only  75  per  cent,  of  the  volume  described  by  the  piston. 
The  stroke  is  usually  equal  to  the  diameter  of  the  air-cylinder,  and  the 
speed  of  piston  varies  from  100  to  200  feet  per  minute.  The  area  of  the 
inlet-valve  openings  is  from  */I5  to  X/I2  of  that  of  the  piston  for  speeds  of 
from  100  to  150  feet  per  minute,  and  from  */^  to  J/9*  for  higher  speeds. 
The  outlet  openings  are  from  Z/I5  to  I/20  of  the  piston,  in  area. 

In  Belgium,  Mr.  Cockerell  employs  Woolf  cylinders,  with  the  beam,  for 
driving  blowing  engines.  In  one  example,  the  engine  is  of  160  horse- 
power; the  cylinders  are  2.79  and  3.94  feet  in  diameter,  adapted  for  an 
expansion-ratio  of  10,  at  regular  work,  to  yield  a  pressure  of  7  inches 
of  mercury,  or  3^  Ibs.  per  square  inch.  The  engines  usually  expand  nine 
times,  with  steam  of  4  atmospheres.  The  intermediate  fall  of  pressure 
between  the  first  and  second  cylinders  is  uniform  throughout  the  stroke, 
equal  to  1.5  Ibs.  per  square  inch.  The  expansion  curves  are  the  same  as 
the  "theoretical  curve."  The  efficiency,  by  the  indicator  applied  to  the 
steam-  and  the  air-cylinders,  is  81  per  cent,  for  a  blast  of  4  Ibs.,  and  83^ 
per  cent,  for  a  blast  of  4^  Ibs.  per  square  inch.3 

ROOT'S  ROTARY  PRESSURE-BLOWER. 

Root's  rotary  blower  is  positive  in  its  action,  and  consists  of  two  revolvers 
on  parallel  axles,  within  a  close-fitting  case,  rectangular  in  section,  with 
semicircular  ends.  The  revolvers  consist  each  of  two  arms,  formed  with  a 
bulbous  expansion  at  each  end;  and  being  geared  together  by  a  pair  of 
spur-wheels  on  their  shafts,  outside  the  case,  they  necessarily  revolve  at  the 
same  speed ;  and  they  work  together  in  such  a  manner  that  the  ends  of  the 
arms  of  one  revolver  enter  or  gear  into  the  middle  of  the  other  revolver. 
Being  very  correctly  fitted,  little  air  is  allowed  to  escape  between  the 
revolvers,  or  between  them  and  the  casing.  By  their  harmonious  revolu- 
tions, one  being  horizontal  whilst  the  other  is  vertical,  the  spaces  below  and 
above  the  revolvers  are  alternately  contracted  and  enlarged  in  such  an 

1  "Blowing  Engines  at  Lackenby  Ironworks,"  by  A.  C.  Hill.     See  Proceedings  of  the 
Institution  of  Mechanical  Engineers,  1871,  1872. 

2  Journal  of  the  Franklin  Institztte,  March,  1876. 

3  Portefeiiille  de  John  Cockerell,  1876,  vol.  iii. 


928  AIR   MACHINERY. 

order  that  whilst  air  is  drawn  into  the  case  at  the  lower  side,  it  is  expelled 
at  the  upper  side.  Four  discharges  of  air  are  thus  performed  for  each 
revolution  of  the  machine,  and  a  steady  current  is  maintained. 

For  blowing  air,  these  machines  are  made  by  Messrs.  Thwaites  &  Carbutt, 
of  from  y%  to  14  nominal  horse-power,  to  supply  from  150  to  10,800  cubic 
feet  of  air  per  minute,  from  delivery  orifices  of  from  2^/2,  to  19  .inches  in 
diameter.  According  to  the  results  of  tests  made  by  a  committee  of 
engineers,  in  the  United  States,  the  efficiency  of  the  blowers  amounted  to 
from  65  to  80  per  cent,  of  the  horse-power  expended  and  applied  to  the 
machine. 

As  mine-ventilators,  Root's  blowers  are  constructed  with  revolvers  of 
from  3  feet  10^  inches  to  25  feet  in  diameter,  and  3  feet  2  inches  to 

13  feet  wide,  making  from  280  to  40  revolutions  per  minute,  delivering  a 
volume  of  air  of  from  45  to  5000  cubic  feet  per  turn,  or  from  12,500  to 
200,000  cubic  feet  per  minute.     The  effective  power  expended  in  delivering 
these  volumes  of  air,  for  an  exhaustion  at  a  6-inch  water-column,  is,  by 
formula  (  14 ),  page  896,  from  15.5  to  189  horse-power,  and  the  dimensions 
of  cylinder  for  a  non-condensing  engine  to  drive  the  ventilators,  vary  from 

14  inches  diameter  with   18  inches  stroke,  to   28  inches  diameter  with 
48  inches  stroke. 


FLOW   OF  WATER. 


FLOW   OF   WATER   THROUGH    ORIFICES. 

The  fundamental  formula  for  the  flow  of  water  by  the  action  of  gravity  is 


0=8  V   h     ................................  (i) 

v  —  the  velocity  in  feet  per  second. 

//  =  the  height  in  feet  through  which  it  freely  falls. 

This  is  the  basic  formula  (  6  ),  for  the  action  of  gravity,  page  279. 

The  quantity  of  water  delivered  per  second,  through  an  orifice  in  the 
side  of  a  vessel,  supposing  that  there  is  no  contraction,  is  expressed  by  the 
formula,  — 

Q  =  8tf>v/^~  ..............................  (2) 

Q  =  the  quantity  in  cubic  feet  per  second. 

a  =  the  normal  sectional  area  of  the  orifice  or  the  stream  in  square  feet. 

But,  in  effect,  the  quantity  is  less  than  is  here  expressed,  by  reason  of  the 
contraction  of  the  outflowing  stream,  and  the  equation  becomes,  for  prac- 
tical use, 


in  which  m  is  a  coefficient,  less  than  i,  the  value  of  which  varies  with  the 
conditions  of  the  orifice. 

When  the  water  flows  through  an  orifice  in  a  thin  plate,  the  average  value 
of  m  is  about  .62,  irrespective  of  the  form  of  the  orifice,  and  the  formula 
becomes,  Q=  4.96  a  *J  h  or,  in  round  numbers, 


(4) 


in  which  7z  =  the  height  in  feet,  measured  to  the  centre  of  the  orifice. 
With  an  approaching  velocity  v',  the  general  formula  (  3  )  becomes 


(s) 


When  adjutages  or  spouts  are  added  to  an  orifice,  the  outflow  is  increased. 
If  an  internal  tube  be  added,  it  is  diminished.     The  average  values  of  the 


930 


FLOW  OF  WATER. 


coefficient  m,  and  the  product  8  m,  in  formulas 
follows  :  — 


3  )  and  (  5  ),  are  as 


FOKM  o,  *,„„«.  VaJo™'"  ' 

Internal  tube,  ..............................................  50 

Thin  plate  simply,  ........................................  62 

Cylinder,  at  least  2  diameters  in  length,  ...........  82 

Converging  cone,  length  =  2>£  diameters,  .........  95 

Vena  contracta,  length  =  l/2  diameter  of  orifice; 
smallest  diameter  =  .785  diameter  of  orifice,... 
Diverging  cone,  length  =  9  diameters,  .............     1.46 


4.0 
5.0 
6.6 
7.6 
g 

11.7 


MR.  J.  F.  BATEMAN'S  EXPERIMENTS  AT  GODBY  RESERVOIR,  IN  I852.1 

Three  rectangular  openings,  6  inches  deep,  and  6  feet  long,  were  made 
in  boards  2^  inches  and  5  inches  thick,  to  the  sections  shown  in 
Figs.  353.  The  forms  of  the  edges,  horizontal  and  vertical,  were  quad- 
rantal,  to  a  radius  of  2^4  inches.  In  No.  i  the  bell-mouth  section  was 


No.  i. 


No.  2. 


No.  3. 


Figs.  3S3-— Godby  Reservoir.— Flow  of  water  through  submerged  openings  in  boards.     Scale 

outwards,  in  No.  2  inwards,  and  in  No.  3  both  outwards  and  inwards.  The 
openings  were  entirely  submerged  on  the  inside,  at  depths  of  from  i  to 
4  feet  to  the  centres  of  the  openings,  and  there  was  a  free  discharge.  The 
following  are  the  values  of  the  coefficient  8  m,  for  formula  (  3  ),  the  coeffi- 
cient for  the  whole  velocity  due  to  the  height  being  expressed  by  8. 

COEFFICIENTS  OF  VELOCITY  OF  DISCHARGE  ( 8  m ),  IN  FORMULA  ( 3  ). 

Deduced  from  the  Results  of  Experiments  at  Godby  Reservoir. 


HEAD. 
feet. 

4 

3-5 
3 

2.5 

2 

1-5 
I 

0.5 


Average  Coefficients  (maximum  limit,  8). 

No.  i   _  No.  2  No.  3 

(2%  inches  thick).  (2%  inches  thick).  (5  inches  thick). 

5.78  7-04  7-60 

5.66  7.04  7.60 

5-67  7-04  7-60 

5.60  7.04  7.80 

5.60  7.04  7.80 

5.50  7.04  7.78 

5.60  6.89  7.30 

q.^o  6.00  6.;; 


Averages,  omitting  the  last,     5.63 


7.02 


7.64 


1  See  Mr.  Bateman's  paper  on  the  Manchester  Water-works,  Proceedings  of  the  Institution 
of  Mechanical  Engineers,  1866. 


FLOW  OF  WATER  THROUGH   ORIFICES.  93! 

Thence  the  values  of  m  and  8  m  are — 

FORM  OF  OPENING  IN  BOARDS  Formulas  (3)  and  (5). 

(Mr.  Bateman).  .       Value  of  m.        Value  of  8  m. 

No.  i.  Quadrantal  outwards, 70  5.6 

No.  2.  „  inwards,  875  7.0 

No.  3.          „  outwards  and  inwards, 94  7.6 

It  is  seen  that  the  values  of  the  coefficients  were  little  affected  by  the 
variations  of  head,  except  when  the  head  was  less  than  about  i  foot,  or 
double  the  height  of  the  aperture. 


MR.  JAMES  BROWNLEE'S  EXPERIMENTS  ON  THE  FLOW  OF  WATER  THROUGH 
A  SUBMERGED  NOZZLE,  CONVERGENT  AND  DIVERGENT.1 

Mr.  Brownlee's  experiments  were  made  with  a  nozzle,  the  sectional 
contour  of  which  may  be  described  as  a  double  trumpet-mouth.  The 
entrance  was  i^  inches  in  diameter,  and  i^  inches  long,  converging 
to  a  diameter  of  .1982  inch  at  the  throat;  whence  it  diverged  to  a 
diameter  of  Is/l6  inch  at  the  other  or  discharging  end,  through  a  length 
of  5.95  inches,  which  was  equal  to  thirty  times  the  diameter  of  the  throat. 
Putting  h^  and  h2  for  the  heads  in  feet  of  water  at  the  entrance  to,  and  the 
exit  from,  the  nozzle,  and  v  for  the  velocity  of  the  water  passing  through 
the  throat,  the  generating  head  is  (ht  -  /i2),  and  the  relations  of  this  head 
and  the  velocity  are  :  — 

1.61     — 

(6) 


These  formulas  differ  from  the  normal  formula  (  i  ),  in  embodying  the 
i.  6  1  power  of  the  velocity  instead  of  the  2d  power,  and  they  indicate  that 
the  velocity  of  discharge  is  greater  than  that  normally  due  to  the  head. 
The  additional  velocity  is  generated  in  consequence  of  a  vacuous  additional 
pressure  at  the  throat,  the  sum  of  which,  and  the  generating  head  (h^  —  ^a), 
is  the  true  head  under  which  the  discharge  takes  place.  The  table  No. 
315  contains,  for  illustration,  a  selection  from  the  experimental  results  of 
Mr.  Brownlee. 

It  appears  that,  under  a  double  generating  head,  the  water  is  driven 
through  the  compound  nozzle,  with  (1>6l\/  2  =)  1.538  times  the  speed 
for  a  given  generating  head;  and  that  a  double  velocity  does  not  require 
four  times,  as  by  the  normal  formula,  but  only  (21-61  =  )  3.05  times  the 
pressure. 

Mr.  Brownlee  attributes  the  great  augmentation  of  velocity  of  flow  through 
the  throat  of  the  compound  nozzle,  to  the  great  length  of  the  divergent 
outlet,  comparatively  to  the  diameter  of  the  throat.  The  principle  of  the 
action  upon  which  the  augmented  flow  is  effected,  is  that  the  velocity  of  the 
outflowing  water  is,  by  the  necessity  of  occupying  the  expanding  capacity  of 

1  Transactions  of  the  Institution  of  Engineers  and  Shipbuilders  in  Scotland,  vol.  xix., 
page  81. 


932 


FLOW  OF  WATER. 


the  nozzle,  rapidly  reduced ;  and  that  a  vacuum  or  reduction  of  back-pres- 
sure is  induced  at  the  throat,  which,  added  to  the  generating  head,  makes 
up  the  true  head  to  which  the  velocity  is  due. 

Table  No.  315. — FLOW  OF  WATER  THROUGH  COMPOUND  OR  DOUBLE- 
CONICAL  NOZZLES  (Mr.  Brownlee's  Experiments). 

The  Heads  are  expressed  in  feet  of  water. 


Generat- 
ing Head, 
\-k* 

Vacuum  at 
Throat  of 

Nozzle. 

True  Head, 
or  Sum 
of  the 
Generating 
Head  and 

Velocity 
due 

to  the 
Generating 

TT           J 

Velocity 
due  to  the 
True  Head. 

Experi- 
mental 
Velocity. 

Velocity  by 
Formula 

(6). 

the  Vacuum. 

Head. 

feet. 

feet. 

feet. 

ft.  $  second. 

ft.  $1  second. 

ft.  $  second. 

ft.  &  second. 

.25 

.52 

•77 

4-01 

7.04 

6.66 

6.77 

.50 

1-3 

1.8 

5.67 

10.76 

10.23 

10.42 

•75 

2.4 

3-15 

6.95 

14.24 

13-6 

13-39 

3-5 

4-5 

8.02 

17.02 

16.34 

16.02 

2 

8.2 

IO.2 

n-35 

25.63 

24.74 

24.64 

3 

14.0 

17 

13-9 

33-09 

31-95 

31-7 

4 

19.8 

23-8 

16.05 

38.84 

37-9 

37-9 

5 

26.0 

31 

17.94 

44.69 

43-45 

43-52 

6 

31-1 

37-1 

19.66 

48.88 

48.14 

48.74 

FLOW  OF  WATER  OVER  WASTE-BOARDS,  WEIRS,  &c. 

To  find  the  discharge  of  water  over  waste-boards,  &c.,  the  general  for- 
mula is, — 


Q  = 


H    - 


Q  =  the  quantity  in  cubic  feet  discharged  per  second. 
m  =  a  coefficient. 

/=  the  width  of  the  notch  or  overflow,  in  feet. 
H  =the  height  in  feet  of  still-water  above  the  edge  of  the  notch  or  board. 

h  =  the  height  in  feet  of  still-  water,  above  the  level  of  the  water  as  it  flows 
over  the  board. 


When  the  coefficient  m  =  .62,  the  equation  becomes,  by  reduction, 


(9) 


FLOW  OF  WATER  IN  CHANNELS,  PIPES,  AND  RIVERS. 

The  friction  of  fluids  upon  solid  surfaces  is  independent  of  the  pressure. 
It  is  proportional  to  the  area  of  the  surface,  or  to  the  area  of  the  sides  and 
bottom  directly,  and  to  the  volume  of  moving  water  inversely;  or,  in  brief, 
it  is  as  the  length  of  the  contour  or  wetted  perimeter  of  the  conduit,  ct 


FLOW  OF  WATER  IN   CHANNELS,   PIPES,   AND  RIVERS.        933 

divided  by  the  sectional  area  a,  of  the  current,  or  to  - .    It  is  proportional 

a 

to  the  square  of  the  velocity  nearly,  or  to  mv*.  The  accelerating  force 
is  equal  to  -j  x  g,  in  which  h  is  the  height,  /  the  length  of  the  slope  of  the 
channel,  and  g  is  gravity,  or  32.2.  Then,  gx  —  =  —  xmv2-,  and, 


10 


The  quotient  —  is  the  hydraulic  mean  depth,  or  mean  radius,  and  the 
c  i  -- 

elocity  is  proportional  to  \/  —. 


Mr.  Downing  deduces  from  experiment  on  the  flow  of  water  in  pipes, 
the  formula,  — 

Velocity  of  water  in  a  channel  or  pipe. 


/ha  /ha  , 

V  -   \/  — —  X  X  10,000  =  IOO  A/   — —  x  (  II  ) 

I        c  I        c 


a 


In  pipes,  --  —  -  -  —  ;  and,  when  pipes  are  filled  with  the  flowing  water,  the 
c        4 

formula  (  1  1  )  becomes,  by  substitution, 

Velocity  of  water  in  a  full  pipe. 


v  =  the  velocity,  in  feet  per  second. 
h  —  the  head,  in  feet. 

/  =  the  length,  in  feet. 
d=  the  diameter,  in  feet. 

c  =  the  wetted  perimeter,  in  feet. 

a  =  the  sectional  area  of  the  current,  in  square  feet. 

Q  =  the  quantity  of  water  discharged,  in  cubic  feet  per  second. 

D  =  —  =  the  hydraulic  mean  depth. 
f—  the  fall,  in  feet  per  mile. 

The  formula  (12)  is  nearly  identical  with  the  formula  employed  by 
Mr.  Hawksley,  for  the  flow  of  water  in  a  smooth  pipe  of  small  and  uniform 
diameter  :  — 


-xd  ................................  (13) 

Quantity  of  water  discharged  by  a  channel  or  a  pipe. 

f^  /ha  /  h      T^  /\ 

Q=ioo#  V   —  /~x  —  =  100  fl/y/  —  —  x  D  ......................  (  14  ; 


934  FLOW  OF  WATER. 

When  the  pipe  runs  full  under  pressure,  Q  =  .7854  d2  x  50  /y/  —  —  ;  from 
which  Mr.  Downing  deduces  the  formulas : — 

Quantity  discharged  through  a  pipe  running  full  under  pressure. 

(Cubic  feet  per  second)   Q=  39.27  ^/—-xd5 (  15  ) 

(Cubic  feet  per  minute)    Q=  2356    /y/ -L-*ds (  16  ) 

(Cubic  feet  per  minute     Q=  J7^ ...  .  (17) 

and  d  in  inches )    ^  V      / 

(Cubic  feet  per  minute,    n  _ 


d  in  inches,  /in  yards) 


(Gallons    per   minute,      n  dsk  /       N 

d  and  /as  above....)    Q=  I?-°3  V         ........................  ('9) 

.) 
Mr.  Downing  further  deduces  from  formula  (  1  1 


_ 

(Gallons     per     hour       Q  */£*...  .  (  20  ) 

d  and  /as  above....)    v 


_  *  (  T*  \ 

~     <7  5  V  ~~  ' 


(Feet  per  second )     z/=    .92  >y/  2/D  : (21) 

(Feet  per  minute )     v=     55  \/ 2/D  (22) 

.    The  average  velocity  in  an  open  channel  is  about  4/  5  ths  of  the  maximum 
velocity,  which  is  usually  attained  at  the  centre,  near  the  surface.1 

Limits  of  Velocity  at  the  Bottom  of  a  Channel. 
Mr.  Beardmore  gives  the  limits  of  velocity  at  the  bottom,  thus : — 

30  feet  per  minute  does  not  disturb  clay,  with  sand  and  stones. 

40  do.  do.     moves  coarse  sand. 

60  do.  do.  moves  fine  gravel,  size  of  peas. 

120  do.  do.  moves  i-inch  rounded  pebbles. 

1 80  do.  do.  moves  angular  stones,  about  i^  inches  in  diameter. 

CAST-IRON  WATER-PIPES. 

Water-pipes  are  made  to  resist  incidental  stress,  in  addition  to  the  normal 
stress  by  internal  pressure.  The  proper  thickness  of  cast-iron  pipes  has 
been  expressed  by  numerous  empirical  formulas,  widely  divergent.  The 
following  simple  formula  is  here  deduced  from  Mr.  Bateman's  practice. 

In  sixteenths  of  an  inch,...  t=     4  +  - — (  23  ) 

600 

In  inches  and  decimals,...  /=  .25  + (  24  ) 

9600 

Do.     do /=.25  +  _^ (25) 

4250 

1  See  Mr.  Downing's  work,  "Elements  of  Practical  Hydraulics,"  from  which  the  above 
formulas  have  been  derived. 


CAST-IRON   WATER-PIPES. 


935 


/  =  the  thickness  of  the  pipe,  in  inches  and  decimals,  or  in  sixteenths  of 

an  inch. 

H  =  the  head  of  pressure,  in  feet  of  water. 
p  =  the  interior  pressure,  in  pounds  per  square  inch. 
d=  the  inside  diameter  of  the  pipe,  in  inches. 

Note.—  The  pressure  in  Ibs.  per  square  inch,  is  equal  to  the  product  of  the 
head  in  feet  of  water,  by  .443. 

Mr.  Hawksley's  formula  for  the  thickness  of  pipes  under  pressure,  is,  — 


/  =  thickness,  F  —  factor  of  strength,  p  —  pressure,  ^-diameter,  s-  tensile 
strength  of  material,  C  =  a  practical  constant  correction  for  imperfections  ot 
process,  method,  and  workmanship. 

For  the  usual  head  of  300  feet  of  water,  formula  (  24  )  becomes,  — 


in  inches  ......  ^=.25+ 


or   /=. 


(  27  ) 
(28) 


The  following  are  special  divisors  and  multipliers  to  be  employed  in 
formulas  (  27  )  and  (  28  )  for  various  heads:  — 

Head  in  Feet. 

100 
ISO 
2OO 
250 
300 
350 
4OO 
500 
750 
1000 

Socket-end.  —  For  a  series  of  water-pipes  cast  at  Woodside  Ironworks,  it 
is  calculated,  from  the  sections,  that  the  equivalent  length  of  pipe,  of  equal 
weight,  for  a  socket-end,  varies  from  7.2  inches  for  2^-inch  pipes,  to  8.6 
inches  for  24-inch  pipes.  Hence  the  formula  for  the  equivalent  length  of 
pipe  for  the  socket  for  any  diameter  :  — 

Equivalent  length  in  inches,  =  7  +  -  —  ........................   (  29  ) 


»                       Divisor  in  (27). 
96 

Equivalent  Pressure 
Multiplier  in  (28).                        in  Ibs.  per 
Square  Inch.      I 
OIO4.                                       AAI  The 

64 

016 

66  5 

4.8 

O2  1 

886 

O26 

j  jo  7 

•22 

I  ^2  Q 

27 

O37 

1J^-V    •>•> 
155-0    „ 

177  2 

24. 

O4.2 

IQ 

*//•*      )5 

078 

332.2      „ 

Q.6      

.104. 

in  feet,     =.6  +  -7:—; 
ibo 


(3°) 


in  which  d  is  the  diameter  of  the  pipe  in  inches. 

The  table  No.  316  gives  the  thickness  and  the  weight  of  cast-iron  water- 
pipes  of  given  diameters  for  a  working  head  of  300  feet  of  water,  or  133  Ibs. 
per  square  inch.  The  bursting  strengths,  taking  the  ultimate  tensile  resist- 
ance of  the  metal  at  7  tons  per  square  inch,  and  the  factors  of  safety,  are 
given  in  the  last  two  columns. 


936 


FLOW  OF  WATER. 


Table  No.  316. — CAST-IRON  PIPES: — THICKNESS,  WEIGHT,  AND  STRENGTH. 


Diameter. 

Thickness 
by  Formula 

(24)  or  (25). 

Nearest 
Thickness 
in  six- 
teenths of 
an  Inch. 

Net  Weight 
per  Foot  run, 
for  Thickness 
in  Column  3. 

Length  of 
Pipe  equal 
in  Weight 
to  the 
Socket. 

Weight  of  a 
o-feet  Length 
of  Pipe. 

Bursting 
Pressure 
per  Square 
Inch, 
reckoned  on 
Column  3. 

Factor  of 
Safety,  for 
Normal 
Pressure  of 
300  Feet  of 
"Water,  or 
133  Ibs.  per 
Square 
Inch. 

inches. 

inches. 

whole 
sixteenths. 

pounds. 

feet. 

cwts. 

Ibs. 

times. 

2 

•31 

S/i6 

7.09 

.60 

(6  ft.)  .418 

4900 

36 

2^ 

•33 

H 

10.6 

.61 

(6  ft.)  .625 

3920 

30 

3 

•35 

£ 

12.4 

.62 

1.  06 

3920 

30 

4 

•375 

3/s 

16.1 

.62 

1.38 

2940 

22 

5 

.41 

7/i6 

23-4 

•63 

2.01 

2744 

21 

6 

•45 

7/x6 

27.7 

•63 

2.38 

2290 

17 

7 

•47 

M 

36.8 

.64 

3-17 

2240 

17 

8 

•5° 

% 

41.7 

.64 

3-59 

1960 

15 

9 

•53 

9/i6 

52.8 

•65 

4-55 

1960 

15 

10 

.56 

9/i6 

58.3 

.66 

5-03 

1764 

13 

ii 

•59 

9/i6 

63-9 

.66 

5-51 

1604 

12 

12 

.62 

# 

77-5 

.67 

6.69 

1633 

12 

13 

•65 

# 

83.6 

•67 

7.22 

1508 

II 

H 

.70 

«/i6 

99.1 

.68 

8.56 

1540 

12 

15 

•7i 

«Afi  • 

105.9 

.68 

9.15 

1440 

II 

cwts. 

16 

•75 

^ 

1.  10 

.69 

10.66 

1470 

II 

18 

.81 

J3/i6 

1-43 

.70 

13-87 

1415 

10.6 

20 

.87 

# 

i.  60 

•7i 

15-54 

1372 

10.3 

21 

.90 

# 

1.68 

.72 

16.33 

1307 

10 

24 

•99 

2.19 

•73 

21.31 

1307 

10 

27 

1.09 

x/x6 

2.61 

•75 

25.45 

1234 

9-3 

30 

1.18 

3/i6 

3-24 

•77 

31-65 

1241 

9-3 

33 

1.27 

X 

3-75 

.78 

36.67 

1190 

8.9 

36 

1.36 

H 

4.50 

.80 

44.10 

1198 

8-9 

39 

1-45 

7/i6 

5.11 

.82 

50.18 

1156 

8.7 

42 

i-55 

9/i6 

5.98 

•83 

58.78 

1107 

8.8 

45 

1.65 

K 

6.67 

•85 

65.70 

H33 

8.5 

48 

1.74 

¥ 

7.63 

.87 

75.31 

H43 

8.6 

Note  to  table. — Flanges. — The  additional  weight  for  a  pair  of  flanges   is  reckoned  as 
equivalent  to  that  of  a  lineal  foot  of  pipe ; — equal  to  1 1  per  cent,  extra  for  Q-feet  lengths. 

Gas-pipes. — Mr.  Thomas    Box   gives   the   following   thickness   for   gas- 
Table  No.  317. — THICKNESS  OF  CAST-IRON  GAS-PIPES. 


pipes:1— 


Diameter. 

Thickness. 

Diameter. 

Thickness. 

Diameter. 

Thickness. 

Diameter. 

Thickness. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

inches. 

1% 

.27 

5 

•37 

10 

.46 

24 

.64 

2 

•29 

6 

•39 

12 

•49 

30 

.69 

2X 

•3 

7 

.41 

15 

•53 

36 

•75 

3 

•32 

8 

•43 

IS 

•57 

4 

•35 

9 

•45 

21 

.6 

Practical  Hydraulics,  1867. 


WATER-WHEELS. 


The  work  of  water-wheels  is  done  by  the  force  of  gravity  acting  on 
water.  The  total  work  in  a  fall  of  water  is  expressed  by  the  product  of  the 
weight  of  water,  w,  and  the  height  of  the  fall,  h;  or  by  w  h;  and,  in  order 
that  the  whole  of  this  work  should  be  realized  by  the  wheel,  the  water  must 
enter  the  machine  without  shock,  and  leave  it  without  velocity.  But  there 
is.  unavoidably,  a  residual  velocity  v't  and  the  loss  of  work  due  to  this 

z/2 
velocity,  is  w =  w  h ',  in  which  h '  is  the  head  due  to  the  residual 

velocity.     The  part  of  the  head  expended  in  effective  work,  and  upon 

„.  '  2 

internal  resistances,  is  therefore  h  —  Ji=h  —  w  — . 

2^ 

There  are  two  classes  of  wheels ; — first,  those  which  turn  on  a  horizontal 
.axis;  second,  those  which  turn  on  a  vertical  axis. 


WHEELS    ON    A    HORIZONTAL    AXIS. 

UNDERSHOT-WHEELS,  WITH  RADIAL  FLOATS  OR  BUCKETS. 

These  are  constructed  from  10  to  25  feet  in  diameter;  the  floats  are  from 
14  to  1 6  inches  apart  at  the  circumference,  and  from  24  to  28  inches  deep. 
Putting  v  and  v'  for  the  initial  and  final  velocities  of  the  water,  which  flows 
under  the  wheel,  v'  is  sensibly  equal  to  the  velocity  of  the  middle  of  the 

float,  and  the  maximum  effect  is  obtainable  when  v'  =  —,  when  the  effici- 

2 

ency  is  50  per  cent.  Smeaton  tried  velocities  v'  of  from  .34?'  to  .52  vt 
mean  .43  v\  and  obtained  an  efficiency,  with  models,  of  from  29  to  35, 
mean  33  per  cent.  By  the  best  modern  experiments,  the  efficiency  is 
usually  from  27  to  30  per  cent.  Experimentally,  40  per  cent,  is  the  maxi- 
mum. Probably,  it  cannot  be  exceeded,  because  the  final  velocity  of  the 
water  is  not  reduced  to  y2  v. 

PONCELET'S  UNDERSHOT-WHEEL. 

The  floats  are  curved, — usually  a  portion  of  a  circle, — and  so  placed 
that  the  hollow  of  the  curve  is  presented  to  the  entering  water,  the  edge 
of  the  float  being  set  at  an  angle  of  30°  to  the  circumference  of  the  wheel. 
There  are  36  floats  in  wheels  of  from  10  to  13  feet  in  diameter,  and  48 
floats  for  diameters  of  from  20  to  23  feet.  If  the  water  could  enter  the 
wheel  without  shock,  tangentially  to  the  floats,  the  velocity  of  the  floats 
being  half  the  velocity  of  the  water,  the  water  would  ascend  the  float, 
and  would  then  descend  by  the  force  of  gravity,  and  drop  into  the  tail-race 


938  WATER-WHEELS. 

with  a  final  forward  velocity  =  o.  The  efficiency,  under  these  circumstances, 
would  be  100  per  cent.  But  the  conditions  of  practice  do  not  admit  of  a 
tangential  entrance,  and  the  efficiency  is  not  more  than  65  per  cent,  for 
falls  of  4  feet  and  less,  60  per  cent,  for  falls  of  from  4  feet  3  inches  to 
5  feet,  and  from  55  to  50  per  cent,  for  falls  of  from  6  feet  to  6^  feet. 
These  efficiencies  are  materially  greater  than  that  of  the  undershot-wheels 
with  radial  floats ;  and  the  experience  of  the  Poncelet-float  conspicuously 
demonstrates  the  essential  importance  of  providing  graduated  entrances,  and 
avoiding  shocks,  concussions,  or  eddies  in  the  water.  The  most  favourable 
ratio  of  the  velocity  of  the  floats  to  that  of  the  water,  is  55  per  cent.  The 
distance  between  the  inner  and  outer  circumferences  that  limit  the  floats 
should  be  at  least  a  fourth  of  the  head;  Poncelet  advises  a  third  of  the 
head. 

PADDLE-WHEEL  IN  AN  OPEN  CURRENT. 

Experience  indicates  that  the  most  suitable  ratio  of  the  velocity  of  the 
floats  to  that  of  the  current  is  40  per  cent.  The  depth  of  the  floats  should 
be  from  ^th  to  I/5ih  of  the  radius;  it  should  not  be  less  than  12  or 
14  inches.  The  diameter  is  usually  from  13  to  i6ft  feet,  with  12  floats; 
but  it  is  thought  that  there  might  be  an  advantage  in  applying  18  or  even 
24  floats.  The  floats  should  be  completely  submerged  at  the  lower  side, 
but  not  more  than  2  inches  under  water. 

BREAST-WHEEL. 

This  wheel  receives  the  water  at  a  level  a  little  below  that  of  the  axis.  In 
practice,  the  efficiency  is  70  per  cent,  when  the  height  of  the  fall  approaches 
8  feet,  and  50  per  cent,  for  a  fall  of  4  feet.  For  a  well  constructed  wheel, 
slow-moving,  M.  Morin  found  an  exceptional  efficiency  of  93  per  cent. 

Sir  William  Fairbairn  states  that  the  efficiency  of  high  breast-wheels  is 
75  per  cent,  moving  at  the  rate  of  5  feet  per  second  at  the  periphery.  The 
usual  velocity  adopted  by  him  for  high  and  low  falls,  was  from  4  to  6  feet 
per  second;  for  a  minimum  velocity,  3  feet  6  inches  per  second,  for  falls 
of  from  40  to  45  feet;  for  a  maximum  velocity,  7  feet  per  second,  for  falls 
of  5  or  6  feet. 

The  water  should  be  delivered  to  the  wheel  at  a  low  velocity;  or,  if  the 
velocity  is  considerable,  the  delivery  should  be  at  a  tangent  to  the  edge  of 
the  float.  The  most  suitable  velocity  of  the  floats  is  4  jj(  feet  per  second ; 
the  velocity  should  not  exceed  the  limits  of  from  3  to  6^4  feet  per  second. 
The  depth  of  water  over  the  sliding  gate  should  be  from  8  to  10  inches, 
measured  from  still  water.  The  diameter  should  be  at  least  11^2  feet;  it 
is  seldom  more  than  from  20  to  23  feet.  These  diameters  are  suitable  for 
falls  of  from  3  to  6  or  even  8  feet.  The  distance  apart  of  the  floats  should 
be  i y$  to  i  y^  times  the  head  over  the  gate,  for  slow  wheels;  for  quick  wheels, 
a  little  more.  The  depth  of  the  floats  should  be  a  little  more  than  2:3  feet. 
Normally,  the  interior  capacity  between  two  floats  should  be  nearly  double 
the  volume  of  the  water  there. 

OVERSHOT-WHEEL. 

The  water  is  delivered  nearly  at  the  top  of  the  wheel.  The  chief  causes 
of  loss  of  head  are,  first,  the  relative  velocity  of  the  water  when  it  enters 
the  wheel,  and  the  velocity  which  it  possesses  at  the  moment  it  falls  to 


WHEELS  ON   A  VERTICAL  AXIS.  939« 

the  level  of  the  tail-race.  Such  wheels  answer  well  for  heads  of  from 
13  to  20  feet.  For  heads  of  less  than  10  feet,  breast-wheels  are  preferred. 
The  velocity  of  the  floats  should  not  be  less  than  3  feet  per  second;  it  may 
be  6^  feet  per  second  for  small  wheels,  and  10  feet  for  larger  wheels,  without 
sensibly  affecting  the  efficiency.  The  efficiency  at  a  low  speed  may  rise  tc» 
80  per  cent.;  but  ordinarily,  with  velocities  of  from  3  to  6^  feet,  the  effi- 
ciency varies  from  70  to  75  per  cent.  At  higher  speeds,  and  when  the 
buckets  are  more  than  two-thirds  filled,  the  efficiency  is  only  60  per  cent. 
For  wheels  employed  in  driving  hammers,  of  from  10  to  13  feet  in  diameter, 
having  a  velocity  of  from  13  to  16  feet  per  second,  the  efficiency  occasion- 
ally falls  to  37  per  cent.  This  low  efficiency  is  due  to  the  impetuosity  of 
the  fall  of  the  water  into  the  quick-moving  buckets,  whence  the  water  is 
thrown  out,  partly  by  reaction,  partly  by  centrifugal  force.  The  capacity 
of  the  buckets  should  be  three  times  the  volume  of  the  charge  of  water; 
they  may  be  10  or  n  inches  deep,  and  12  or  14  inches  apart.  With  a 
velocity  of  4  feet  per  second,  i  cubic  foot  of  water  per  foot  of  breadth  of 
the  wheel,  may  be  consumed  per  second.  The  stream,  as  it  leaves  the 
gate,  is  rarely  more  than  4  or  5  inches  deep ;  it  is  often  less  than  2  inches, 
deep. 


WHEELS   ON  A  VERTICAL  AXIS. 

TUB-WHEEL. 

The  old-fashioned  spoon-wheel  or  tub-wheel,  consists  of  a  number  of 
paddles  fixed  on  a  vertical  axis,  revolving  within  a  cylindrical  well  of 
masonry,  with  very  little  clearance.  The  paddles  are  slightly  concave, 
and  are  struck  in  the  hollow  side  by  a  horizontal  current  from  a  reservoir 
at  a  considerable  head;  the  current  enters  the  well  tangentially,  and,  after 
having  expended  its  force,  it  falls  between  the  open  paddles  to  the  bottom 
of  the  well.  The  maximum  efficiency  is  calculated  to  be  that  due  to  a 
velocity  of  the  centre  of  the  paddles,  when  they  are  struck,  equal  to  one- 
third  of  the  velocity  of  the  current;  and  the  efficiency  is  30  per  cent.  In 
practice,  the  efficiency  varies  from  1 5  to  40  per  cent. 

WHITELAW'S  WATER-MILL. 

Mr.  James  Whitelaw,  of  Paisley,  developed  the  principle  of  Barker's  rnill,- 
and  produced  an  efficient  motor.  Barker's  mill  consisted  of  two  hollow 
radial  arms  revolving  on  a  central  pipe,  through  which  water  under  pressure 
passed  to  the  extremities  of  the  arms,  and  was  ejected  through  an  orifice 
at  the  end  of  each  arm,  in  an  opposite  direction;  and  so  producing  rotatory 
motion.  In  Whitelaw's  mill,  the  arms  taper  from  the  centre  towards  the  cir- 
cumference, and  they  are  curved  in  such  a  manner  as  to  allow  the  water  to 
pass  from  the  central  openings  to  the  orifices,  in  directions  nearly  straight 
and  radial,  when  the  machine  runs  at  its  proper  speed;  so  that  very- 
little  centrifugal  force  is  imparted  to  the  water  by  the  revolution  of  the 
arms,  and  that,  therefore,  a  minimum  of  frictional  resistance  is  opposed 
to  the  motion  of  the  water.  A  model,  15  inches  in  diameter,  measured  to 
the  centres  of  the  orifices,  with  a  central  opening  6  inches  in  diameter,  and 
two  orifices  of  discharge,  each  2.4  inches  by  0.6  inch, — was  tested  under 


940  WATER-WHEELS. 

.a  head  of  10  feet,  making  387  revolutions  per  minute.  The  efficiency 
.amounted  to  73.6  per  cent.  At  324  revolutions,  the  efficiency  was  71  per 
cent.  According  to  the  results  of  tests  of  another  model  mill,  made  by 
competent  engineers,  the  efficiency  amounted  to  76  per  cent.,  when  the 
speed  of  the  orifices  was  equal  to  that  due  to  the  height  of  the  fall. 

A  water-mill,  onWhitelaw's  system,  9.55  feet  in  diameter,  having  circular 
orifices  4.944  inches  in  diameter,  with  a  fall  of  25  feet,  was  erected  on  the 
Chard  Canal  in  1842,  for  the  purpose  of  hauling  boats  up  an  inclined  plane. 
The  net  work  done  by  the  machine  represented  an  efficiency  of  67.3  per 
cent.,  with  the  resistance  of  the  gearing  in  addition.  It  was  estimated  that 
.the  actual  duty  of  this  mill  amounted  to  75  per  cent.1 

TURBINES. 

Turbines,  like  Whitelaw's  mill,  apply  the  force  of  water  by  impact  and 
reaction  combined;  but  they  comprise  an  additional  feature  not  embodied 
in  Whitelaw's  mill,  the  employment  of  guide-blades  to  change  the  direc- 
tion of  the  water  descending  under  a  head,  so  as  to  cause  it  to  enter 
tangentially  between  the  blades  of  the  wheel.  The  blades  of  the  wheel  are 
so  curved  as  to  receive  the  water  without  shock,  and  to  discharge  it  hori- 
zontally. Turbines  are  of  three  kinds; — outward  flow,  downward  flow,  and 
inward  flow. 

OUTWARD-FLOW  TURBINES. 

Fonrneyron  Turbine. — This  turbine  acts  with  an  outward  flow;  that  is  to 
say,  the  water  enters  from  above  through  a  central  opening,  and  is  guided 
by  curved  blades,  to  be  discharged  laterally  at  the  base  of  a  circular  chamber, 
or  well,  equally  at  all  parts  of  the  circumference,  into  the  buckets  or  curved 
blades  of  the  wheel.  The  wheel  is  annular,  and  closely  surrounds  the 
circular  chamber;  thus  it  receives  the  whole  of  the  water  at  its  maximum 
velocity,  and  it  is  the  function  of  the  curved  blades  to  receive  and  transmit 
the  force  of  the  water,  and  to  discharge  the  water  at  the  outer  circumference 
of  the  wheel,  with  the  least  possible  residual  velocity. 

When  the  supply  of  water  is  insufficient  for  working  the  turbine  at  its 
full  power,  the  exit  openings  from  the  well  are  partially  closed  by  a  cylin- 
drical sluice  which  is  lowered  upon  them  to  the  required  extent. 

The  efficiency  is  reduced  in  proportion  as  the  sluice  is  lowered,  for  the 
-action  of  the  water  on  the  wheel  is  less  favourably  exerted.  M.  Morin 
tested  a  Fourneyron  turbine,  2  metres,  or  6.56  feet  in  diameter,  and  he 
found  that,  in  this  way,  the  efficiency  varied  from  a  maximum  of  79  per  cent, 
to  24  per  cent.,  when  the  supply  of  water  was  reduced  to  a  fourth  of  the 
full  supply.  In  practice,  the  radial  length  of  the  buckets  or  floats  of  the 
wheel  is  a  fourth  of  the  radius,  for  falls  not  exceeding  6  */£  feet,  three-tenths 
for  falls  of  from  6^  to  19  feet,  and  two-thirds  for  higher  falls. 

Boy  den  Turbine. — Mr.  Boyden,  of  Massachusetts,  designed  an  outflow- 
turbine  of  75  horse-power,  which  realized  an  efficiency  of  88  per  cent.  The 
peculiar  features,  as  compared  with  Fourneyron's  turbine,  are,  ist,  and  most 
important,  the  conducting  of  the  water  to  the  turbine  through  a  vertical  trun- 
cated cone,  concentric  with  the  shaft.  The  water,  as  it  descends,  acquires  a 
gradually  increasing  velocity,  together  with  a  spiral  movement  in  the  direction 

1  See  Description  of  Whitelaw  &  Stirrafs  Patent  Water-mill,  1843. 


TURBINES.  941 

of  the  motion  of  the  wheel.  The  spiral  movement  is,  in  fact,  a  continua- 
tion of  the  motion  of  the  water  as  it  enters  the  cone.  20!.  The  guide-plates 
at  the  base  are  inclined,  so  as  to  meet  tangentially  the  approaching  water. 
3d.  A  "  diffuser,"  or  annular  chamber  surrounding  the  wheel,  into  which  the 
water  from  the  wheel  is  discharged.  This  chamber  expands  outwardly, 
and  thus  the  escaping  velocity  of  the  water  is  eased  off  and  reduced  to 
a  fourth  when  the  outside  of  the  diffuser  is  reached.  The  effect  of  the 
diffuser  is  to  accelerate  the  velocity  of  the  water  through  the  machine  ;  and  the 
gain  of  efficiency  is  3  per  cent.  The  diffuser  must  be  entirely  submerged. 

RULES  FOR  OUTWARD-FLOW  TURBINES. 

Lieutenant  F.  A.  Mahan,  U.S.,  has  deduced,  from  the  practice  of  Mr.. 
Boyden,  the  following  rules  for  proportioning  outward-flow  turbines.  He 
has  also  deduced  the  formulas  which  follow  from  the  results  of  Mr.  Francis' 
experiments  on  Mr.  Boyden's  turbines  at  the  Tremont  Mills,  Lowell.1 

Rtiles  for  Proportioning  Outward-flow  Turbines. 

For  falls  of  from  5  feet  to  40  feet,  and  diameters  not  less  than  2  feet  :  — 
i  st.  The  sum  of  the  shortest  distances  between  the  buckets  should  be 
equal  to  the  diameter  of  the  wheel.  2d.  The  height  of  the  orifices  at  the 
circumference  of  the  wheel  should  be  equal  to  one-  tenth  of  the  diameter 
of  the  wheel.  3d.  The  width  of  the  crowns  should  be  four  times  the 
shortest  distance  between  the  buckets.  4th.  The  sum  of  the  shortest  dis- 
tances between  the  curved  guides,  taken  near  the  wheel,  should  be  equal 
to  the  interior  diameter  of  the  wheel. 

For  falls  greater  than  40  feet,  the  2d  rule  should  be  modified  :  the  height 
of  the  orifices  should  be  smaller  in  proportion  to  the  diameter  of  the  wheel. 

On  the  basis  of  these  rules,  an  efficiency  of  75  per  cent,  may  be 
obtained. 

Formulas  for  the  proportions  and  performance  of  Outward-flow  Turbines. 


3 

(  4  ) 
H-.ioD   .......................................  (  5  ) 

N  =  3(D  +  io)  .................................  (  6  ) 

-8   .............  ...............................   <  7  > 


^  =  .50  N  to  .75  N   ...........................  (I0) 

;'  "-J  ...........................................  <»> 

1  Water-  Wheels  and  Hydraulic  Motors.     New  York,  1876. 


942  WATER-WHEELS. 

D  =  the  exterior  diameter  of  the  wheel. 

</=the  interior  diameter  of  the  wheel. 

H=the  height  of  the  orifices  of  discharge  at  the  outer  circumference. 
W  =  the  width  of  the  crown  of  the  buckets. 

N=the  number  of  buckets. 

72  =  the  number  of  guides. 

7e/  =  the  shortest  distance  between  two  adjacent  buckets. 
•z£/=:the  shortest  distance  between  two  adjacent  guides. 
HP— the  actual  horse-power  of  the  turbine. 

^=the  height  of  the  fall  acting  on  the  wheel. 

Q=the  quantity  of  water  expended  by  the  turbine,  in  cubic  feet  per  second. 

V  — the  velocity  due  to  the  fall. 

V'— the  velocity  of  the  water  passing  through  the  narrowest  sections  of  the 
wheel. 

2/=the  velocity  of  the  interior  circumference  of  the  wheel. 

C  =  the  coefficient  for  V  in  terms  of  V,  or  — . 

The  dimensions  are  in  feet;  and  the  velocities  in  feet  per  second. 
These  formulas  are  only  to  be  taken  as  guides  for  practice;    not  as 
established  proportions. 

DOWNWARD-FLOW  TURBINES. 

Fontaines  Turbine. — In  turbines  constructed  with  downward  flow,  the 
wheel  is  placed  below  an  annular  series  of  guide-blades,  by  which  the  water 
is  conducted  to  the  wheel.  The  water  strikes  the  curved  floats  of  the  wheel, 
and  it  falls  vertically,  or  nearly  so,  into  the  tail-race.  The  principle  of  action 
is  the  same  as  that  of  the  outward- flow  turbine;  but  the  centrifugal  action 
of  the  latter  is  avoided,  and  the  downward  flow  is  more  compact. 

The  Fontaine  turbine  yields  an  efficiency  of  70  per  cent.,  when  fully 
•charged.  When  the  supply  of  water  is  shut  off  to  three-fourths,  by  the 
sluice,  the  efficiency  is  5  7  per  cent.  The  best  velocity  at  the  mean  circum- 
ference of  the  wheel,  is  equal  to  55  per  cent,  of  that  due  to  the  height  of 
fall.  It  may  vary  a  fourth  of  this  either  way,  without  materially  affecting 
the  efficiency.  An  efficiency  of  70  per  cent,  is  guaranteed  by  manufac- 
turers. 

Jonval  Turbine. — This  turbine  is,  essentially,  the  same  as  Fontaine's;  but, 
for  convenience,  it  is  placed  at  some  height  above  the  level  of  the  tail- 
water,  within  an  air-tight  cylinder,  or  "  draft-tube,"  so  that  a  partial  vacuum 
•or  reduction  of  pressure  is  induced  under  the  wheel,  and  the  effect  of  the 
wheel  is  by  so  much  increased.  The  resulting  efficiency  is  the  same  as  if 
the  wheel  were  placed  at  the  level  of  the  tail-race;  and  thus,  whilst  the 
turbine  may  be  placed  at  any  level,  advantage  is  taken  of  the  whole  height 
of  the  fall. 

The  efficiency  under  a  full  charge  is  72  per  cent.  The  best  velocity  at 
the  exterior  of  the  wheel,  is  70  per  cent,  of  that  due  to  the  total  fall. 

Turbine  by  the  North  Moor  Foundry. — The  water  is  conducted  by  a  pipe 
into  a  spiral  water-chamber  surrounding  the  wheel,  from  which  it  is  guided 
through  the  guides  or  water-ports  horizontally  on  to  the  middle  of  the 
circumference  of  the  wheel,  so  as  to  enter  the  curved  wings  or  buckets 
without  any  shock.  After  traversing  them,  it  passes  off  vertically,  or 
nearly  so,  on  both  sides  of  the  wheel,  above  and  below,  to  the  tail-race. 
The  buckets  are,  for  this  purpose,  constructed  in  two  rings,  with  right 


TANGENTIAL  WHEELS.  943 

and  left  curvatures,  meeting  at  the  middle,  at  an  acute  angle,  where  the 
water  enters  and  is  divided  equally  between  them.  The  flow  is  partially 
inwards,  combined  with  vertical  flow. 

INWARD-FLOW  TURBINES. 

Vortex  Wheel,  or  Inward-flow  Turbine. — The  vortex  wheel  is  made  with 
radiating  vanes,  and  is  surrounded  by  an  annular  case,  closed  externally, 
and  open  internally  to  the  wheel,  having  its  inner  circumference  fitted  with 
four  curved  guide-passages.  The  water  is  admitted  by  one  or  more  pipes 
to  the  case,  and  it  issues  centripetally  through  the  guide-passages  upon  the 
circumference  of  the  wheel.  The  water  acting  against  the  curved  vanes  of 
the  wheel,  the  wheel  is  driven  round  at  a  velocity  dependent  on  the  height 
of  the  fall,  and  the  water,  having  expended  its  force,  passes  towards  and  out 
at  the  centre.  This  wheel  has  realized  an  efficiency  as  high  as  77^  per 
cent.  It  was  originally  designed  by  Professor  James  Thomson. 

Swain  Turbine. — In  the  Swain  turbine  are  combined  an  inward  and  a 
downward  discharge.  The  receiving  edges  of  the  floats  of  the  wheel  are 
vertical,  opposite  the  guide-blades,  and  the  lower  portions  of  the  edges  are 
bent  into  the  form  of  a  quadrant.  Each  float  thus  forms  with  the  surface 
of  the  adjoining  float  an  outlet  which  combines  an  inward  and  a  downward 
discharge.1  A  Swain  turbine,  72  inches  in  diameter,  was  tested  at  Boott 
Cotton  Mill,  U.S.,  by  Mr.  J.  B.  Francis,  for  several  heights  of  gate,  or 
sluice,  from  2  to  13.08  inches,  and  circumferential  velocities  of  wheel  ran- 
ging from  60  per  cent,  to  80  per  cent,  of  the  respective  velocities  due  to  the 
heads  acting  on  the  wheel.  For  a  velocity  of  60  per  cent.,  and  for  heights 
of  gate  varying  within  the  limits  already  stated,  the  efficiency  ranged  from 
47^3  to  76}^  per  cent,  and  for  a  velocity  of  80  per  cent,  it  ranged  from 
37/4  to  83  per  cent  The  maximum  efficiency  attained  was  84  per  cent, 
with  a  12-inch  gate  and  a  velocity-ratio  of  76  per  cent;  but  from  Q-inch  gate 
to  i3-inch  gate, — say,  from  two-thirds  gate  to  full  gate, — the  maximum  effici- 
ency varied  within  very  narrow  limits — from  83  to  84  per  cent., — the  velocity- 
ratios  being  72  per  cent,  for  the  p-inch  gate,  and  76^  per  cent  for  full 
gate.  At  half-gate,  the  maximum  efficiency  was  78  per  cent.,  when  the 
velocity-ratio  was  68  per  cent.  At  quarter-gate,  the  maximum  efficiency 
was  6 1  per  cent.,  and  the  velocity-ratio  66  per  cent. 


TANGENTIAL   WHEELS. 

Wheels  to  which  the  water  is  applied  at  a  portion  only  of  the  circumfer- 
ence, are  called  tangential  wheels.  They  are  specially  suited  for  very 
high  falls,  where  a  considerable  diameter  and  high  tangential  velocity  may 
be  combined,  with  a  moderate  speed  of  revolution.  The  Girard  turbine 
belongs  to  this  class.  It  is  employed  at  Goeschenen  station,  for  the  works 
of  the  St.  Gothard  tunnel;  and  it  works  under  a  head  of  279  feet.  The 
wheels  are  7  feet  10}^  inches  in  diameter;  they  have  80  buckets,  and  their 
regular  speed  is  160  turns  per  minute,  with  a  maximum  charge  of  water  of 
67  gallons  per  second.  It  is  said  that  the  Girard  turbines  employed  at  the 
Paris  water-works  have  yielded  an  efficiency  as  high  as  87  per  cent.;  ordi- 
narily, the  efficiency,  it  is  said,  is  from  75  to  80  per  cent. 

1  Journal  of  the  Franklin  Institute,  April,  1875.  See  also  a  notice  in  the  Proceedings 
of  the  Institution  of  Civil  Engineers,  vol.  xli.,  page  334. 


MACHINES    FOR    RAISING    WATER. 


PUMPS. 

The  effective  work  done  by  a  pump,  in  raising  water,  is  equal  to  the  pro- 
duct of  the  weight  of  water  lifted,  by  the  height  through  which  it  is  raised. 
The  efficiency  of  the  pump  is  the  ratio  of  the  effective  work  to  the  whole 
work  expended  in  driving  the  pump. 

RECIPROCATING  PUMPS. 

The  efficiency  of  well-proportioned  reciprocating  pumps  is  from  75  to 
85  per  cent.;  ordinarily,  it  is  not  more  than  75  per  cent.,  and  is  often 
less  than  that.  In  well-made  pumps,  in  good  order,  the  volume  of  water 
passed  is  96  or  97  per  cent,  of  the  volume  described  by  the  piston;  but, 
for  ordinary  pumps,  it  is  only  from  80  to  90  per  cent.  When  the  speed  of 
the  pump  is  very  rapid,  the  volume  of  water  may  be  equal  to  or  greater  than 
the  volume  described  by  the  piston.  In  such  instances,  the  column  of  water 
continues  in  motion  after  the  piston  has  arrived  at  the  end  of  its  stroke. 
Ordinary  well-made  pumps  may,  with  certainty,  draw  water  from  any  depth 
not  exceeding  27  feet.  M.  Claudel  gives  the  results  of  experimental  tests 
of  various  pumps : — 

Ratio  of  Volume  of  Water 
Efficiency.  to  that  described 

by  the  Piston. 

Fire-engines  : — Height,  loto  16  feet,         20.7  per  cent 91  per  cent. 

Worked  with  hose,..          35.8        „  91         „ 

Pumps  for  drainage,  50  to  69        „  93        „ 

Pumps  in  towns,  double-acting, 701075         „  901095         „ 

Mr.  R.  Davison1  gives  the  duty  of  6-inch  three-throw  pumps  employed  in 
raising  water  at  a  brewery,  with  the  indicator  horse-power  consumed,  from 
which  the  following  deductions  are  made.  A  barrel  of  water  holds  36 
gallons : — 

Water  per  Hour  L;ft  Gross  Power.  N*  Power  Effidency 

120  barrels, 165  feet,  4.7    H.P.  3.6  H.P.,  or  77  per  cent. 

160  barrels, 140    „  6.2      „  4.07        „      or  65.6      „ 

80      »       54    „  1.0      „  .785      „      or  78.5      „ 

25°       „       48     „  4-87     „  2.18       „      or  45         „ 

Average, 66.5      „ 

^Proceedings  of  the  Institution  of  Civil  Engineers,  vol.  ii.,  year  1843,  page  79. 


RECIPROCATING  PUMPS. 


945 


A  double-acting  pump,  constructed  by  M.  Farcot,  was  tested  by  M.  Tresca. 
It  contains  two  barrels  placed  vertically,  side  by  side,  and  worked  from  a 
shaft  overhead,  with  two  cranks,  driven  through  a  belt  pulley.  The  cranks 
coincide,  so  that  the  buckets  of  the  pumps  reciprocate  simultaneously.  The 
valves  of  the  first  bucket  open  downwards,  and  those  of  the  second  bucket 
upwards.  They  are  made  as  large  as  possible,  and  no  other  valves  are 
required.  The  water  enters  at  the  upper  part  of  the  first  barrel,  and  leaves 
at  the  upper  part  of  the  second  barrel,  passing  from  one  to  the  other  by  a 
connecting  chamber  at  the  bottom.  By  this  arrangement,  the  first  barrel 
forces  the  water,  and  the  second  barrel  draws  the  water,  and  a  continuous 
stream  is  set  up.  The  barrels  of  the  pump  were  18  inches  in  diameter, 
with  a  stroke  of  6  inches.  The  following  are  classified  results  of  performance, 
in  which  the  efficiency  is  expressed  in  parts  of  the  dynametric  power  expended 
in  driving  the  pump: — 


Speed  of  the 
Pumps. 

Total  Head. 

Efficiency. 

Ratio  of  Volume 
of  Water  to  Capa- 
city of  Pump. 

turns  per  minute. 

feet. 

per  cent. 

per  cent. 

33-oo 

I4.IO 

43-i 

96 

42.40 

14.10 

43-i 

97-2 

55.08 

I4.IO 

44-7 

92 

60.55 

16.63 

53-7 

94-5 

Averages,  .... 

14.73 

46.1 

23-75 

23.22 

637 



45.48 

24-93 

53-o 

957 

60.00 

27.32 

53-o 

95-4 

Averages,  .... 

25.16 

56.6 

39.62 

33-54 

66.7 

97.6 

43-75 

33-54 

69 

98.1 

40.50 

33-39 

61.2 

91.4 

55-oo 

35-55 

63-2 

95-4 

28.00 

35-55 

71.4 

91.2 

Averages,  — 

34-31 

66.2 

31.00 

42.80 

73-6 

93-9 

24.33 

45.62 

73-7 

89.8 

52.68 

45.62 

71.0 

95-3 

32-50 

46.28 

66.5 

91.7 

55.00 

46.97 

70.4 

94-8 

50.00 

49-33 

71.0 

95.8 

61.98 

51.00 

68.7 

90.5 

55.00 

75-44 

70.4 

92.5 

Averages,.... 

50.38 

70.7 

From  these  data,  it  is  evident  that  the  efficiency  increases  with  the  height 
of  the  lift, — a  result  which  is  explained  by  the  less  proportion  of  the  con- 
stant resistances  of  the  pumps  to  the  work  done  at  higher  lifts. 

60 


946  MACHINES   FOR  RAISING  WATER. 

CENTRIFUGAL  PUMPS. 

The  Appold  pump,  made  with  curved'  receding  blades,  is  the  form  of 
centrifugal  pump  most  widely  known  and  accepted.  M.  Morin  tested  three 
kinds  of  centrifugal  or  revolving  pumps: — ist,  on  the  model  of  the  Appold 
pump;  2d,  having  straight  receding  blades  inclined  at  an  angle  of  45°  with 
the  radius;  3d,  having  radial  blades.  They  were  12  inches  in  diameter  and 
3^5  inches  long,  and  had  6-inch  central  openings.  Their  efficiencies  were 
respectively  as  follows : — 

1.  Curved  blades,  Efficiency,  48  to  68  per  cent. 

2.  Inclined  blades, „          401043        „ 

3.  Radial  blades,  „  24  „ 

The  height  to  which  water  ascends  in  a  pipe,  by  the  action  of  a  centri- 
fugal pump,  would,  if  there  were  no  other  resistances,  be  that  due  to  the 

v2 
velocity  of  the  circumference  of  the  revolving  wheel,  or  to  — .     The  results 

2S 

of  experiments  made  by  the  author  on  two  pumps,  at  the  International 
Exhibition  of  1862,  yielded  the  following  data,  showing  the  height  to  which 
the  water  was  raised,  without  any  discharge : — 


Diameter  of  pump-wheel 

GWYNNE'S  PUMP 
(blades  partly  radial, 
curved  at  the  ends). 

A  feet 

APPOLD  PUMP 
(curved  blades). 

4  feet  7  inches. 

Revolutions  per  minute,  

if    IV^V^L, 

177 

QC    A 

Velocity  of  circumference,  per  second,... 
Head  due  to  the  velocity,  
Actual  head,  

37.05  feet, 

21-45     „ 
18.21 

yjfLT 

22.9      feet. 
8.194    „ 
5.833 

Do.     in  parts  of  head  due  to  velocity, 

?? 

85  per  cent. 

j      OO        ?? 

71.2  percent. 

Mr.  David  Thomson  made  similar  experiments  with  Appold  pumps  of 
from  1.25  to  1.71  feet  in  diameter,  the  results  of  which  showed  that  the 
.actual  head  was  about  90  per  cent,  of  the  head  due  to  the  velocity. 

M.  Tresca,  in  1861,  tested  two  centrifugal  pumps,  18  inches  in  diameter, 
with  a  9-inch  central  opening  at  each  side.  The  blades  were  six  in  number, 
of  which  three  sprung  from  the  centre,  where  they  were  ^  inch  thick;  the 
alternate  three  only  sprung  at  a  distance  equal  to  the  radius  of  the  opening 
from  the  centre.  They  were  radial,  except  at  the  ends,  where  they  were 
curved  backward,  to  a  radius  of  about  2^  inches;  and  they  joined  the  cir- 
cumference nearly  at  a  tangent.  The  width  of  the  blades  was  taper;  the 
blades  were  5^  inches  wide  at  the  nave,  and  only  2^5  inches  at  the 
ends:  so  designed  that  the  section  of  the  outflowing  water  should  be  nearly 
constant. 

M.  Tresca  deduced  from  his  experiments,  that,  in  making  from  '630  to 
700  revolutions  per  minute,  the  efficiency  of  the  pump,  or  the  actual  duty  in 
raising  water,  through  a  height  of  31.16  feet,  amounted  to  from  34  to  54 
per  cent,  of  the  work  applied  to  the  shaft;  or  that,  in  the  conditions  of  the 
experiment,  the  pump  could  raise  upwards  of  16,200  cubic  feet  of  water 
per  hour,  through  a  height  of  33  feet,  with  about  30  horse-power  applied  to 
the  shaft,  and  an  efficiency  of  45  per  cent. 

According  to  Mr.  Thomson,  the  maximum  duty  of  a  centrifugal  pump 
worked  by  a  steam-engine,  varies  from  55  per  cent,  for  smaller  pumps  to 


ENDLESS-CHAIN   PUMP.  947 

70  per  cent,  for  larger  pumps.  They  may  be  most  effectively  used  for  low 
or  for  moderately  high  lifts,  of  from  15  to  20  feet;  and,  in  such  conditions, 
they  are  as  efficient  as  any  pumps  that  can  be  made.  For  lifts  of  4  or  5  feet 
they  are  even  more  efficient  than  others.  At  the  same  time,  the  larger  the 
pump  the  higher  the  lift  it  may  work  against.  Thus,  an  1 8-inch  pump 
works  well  at  a  2o-feet  lift,  and  a  3-feet  pump  at  a  3o-feet  lift.  A  2i-inch 
fan  at  a  4o-feet  lift  has  not  given  good  results :  high  lifts  demand  very  high 
velocities.1 

The  efficiency  is  influenced  by  the  form  of  the  casing  of  the  pump.  The 
Hon.  R.  C.  Parsons  made  experiments  with  two  1 4-inch  fans  on  Appold's 
form  and  on  Rankine's  form.2  In  Rankine's  fan,  the  blades  are  curved 
backwards,  like  those  of  Appold's,  for  half  their  length;  and  curved  for- 
wards, reversely,  for  the  outer  half  of  their  length.  Plotting  the  results 
of  performance  arrived  at  by  Mr.  Parsons,  the  following  are  the  several 
amounts  of  work  done  per  pound  of  water  evaporated  from  the  boiler, 
reduced  for  a  speed  of  350  turns  per  minute: — 

Work  done  per  pound 
of  water  evaporated.  Ratio. 

foot  pounds. 

Appold  fan,  in  concentric  circular  casing,...  6,250,  as  i 

Do.,        in  spiral  casing, 9,000,  „  1.44 

Rankine  fan,  in  concentric  circular  casing,..  9,700,  „  1.55 

Do.,         in  spiral  casing, 1 2,500,  „  2 

These  data  prove  two  things: — ist,  that  the  spiral  casing  was  better  than 
the  concentric  casing;  2d,  that  Rankine's  ogee-fan  was  more  efficient  by 
one-half  than  Appold's  fan. 

ENDLESS-CHAIN  PUMP  OF  SQUARE  BOARDS. 

A  series  of  square  boards,  or  paddles,  5  or  6  inches  square,  strung 
together;  turning  on  two  centres;  inclined  at  an  angle  of  30°  or  40°  with 
the  horizon.  The  lower  end  is  immersed  in  the  water  to  be  raised,  and  the 
water  is  dragged  by  the  paddles  up  an  inclined  trough  to  the  higher  level. 
Efficiency,  40  per  cent. 

When  the  endless-chain  pump  works  vertically,  the  paddles  pass  through 
a  vertical  pipe,  and  this  arrangement  is  suitable  for  lifts  of  more  than  12  feet. 
Worked  by  from  4  to  8  men,  the  efficiency  of  the  vertical  endless-chain 
pump  is  65  per  cent.,  and  the  volume  of  water  raised  is  s/6 ths  of  the  volume 
described  by  the  paddles. 

NORIA. 

The  noria  is  an  endless  chain  of  buckets  having  a  capacity  of  from  i  ^ 
to  3  gallons,  working  vertically.  The  efficiency  increases  with  the  height, 
because  a  given  excess  of  elevation  above  the  higher  level  is  required  for 
accommodation  for  delivering  the  water.  For  lifts  of  from  3  to  12  feet, 

1  "  Pumps,"  page  156,  in  Mr.  Humber's  Water  Stipply  of  Cities  and  Towns. 

2  See  Mr.  Parsons'  paper  on  "  Centrifugal  Pumps,"  in  the  Proceedings  of  the  Institution 
of  Civil  Engineers  (i875~76)>  vol.  xlvii.,  page  267.     It  is  due  to  the  author  of  this  paper 
to  say  that  he  has  deduced  other  conclusions  than  those  drawn  in  the  text,  from  the 
comparative  results  of  the  experiments  with  Appold's  fan  and  Rankine's  fan. 


948  MACHINES   FOR   RAISING  WATER. 

the  efficiency  is  from  50  to  66  per  cent.     For  higher  lifts,  an  efficiency  of 
from  70  to  80  per  cent,  may  be  realized. 

WATER-WORKS  PUMPING  ENGINES. 

The  indicator-power  of  the  single-acting  beam-engines  at  the  East  London 
Water-works,  according  to  Mr.  Greaves'  experiments,  was  effective  in  actual 
"  pumping  duty,"  or  efficiency,  or  quantity  of  water  raised,  to  the  extent 
of  8 1  per  cent.  The  difference,  19  per  cent.,  was  absorbed  in  nearly  equal 
proportions  by  the  friction  of  the  engine,  including  that  of  the  pole,  on  the 
one  part,  and  the  friction  and  resistance  of  the  pump  on  the  other  part; 
being  each  nearly  10  per  cent,  of  the  "total  load,"  or  indicator-power.1 

Mr.  David  Thomson  gives  data  for  the  "pumping  duty"  of  the  double- 
acting  compound-cylinder  rotative  beam-engines  of  the  Chelsea  Water- 
works, in  which  the  cylinders  are  placed  vertically  side  by  side,  under  one 
end  of  the  beam,  acting  on  the  Woolf  system.  It  appears  that  the  duty  is 
80  per  cent,  of  the  indicator-power.2 

A  pair  of  compound  rotative  beam-engines,  designed  by  Mr.  E.  D.  Leavitt, 
jun.,  have  been  erected  at  the  St.  Lawrence  Water-works,  Mass.,  U.S.3  The 
first  and  second  cylinders  of  each  engine,  are  connected  one  to  each  end  of 
the  beam.  They  are  placed  together  under  the  main  centre,  and  conse- 
quently are  directed  obliquely  each  to  its  proper  end  of  the  beam;  and, 
whilst  the  lower  ends  are  close  together,  the  upper  ends  lie  apart  from  each 
other.  The  connecting-rod  to  the  fly-wheel  shaft  is  connected  to  the  first- 
cylinder  end  of  the  beam;  and  the  rod  to  the  pump,  to  the  second-cylinder 
end.  The  cylinders  are  fitted  with  gridiron  valves,  having  a  large  area  of 
opening,  and  small  movement.  The  pumps  are  of  the  bucket-and-plunger 
type,  first  introduced  by  Mr.  David  Thomson,  in  England.  The  effective 
pressure  in  the  boiler  is  90  Ibs.  per  square  inch.  The  cylinders  are  18  inches 
and  38  inches  in  diameter,  with  8  feet  of  stroke.  They  are  entirely  steam- 
jacketted.  The  clearance  averages,  for  the  first  cylinder,  2.43  per  cent., 
and  for  the  second  cylinder,  1.68  per  cent,  of  the  capacities  of  the  cylinders 
respectively.  The  volume  of  the  connecting  pipe  between  their  upper  ends 
is  9.92  per  cent,  of  that  of  the  cylinder.  The  pump-barrel  is  26^  inches, 
and  the  plunger  is  18  inches,  in  diameter,  with  a  stroke  of  8  feet.  The  fly- 
wheel is  30  feet  in  diameter,  and  weighs  16  tons.  From  the  results  of  the 
trials,  it  appears  that  the  steam  was  cut  off  at  about  30  per  cent,  of  the 
stroke,  with  an  initial  absolute  pressure  of  about  100  Ibs.  per  square  inch. 
The  engines  made  16%  turns  per  minute,  and  yielded  about  195.5  mdi- 
cator  horse-power.  The  water  delivered  per  stroke  amounted  to  95  per 
cent,  of  the  capacity  of  the  pump.  The  total  quantity  of  coal  (Cumberland) 
consumed  was  equivalent  to  1.69  Ibs.  per  indicator  horse-power  per  hour; 
and  to  2.06  Ibs.  per  horse-power  of  duty  at  the  pump.  The  efficiency  of 
the  engine  was  81.94  per  cent.  The  duty  per  100  Ibs.  of  coal  amounted  to 
96,200,000  foot-pounds.  The  quantity  of  water  evaporated  was  8.27  Ibs. 
per  pound  of  coal,  and  (8.27  x  1.69  =  )  14  Ibs.  of  steam  was  consumed  per 
indicator  horse-power. 

1  Article,  "Steam  Engine,"  by  the  author,  in  the  Encyclopedia  Britannica,  8th  edition. 

2  "On  Double-Cylinder  Pumping  Engines,"  by  Mr.  David  Thomson,  in  the  Proceedings 
of  the  Institution  of  Mechanical  Engineers,  1862,  page  268. 

z  Journal  of  the  Franklin  Institute,  November,  1876,  page  312.  Report  by  Messrs. 
W.  E.  Worthen,  J.  C.  Hoadley,  and  J.  P.  Davis;  with  illustrations. 


HYDRAULIC  RAMS. 


949 


The  pumping  duties,  at  the  pump,  of  large  pumping  engines,  whether 
single-acting  or  double-acting,  are  thus  seen  to  average  about  81  per  cent, 
of  the  indicator  horse-power. 

HYDRAULIC  RAMS. 

Hydraulic  rams  are  used  where  there  is  a  considerable  flow  of  water  with 
a  moderate  fall,  to  raise  a  small  portion  of  the  flow  to  a  height  greater  than 
the  fall.  The  outflow  of  water  falling  through  a  pipe,  when  the  lower  end 
of  the  pipe  is  suddenly  closed,  is  suddenly  arrested,  and  the  momentum  of 
the  current  in  the  pipe  is  expended  in  forcing  a  portion  of  itself  through 
another  pipe, — the  delivery-pipe, — into  an  elevated  reservoir.  When  the 
momentum  is  expended,  the  upward  current  ceases  to  flow,  the  valves  in 
the  delivery-pipe  are  closed  against  the  return  of  the  elevated  water;  and 
the  valve  by  which  the  supply  current  was  arrested,  opens  as  it  is  relieved 
of  the  momentary  excessive  pressure,  and  the  outflow  is  resumed.  Thus,  by 
a  succession  of  impulses,  the  water  is  lifted.  Mr.  C.  L.  Hett  recommends 
Daubuisson's  formula  for  the  efficiency : — 


.     (12) 


d=the  quantity  of  water  used,  in  gallons  per  minute. 
//!  =  the  quantity  of  water  raised,  in  gallons  per  minute. 

h  =  the  head  used,  in  feet. 
ht  =  the  lift,  in  feet. 

Mr.  Hett1  gives  the  following  table,  calculated  by  means  of  Daubuisson's 
formula,  showing  the  efficiency  or  percentage  of  duty  due  to  proportions  of 
lift  to  fall,  of  from  4  to  26 : — 


Lift,  when  the 
Fall  =  i. 

Efficiency. 

Lift,  when  the 
Fall  =  i. 

Efficiency. 

Lift,  when  the 
Fall  =  i. 

Efficiency. 

ratio. 

per  cent. 

ratio. 

per  cent. 

ratio. 

per  cent. 

4 

86 

12 

45 

20 

17 

5 

79 

13 

4i 

21 

H 

6 

73 

H 

37 

22 

II 

7 

68 

15 

34 

23 

8 

8 

63 

16 

30 

24 

5 

9 

58 

17 

27 

25 

2 

10 

53 

18 

23 

26 

0 

ii 

49 

19 

20 

Mr.  Hett  recommends,  to  adopt,  allowing  for  contingencies,  only  five- 
sixths  of  the  efficiencies  here  given.  For  the  diameter  of  the  driving  pipe, 
he  gives  Eytelwein's  formula  as  sufficient  for  all  practical  purposes : — 

Diameter  of  pipe  in  inches  =  .058  \f  quantity  of  water  in  gallons.  ...  (  13  ) 
1  The  Engineer •,  January  7,  1876,  page  3. 


950  HYDRAULIC  MOTORS. 

HYDRAULIC    MOTORS. 

HYDRAULIC  PRESS. 

The  action  of  the  hydraulic  press  depends  on  the  principle  that  fluids 
press  equally  in  all  directions ;  and  if  the  pressure  applied  to  the  plunger  of 
the  force-pump  be  multiplied  in  the  ratio  of  the  sectional  areas  or  of  the 
squares  of  the  diameters  of  the  plunger  and  the  ram,  the  product  is  the 
pressure  applied  to  the  ram. 

The  ram  is  packed  with  a  leather  collar,  and  according  to  the  results  of 
experiments  made  by  Mr.  John  Hick,  M.P.,1  the  friction  of  the  collar 
increases  directly  with  the  pressure  and  with  the  diameter;  and  it  is 
independent  of  the  depth  of  the  collar.  The  friction  is  equivalent  to  i  per 
cent,  of  the  pressure  for  a  4-inch  ram,  yz  per  cent,  for  an  8-inch  ram,  and 
%  per  cent,  for  a  1 6-inch  ram.  The  following  formula  is  deduced: — 

Leather,  new  or  badly  lubricated, . . .  f—  .047 1  dp,  (I4) 

Leather  in  good  condition, f=  .0314^/5   (  15  ) 

f  —  the  total  frictional  resistance  of  a  leather  collar. 

d  =  the  diameter  of  the  ram,  in  inches. 

p  =  the  pressure,  in  pounds  per  square  inch. 

ARMSTRONG'S  HYDRAULIC  MACHINES. 

In  a  paper  by  Mr.  Henry  Robinson,2  the  following  data  are  communi- 
cated, on  the  authority  of  Mr.  Percy  Westmacott,  giving  the  coefficients  of 
effect  obtained  in  hydraulic  machines  of  ordinary  make, —  ;. 

Direct  acting, 93  per  cent. 

2  to  r, 80        „ 

4  to  i  76 


6  to 

8  to 

10  to 

12  tO 

14  to 


•72 

.67     „ 
.63     „ 

•59 


54 

16  to  i, 50        „ 

These  coefficients  are  based  on  the  use  of  ordinary  hemp-packing,  and 
with  sheaves  and  wrought-iron  pins,  and  with  no  exceptional  arrangements 
for  lubrication.  But,  where  special  precautions  have  been  taken,  with 
large  sheaves  and  small  hard  steel  pins,  the  efficiency,  multiplying  20  to  i, 
with  a  load  of  17^  tons,  was  as  high  as  66  per  cent.  Well  made  cupped 
leathers,  used  instead  of  hemp-packing,  increase  the  efficiency. 

It  is  considered  that  the  coefficient  of  effect  obtained  from  a  steam- 
engine  pumping  into  an  accumulator,  may  be  taken  at  91.7  per  cent.,  the 
loss  by  friction  amounting  to  8.3  per  cent.  It  is  found  by  experiment  that 
the  difference  of  pressure  with  the  accumulator  (at  700  Ibs.)  rising  or  falling 
is  about  30  Ibs.,  representing  .022  of  effect.  The  compounded  efficiency 
will  therefore  be  ascertained  by  combining  the  efficiency  of  the  engines 
with  the  above  varying  rates  of  efficiency. 

1  Spans1  Dictionary  of  Engineering,  page  1992. 

2  "  On  the  Transmission  of  Motive-power  to  Distant  Points,"  in  the  Proceedings  of  the 
Institution  of  Civil  Engineers,  1876-77,  vol.  xlix. 


FRICTIONAL   RESISTANCES. 


INTERNAL   RESISTANCE   OF    STEAM-ENGINES. 

It  may  be  assumed  that,  taken  generally,  the  efficiency  of  steam-engines, 
that  is  to  say,  the  ratio  of  the  work  transmitted  through  the  engine  to  the 
fly-wheel  shaft,  to  the  effective  work  done  by  the  steam  in  the  cylinder,  in- 
creases with  the  power  of  the  engine.  When  engines  are  in  good  working 
order,  and  are  doing  their  full  duty,  the  efficiency  varies  from  80  per  cent, 
for  smaller  engines,  to  90  per  cent,  for  larger  engines.  In  one  exceptional 
instance, — a  Corliss  engine  in  France, — the  efficiency  amounted  to  93 
per  cent. 


RESISTANCE   OF   TOOLS.1 

The  following  are  results  of  Dr.  Hartig's  experiments  on  the  resistance  of 
tools  :  — 

Single-acting  Shearing  Machines.  —  The  power  necessary  to  drive  such 
tools  when  empty  is  expressed  by  the  formula,  — 


P  =  horse-power. 

t  =  maximum  thickness  of  plate  to  be  cut. 
n  =  the  number  of  cuts  per  minute. 

a  =  the  area  of  surface  cut  or  punched  per  hour,  in  square  inches. 
F  =  (ii66+  1691  /),  a  factor  expressing  the  work  required  to  produce  a  cut 
or  sheared  surface  of  I  square  inch. 

The  power  required  to  do  the  work  itself,  in  addition  to  that  required  to 
drive  the  tool  when  empty,  is 


c 

33,000  x  60     1,980,000 

For  example,  a  shearing  machine,  cutting  4648  square  inches  of  surface  per 
hour,  in  plates  0.4  inch  thick,  would  absorb  0.68  horse-power  empty,  and 
4.3  horse-power  in  effective  work;  total,  say,  5  horse-power. 

1  The  above  particulars  of  the  resistance  of  tools  are  abstracted  from  a  notice  of  the 
results  of  Dr.  Hartig's  experiments,  vs\  Engineering  for  October  and  December,  1874. 


952  FRICTIONAL   RESISTANCES. 

Plate-bending  Machines. — The  net  work  required  to  bend  a  plate  or  a  bar, 
is  expressed  by  the  formula, — 

F  =  85»ooo  J*8^  for  cold  wrought-iron  plates  (  3  ) 

F  -  ££i3°^_l^l/5  for  red-hot  iron  plates (4) 

fr,  /,  and  /  =  the  breadth,  thickness,  and  length  of  the  plate,  in  inches, 
r=the  radius  of  curvature,  in  inches. 
F  =  the  net  work  of  bending  the  plate. 

The  power  required  to  drive  large  plate-bending  rolls,  when  empty,  is 
between  0.5  and  0.6  horse-power. 

Circular  Saws. — The  horse-power  required  to  drive  circular  saws  running 
empty,  is 

*'-<££ (5) 

32,000 

d  =  the  diameter  of  the  saw,  in  inches. 
n  =  the  number  of  revolutions  per  minute. 

The  net  power  required  to  cut  with  a  circular  saw,  is  proportional  to  the 
cubic  contents  of  the  material  removed.  For  a  saw  for  cutting  hot  iron, 
moving  at  a  circumferential  speed  of  7875  feet  per  minute,  and  making  a 
cut  0.14  inch  wide,  the  power  is  expressed  by  the  formulas — 

Pc  =  0.702  A,  for  red-hot  iron  (  6  ) 

Pc  =  1.013  A,  for  red-hot  steel  (  7  ) 

A  =  the  sectional  area  of  surface  cut  through,  in  square  feet. 

Work  of  Ordinary  Cutting  Tools,  in  Metal. — Materials  of  a  brittle  nature, 
as  cast-iron,  are  reduced  most  economically  in  power  consumed,  by  heavy 
cuts ;  whilst  materials  which  yield  tough  curling  shavings  are  more  economi- 
cally reduced  by  thinner  cuttings.  The  following  formulas  apply  to  light 
cutting  work : — 

The  power  required  to  plane  away  cast-iron  is, — 

Planing  cast-iron. P  =  W  (.01515+ — )  ..    ..  (8) 

v      o:>     1 1, 000^ 

W  =  the  weight  of  cast-iron  removed  per  hour,  in  pounds. 
s  =•  the  average  sectional  area  of  the  shavings,  in  square  inches. 

For  planing  steel,  wrought-iron,  and  gun-metal,  with  cuts  of  an  average 
character, — 

Planing  steel, P  =  o.ii2\Y, (9) 

„       wrought-iron, P=    .052  W, (  10  ) 

„       gun-metal,  P=    .0127  W, (  n  ) 

For  turning  off  metals,  the  power  required  is  less   than  for   planing 


RESISTANCE  OF  TOOLS.  953 

them  off,  and  it  was  found  that  the  power  was  greater  for  smaller  diameters 
than  for  larger  diameters. 

Turning  cast-iron,  .........   P  =  .o3i4W,  .........   (12) 

„        wrought-iron,.  .  .  .   P  =  .o32yW,  .........   (13) 

„        steel,  ..............  P  =  .o47W,  ...........   (14) 

For  drilling  metals,  the  power  required  to  remove  a  given  weight  of 
material  is  greater  than  in  planing.  Volume  is  taken  instead  of  weight 
in  the  formulas:  applicable  to  holes  of  from  0.4  to  2  inches  in  diameter:  — 


Drilling  cast-iron,  dry,  ..............  P  =  <?  (.Oi68  +  '-)  ...   (  15) 

„       wrought-iron,  with  oil,...  P  =  $  (.01  68  +  1^-9)  ....  (  16  ) 

q  -  the  volume  removed,  in  cubic  inches  per  hour. 
d-=  the  diameter  of  the  hole. 

In  the  use  of  shaping  machines,  the  resistance  is  greater  for  notch-cuts 
than  for  cuts  on  flat  surfaces,  and  for  skin-cuts  than  for  under-cuts  :  — 

Shaping  cast-iron  skin-cuts  .....................  P  =  .  1087  W,  .  .  .  (17 

„  „        under-cuts,  ..................  P  =  .o6o4  W,...  (  18 

„  „        wheel-teeth  (notch-cuts),.  P  =  .n8W,  .....  (19 

Power  required  to  drive  Ordinary  Cutting  Tools  when  empty.  —  For  lathes, 
the  power  varies  with  the  number  of  shafts  between  the  driving  shaft  and 
the  main  spindle  :  — 


Light  Lathes,  Empty.  Heavy  Lathes, 

o, ..   P  =  .o5  +.0005  n P  =  o.25+    .0031  ;z....  (  20 

i  or  2, P  =  .o5  +.0012  n P  — 0.25+   .053  n  ....  (  21 

3  or  4, P  =  . 05  +.05/2 P  =  0.25 +0.18/2     (22 

n  =  the  number  of  turns  of  the  lathe-spindle  per  minute. 


For  drilling  machines,  when  empty,  the  power  varies  according  to  the 
construction  of  the  machines,  with  or  without  intermediate  gearing : — 


a.  Drill,  without  gearing, P  =  .0006  n^  +  .0005  n2.. 

b.  „     with  gearing  for  the  spindle,  P  =  .0006 «,  +.001  n2... 

c.  „     radial  drills,  without  gearing,  P  =  .0006  nx  +  .004  /z2 . . . 

d.  „  ,,          with  gearing, P  =  .04  +  .0006^  +  .  004  «2  .. 

ns  =  the  number  of  turns  per  minute  of  the  gearing  shaft, 
/z,  =  the  number  of  turns  of  the  drill. 


23) 

24) 

25) 
26) 


For  a  slotting  machine,  having  a  stroke  of  8  inches,  and  a  tool-holder  and 
:slide  weighing  93^  Ibs.,  running  empty,  — 

(27) 


4OOO 


n  =  the  number  of  strokes  per  minute. 
s  =  the  stroke  in  inches. 


954  FRICTIONAL   RESISTANCES. 

For  shaping  machines,  the  movements  of  which  are  slow,  the  power 
required  for  moving  the  machine,  when  empty,  is  only  from  10  to  15  per 
cent,  of  the  whole  power  required  to  work  the  machine. 

Screw-cutting  Machines.  —  The  power  required  by  one  of  Sellers'  tools  for 
cutting  screws  on  shafts  or  on  bolts,  and  for  tapping  nuts,  is  expressed  by 
the  formulas  :  — 


Screwing  (network),..  ..........  p  =     __  .....................  (28) 

64 

7  //3 

Tapping         „          .........  ,.  p  =  -—   .......................  (  29  ) 

d  —  the  diameter  in  inches. 

/=  the  length  in  feet  cut  per  hour. 

The  additional  power  required  for  driving  a  screwing  machine  of  medium 
size,  is  about  one-fifth  of  a  horse-power. 

Wood-cutting  Machines.  —  The  power  required  for  driving  circular-saws, 
when  empty,  has  been  given  by  formula  (  5  ),  page  952.  The  net,  or 
additional,  power  required  to  do  the  work,  is  proportional  to  the  cubic  con- 
tents of  the  wood  reduced  to  sawdust,  at  the  rate  of  i  horse-power  for 
i  cubic  foot  per  hour  of  soft  wood,  or  for  half  a  cubic  foot  of  hard  wood. 

A     - 

Circular-saw,  hard  wood,  .........  Pc  =  ——  ...................  (  3°  ) 

„         softwood,  ..........  P,  =  ^  ...................  (31) 

A  =  the  sectional  area  of  surface  in  square  feet  cut  through  per  hour. 
c  =  the  width  of  the  cut,  in  inches. 

For  saw-frames,  cutting  dry  pine  timber,  the  net  power  required  to  do 
the  work,  exclusive  of  the  work  to  drive  the  machine  when  empty,  is  as 
follows  :  — 

Saw-frame,  pine,  ......  Pc  =  .  00428  +  .0065-—  ...............  (  32  ) 

S  =  the  stroke  of  the  saws,  in  feet. 

c  =  the  width  of  the  cuts,  in  inches. 
f  =  the  feed  per  cut,  in  inches. 
P,.  =  the  horse-power  required  per  square  foot  cut  through  per  hour. 

For  band-saws,  the  power  required  for  the  work  itself  is  :  — 

Band-saw,  pine,  .......  Pc=.oo34   +°^~  ..............  (33) 

-*  ........  P--4S3  +  ..............  (34) 

..............  (35) 


v  =  the  velocity  of  the  band-saw,  in  feet  per  minute. 
f=  the  rate  of  feed,  in  feet  per  minute. 
c  =  the  width  of  the  cut,  in  inches. 


RESISTANCE  OF  WOOD-CUTTING  MACHINES.  955. 

In  planing  and  moulding  machines,  which  are  driven  at  high  speeds,  a 
large  proportion  of  the  total  power  is  absorbed  in  driving  the  machine  itself. 
The  formula  is,  — 


N  =  the  sum  of  the  turns  per  minute  made  by  the  several  shafts. 

The  net  power  required  for  moulding  and  shaping  wood,  is  given  by  the 
formulas  :  — 


Pine,  .................................  P  =  .o566    +  .....  (37) 

Red  beech,  using  cutters,  .........  p  =  .  08895  +  '-^p-  .....  (38) 

„         using  cutting  discs,..  P  =  .0895    +9.138  s  ....  (  39  ) 

P  =  horse-power  required  to  produce  i  cubic  foot  of  shavings  per  hour. 
h  =  the  height  of  wood  cut  down  to  form  the  moulding,  in  inches. 
.$•  =  the  average  sectional  area  of  the  shavings,  in  square  inches. 

In  drilling  timber  with  holes  of  from  0.4  to  4  inches  in  diameter,  for 
depths  up  to  6  inches  :  — 


Drilling  pine,  ............  P  =  ?  (.000125  +  '---)  ...  (40) 

„         alder,  ...........  P  =  ?  (.000472  +  -^^)  ...  (41  ) 

white  beech,...  P  =  ?  (.003442  +  '°0^495)  ...  (  42  ) 

Grindstones,  —  To  drive  grindstones  empty,  the  power  is  expressed  by  the 
formulas  :  — 

Large  grindstones  empty,  ...  P  =  .0000409  dv  .........  (  43  ) 

or  P=  .000128  d*n  .........  (44) 

Small  fine    „  „      ...  P  =  0.16  +  .  0000895  dv  .........  (45) 

or  P  =  0.16  +  .  00028  d*n  ..........  (  46  ) 

The   coefficients  of  friction   between   grindstones   and   metals   are   as 
follows  :  — 

Coarse  Grindstones,  Fine  Grindstones, 

at  high  speeds.  at  low  speeds. 

0.72 
i.oo 
0.94 

(  47  ) 

p  =  the  pressure  between  the  material  and  the  stone. 

^  =  the  circumferential  velocity  of  the  stone,  in  feet  per  minute. 

K  =  the  coefficient  of  friction. 


For  wrought  iron  .     ... 

.AA 

For  steel 

2Q 

Grindstones  net  work 

5> 
P 

_/Kz> 

33,000 

•956 


FRICTIONAL  RESISTANCES. 


RESISTANCE   OF   COLLIERY  WINDING   ENGINES. 

For  working  shafts  from  246  to  580  yards  deep,  in  the  county  of  Durham, 
the  following  are  particulars  of  dimensions  and  performance  of  engines  and 
winding  gear:1 — 

DIMENSIONS. 


No. 

DIRECT-ACTING  ENGINE. 

Cylinder. 

Speed 
of 
Piston. 

Steam 
in 
Boiler. 

Drums. 

Mean 
Dia- 
meter. 

in.       in. 

ft.  per 

minute. 

Ibs. 

feet. 

I 

i  cyl.,  vertical,  condensing 

65x84 

I76 

19 

flat 

26 

2 

I       11                      11                            11 

68x84 

224 

20 

2S% 

3 

2    „    horizontal,  non-condensing 

40x72 

253 

— 

conical     21 

4 

2      11                      11                            11 

34x72 

291 

40 

17;^ 

5 

I       11                      11                            11 

48x72 

232 

— 

flat 

21$ 

PERFORMANCE. 


No. 

Gross  Engine 
Power  per 
Minute. 

Duty,  Coal 
Raised,  per 
Minute. 

Effi- 
ciency. 

Ropes. 

Average 
Speed  of 
Ropes. 

foot-pounds. 

foot-pounds. 

per 
cent. 

in.          in. 

feet  per 
minute. 

I 

10,342,350 

7,729,344 

75 

flat,  iron,  6l/z  x  7/& 

IO2O 

2 

15,403,000 

9,744,000 

63 

11        11     6      x  # 

1180 

3 

7,470,000 

4,077,000 

55 

round,  steel,     1% 

1689 

4 

4,375>596 

2,700,000 

62 

round,  iron,      i^ 

1302 

5 

5,281,107 

3,956,178 

75 

•  — 

1300 

RESISTANCE   OF   WAGGONS   IN   COAL   PITS. 

The  average  resistance  of  waggons  used  in  the  Midland  coal  pits,  having 
four  1 5-inch  wheels  at  2i-inch  centres,  is  Vsoth  of  the  weight,  or  45  Ibs.  per 
ton.  The  bodies  of  the  waggons  are  4  feet  3  inches  by  3  feet  wide,  and 
20  inches  deep.  On  a  roadway  having  an  average  fall  of  i  in  30,  worked 
by  an  endless  chain,  passed  over  a  7^-feet  grooved  pulley  at  the  end 
of  the  course,  the  gravitation  of  full  waggons  descending,  supplies  sufficient 
hauling  power  to  overcome  all  the  wheel-friction  and  take  up  the  empty 
waggons.  The  greatest  traverse  is  from  5000  to  6000  feet;  the  speed  is 
3  miles  per  hour,  and  the  tubs  or  waggons  are  attached  to  the  rope  at  intervals 
•of  from  20  to  25  yards.2 

1  See  paper  by  Mr.  G.  H.   Daglish,  on  Winding  Engines,  in  the  Proceedings  of  the 
Institution  of  Mechanical  Engineers,  1875,  page  217. 

2  Mr.  G.  Fowler,  Proceedings  of  the  Institution  of  Mechanical  Engineers,  1870. 


RESISTANCE   OF   FLAX-MILL   MACHINERY. 


957 


RESISTANCE  OF  MACHINERY  OF  FLAX  MILLS. 

M.  E.  Cornut,  in  1871-72,  tested,  by  the  indicator,  the  resistance  of  the 
steam-engines,  shafting,  and  machinery  of  the  flax  mill  at  Hamegicourt.1 
There  were  two  Woolf  engines,  with  vertical  cylinders  and  beam,  condens- 
ing, which  made,  at  regular  speed,  25  turns  per  minute. 

First  cylinder  of  each  engine, 12.9  inches  diameter,  44.3  inches  stroke. 


Second 


22.0 


59.8 


The  power  required  to  drive  the  machinery  was  very  variable.  It  varied 
15  or  20  per  cent,  according  to  the  lubricant  employed,  and  the  mode  of 
lubrication : — 

Atmospheres. 

With  vegetable  oil  and  hand  oiling,  steam  pressure  required,  5  to  5X- 
With  mineral  oil  and  continuous  oiling,     „  „          4^  maximum. 

Making  a  difference  in  pressure  of  at  least  15  per  cent.  In  accordance 
with  this  result,  it  was  found,  by  direct  test,  that,  in  lubricating  with  vege- 
table oil  (huile  grasse),  2.90  horse-power  per  100  spindles  (wet  system),  was 
consumed,  and  only  2.44  horse-power  with  mineral  oil,  making  a  difference 
of  1 6  per  cent.  But  the  modes  of  lubrication  are  not  distinguished.  Finally, 
with  vegetable  oil,  the  belts,  if  tight,  were  broken  at  starting  on  Monday 
mornings,  after  a  day's  stoppage;  but,  with  mineral  oil,  there  were  no 
breakages. 

The  quantity  of  mineral  oil  consumed  per  day,  in  lubricating  the  same 
three  pedestals,  was, — 

By  hand  oiling, 29.0  grains. 

By  continuous  oiling, 16.2      „ 

Showing  a  reduction  of  44  per  cent,  by  continuous  oiling. 

To  test  the  increase  of  resistance  by  want  of  lubrication,  the  engines,, 
shafting,  and  belts,  were,  one  day  in  March,  1871,  started  empty  (a  vide), 
at  4  A.M.  At  6  A.M.,  30.08  horse-power  was  indicated.  All  the  lubricators 
were  then  removed,  except  two  next  the  engines,  and  the  oil  that  remained 
was  cleared  off  the  journals.  The  engines  and  shafting  continued  in 
motion,  and  observations  on  the  power  expended  were  made  at  intervals. 


Time 

Observation. 

Elapsed 
after  Remov- 
ing the 

Indicator  Horse- 
Power. 

Augmentation  of 
Resistance. 

Lubricators. 

hours. 

per  cent. 

ISt 

0 

30.08 

— 

2d 

2 

31.60 

$X 

3d 

3 

3347 

10.9 

4th 

4 

35-34 

1745 

5th 

6 

37-33 

30.78 

1  Essais  Dynamometriques,  Lille,  1873.     These  experiments  were  carefully  and  intelli- 
gently conducted,  and  M.  Cornut's  conclusions  appear  to  be  worthy  of  confidence. 


958  FRICTIONAL  RESISTANCES. 

At  this  time,  the  journals  began  to  heat,  and  the  experiment  was  ended. 
Taking  the  first  three  observations  to  represent  ordinary  practice  in 
hand-oiling,  a  variation  is  manifested,  of  from  5  to  10  per  cent,  of  re- 
sistance. 

From  comparative  tests  made  at  6  A.M.  and  at  n  A.M.,  respectively 
^  hour  and  5  y2  hours  after  starting  in  the  morning,  the  total  resistance 
of  the  engines  and  machinery  in  ordinary  work,  for  the  second  test,  was 
from  8  to  9  per  cent,  less  than  for  the  first  test.  The  temperature  in  the 
workshops  had  risen  10°  or  11°  F.  in  the  interval. 

The  resistance  of  the  engines  and  machines,  separately,  was  tested  during 
the  period  from  August,  1871,  to  February,  1872: — 

The  engines,  shafting,  and  belts  only,  making  25  turns  of  the  engine  per 
minute,  expended  an  average  of  30.41  indicator  horse-power.  The  maxi- 
mum power  was  32.10  H.P.,  in  December,  1871. 

Four  cards  consumed  from  9.10  to  7.40 ; — average  8.42  H.P. 

14  drawing-frames  of  29  heads,  or  156  slivers,  consumed  from  6.82  to 
8.04; — average  7.19  H.P. 

6  roving-frames,  of  330  spindles,  9.04  to  8.25; — average  8.67  H.P.,  or 
2.627  H.P.  per  100  spindles. 

4  combing-machines,  2.228  H.P. 

20  spinning-frames,  dry,  1480  spindles,  47.50  H.P. 

20  spinning-frames,  wet,  2080  spindles;  spinning  Nos.  25  to  30,  46.59 
H.P.,  or  2.24  H.P.  per  100  spindles;  Nos.  40  to  70,  35.82  H.P.,  or  1.72 
H.P.  per  100  spindles.  For  several  particular  numbers,  the  power  was 
determined  to  be  as  follows : — 

No.  16, 3.200  H.P.  per  100  spindles. 

„      20, 2.760  „  „ 

„  -25, 2.262  „  „ 

»   28, 2.190  „  „ 

„   30, 2.140  „  „ 

„  40, 1.917  „ 

M.  Cornut  deduced  from  those  data,  that  the  horse-power  per  100  spindles 
varied  inversely  as  the  square  root  of  the  number. 

The  average  indicator-power  that  would  have  been  required  for  driving 
the  whole  of  the  machinery  in  full  work,  was  148.42  horse-power.  The 
actual  total  average  power  required  was  only  131.23  horse-power,  or  88  per 
cent,  of  the  power  for  full  work;  being  17.19  horse-power  less  than  for  the 
whole  of  the  machines.  This  reduction  represents  the  power  for  the 
proportion  of  machines  at  rest. 

The  power  required  to  drive  the  machines  when  empty  was  also  mea- 
sured. 

The  table  No.  318  contains  particulars  of  the  horse-power  required  for 
each  machine,  at  work,  and  empty.  The  indicator-power  may  be  divided 
thus  :— 

Steam-engine,  shafting,  and  belts, 30  I. H.P.  or  20  per  cent. 

Preparing  and  spinning  machinery,  &c., 120      „       or  80       „ 

150      „          100       „ 


RESISTANCE   OF   WOOLLEN-MILL  MACHINERY. 


959 


Table  No.  318. — FLAX  MILL  AT  HAME'GICOURT — HORSE-POWER  REQUIRED 
TO  DRIVE  THE  ENGINES,  SHAFTING,  AND  MACHINERY. 

(From  M.  Cornut's  data.) 


Indicator  Horse-Power. 

Efficiency 

DESCRIPTION. 

Total  at 
Work. 

For  One  Machine  at 
Work. 

For  One 
Machine 
Empty. 

of  the 
Machines. 

Engines,  shafting,  and  belts,.. 
A  Cards  . 

H.-P. 
30.41 

8  42 

H.-P. 
2  IOC 

H.-P. 
T  A^.'i. 

per  cent. 
M 

1  4  Drawing  frames  (29  heads  ) 
or  156  slivers),  j 

7.19 

.0934  per  sliver 

i«4Zj 
.0794 

15 

4  Combing  machines 

2  22 

C  C  C 

T  C  T 

7C 

6  Roving  frames  (330  spindles) 
20  Spinning    frames,    dry 
(1480  spindles),  

778 

47.50 

O33 

2.627  ?•  i  oo  spindles 

3-21                  „ 

.151 

2.434 
2.515 

/  j 
7-3 

21.6 

20  Spinning    frames,    wet 
(2080  spindles) 

46.59 

2.24             „ 

1.613 

19 

Total    horse-power,   all    at  } 
work,     calculated      from  j- 
separate  experiments,  ) 
Total  hor.-power  that  would  } 
have    been    actually  ex-  > 
pended,  all  at  work  ) 

151.00 

148.42 

Estimation  of  Horse-power  required  for  a  Flax  Spinning-mill. — Let  the 
power  be  expressed  in  terms  of  the  number  of  spindles  driven  by  i  indi- 
cator horse-power.  M.  Cornut  gives  the  following  data : — 

M.  Feray  d'Essone, 55  spindles  per  H.P.  for  No.  40. 

20        „  „          for  No.  6. 

English  estimate          \    25    spindles  per    H.P.,  for   No.    35   long  staple, 

No.  1 8  tow. 
40  spindles  per  H.P.,  for  Nos.  35  to  51  long  staple, 

Nos.  1 6  to  29  tow. 

26  spindles  per  H.P.,  for  Nos.  25  to  30  long  staple, 
Nos.  14  to  23  tow. 


(2000  to  1 2,000  spindles), 
Dr.  Hartig, 

Do 

M.  Cornut: — 


2080  spindles,  wet, 34.4  spindles  per  H.P.,  long  fibre. 

640        „         dry, 20.1        „  „  „ 

840        „  „    14-5        »  »  tow. 


3560 


23-7 


RESISTANCE  OF  MACHINERY  OF  WOOLLEN  MILLS. 

Dr.  Hartig,  of  Dresden,  tested  the  machinery  of  a  woollen  mill,  by  means 
of  a  dynamometer,  which  was  applied  directly  to  each  machine.  The  prin- 
cipal results,  showing  the  actual  horse-power  expended  in  driving  each 
machine,  when  empty  and  when  at  work,  are  given  in  table  No.  319. 


960 


FRICTIONAL   RESISTANCES. 


Table  No.  319. — WOOLLEN  MACHINERY — HORSE-POWER  REQUIRED 

TO  DRIVE  THE  MACHINES. 

(From  Dr.  Hartig's  data. ) 


DESCRIPTION. 

Ordinary 
Speed  in 
Turns  per 
Minute. 

Indicator  Horse-power. 

Empty. 

At  Work. 

Effi- 
ciency. 

At  Work, 
Resistance 
of  Shafting 
included.   ' 

H.P.        | 

1.  00 

1-75 
.70 
I.OO 

1  75 

.003 
.007 

•075 

•13 

•55 
2.25 
1.50 
2.75 

3-25 
2-75 

1.70 

•75 

•75 
2-75 

•45 
.90 
.60 

•25 
.40 
.90 

Washing  machine,  two  cylinders,  .... 
Centrifugal  pump  for  this  machine,  ... 
Hydro-extractor,  

turns. 

35 
300 

IOOO  to  I2OO 

350  to  500 

300 

350  to  450 
no 
2500 

I  IOO 

17 

40  to  45 
40 

IOO 

45 
no 

IOO 

45 

blows. 
125 

116 

turns. 
90 
IOO 

IOO 
IOO 

650 

IOOO 

250 

250 

H.P. 

•75 
1.47 

•47 
.42 

•34 

.27 

.005 

•13 

.19 

•17 

•30 
.16 

•53 
•43 
.19 

.20 
.11 

•17 

•52 
•25 

•37 

H.P. 
.223 

•77 
i.  20 

'is 

•95 
•58 
•43 

.0027 
.006 

.071 

•50 

2-54 
i-59 
2-74 
3-40 
3.26 

1.64 
1.99 

•73 
2.03 

2.45 
.61 
•31 

1.03 

&  cent. 

39 
17 
29 

55 
36 

13 

74 
93 
89 
73 

95 

78 
74 

74 
86 

94 

93 
14 

22 
64 

Burring  machine                  . 

Scutcher,  continuous  feed,  
Opening  machine,  

Scribbler  card,  40  inches  wide,  
Intermediate  card,         ,,             
Finishing  card,               ,  ,             
i  spindle,  spinning,  . 

I  spindle,  doubling,  

Sizing  and  warping  machine  (with-  ) 
out  ventilator),  ....                     ..  ( 

Loom    7  54  feet  wide 

Scouring  machine,  for  2  pieces,  
Fulling  mill,  i  cylinder,  fornouveautes, 
Do.         3  cylinders,  for  cloth,  ... 

Double  fulling  mill,  for  noiiveautes,... 
Do.             do.     for  cloth,      

Single    fulling   stock,    two   stamps 
(Dobb's),   

Double   fulling  stock,  two  stamps 
(Spranger's), 

Gig  (Laineur),  single,  without  spreader, 
Do.         double,  
Machine   for  dressing    the   reverse  ) 
side,  with  cylinders,                      \ 

Machine   for  dressing   the   reverse 
side,  with  eccentrics,  

Wringing  machine,  lengthwise,  
Do.           do.       across,. 

Brushing  machine,  one  cylinder,  
Do.         do.        two  cylinders,  — 

RESISTANCE  OF  MACHINERY  FOR  CONVEYANCE  OF  GRAIN. 

Conveyance  of  Grain  horizontally  by  Screws  and  by  Bands. — A  1 2-inch 
screw,  having  4  inches  pitch,  turning  in  a  trough,  with  a  clearance  of  ^  inch, 
revolving  with  the  speed  of  maximum  effect,  60  turns  per  minute,  discharged 
6^  tons  of  grain  per  hour,  expending  .04  horse-power  per  foot  run.  The 
sectional  area  of  the  body  of  grain  moved  was  49  per  cent,  of  that  of  the 
screw.  At  speeds  above  60  turns  per  minute,  the  grain  did  not  advance, 
but  revolved  with  the  screw. 


RESISTANCE  TO  TRACTION   ON   COMMON   ROADS.  961 

A  i2-inch  screw,  having  a  1 2-inch  pitch,  delivered  34  tons  per  minute, 
at  70  turns  per  minute,  expending  .125  horse-power  per  lineal  foot,  or 
3  7  per  cent,  less  power  for  equal  weights  of  grain.  The  sectional  area  of  the 
grain  was  72  per  cent,  of  that  of  the  screw. 

An  endless  band,  18  inches  wide,  travelling  at  about  9  feet  per  second, 
delivered  70  tons  of  grain  per  hour;  power  expended,  .014  horse-power  per 
foot  run. 

To  convey  50  tons  per  hour  through  100  feet,  the  power  expended  by 
the  screw  was  18.38  H.P. ;  by  the  endless  band,  1.02  H.P. 

Grain  has  been  raised  by  a  dredger  30  feet  long;  efficiency,  50  per  cent1 

Mr.  Davison  states  that  95  feet  of  horizontal  Archimedian  screw, 
15  inches  in  diameter,  with  an  elevator  lifting  65  feet,  convey  40  quarters 
of  malt  per  hour,  for  an  expenditure  of  3.13  indicator  horse-power. 


RESISTANCE  TO   TRACTION   ON   COMMON   ROADS. 

From  the  results  of  recent  and  carefully  conducted  experiments  made  by 
M.  Dupuit,  he  made  the  following  deductions  as  to  the  resistance  to  traction 
on  macadamized  roads  and  on  uniform  surfaces  generally : — 

1.  The  resistance  is  directly  proportional  to  the  pressure. 

2.  It  is  independent  of  the  width  of  tyre. 

3.  It  is  inversely  as  the  square  root  of  the  diameter. 

4.  It  is  independent  of  the  speed. 

M.  Dupuit  admits  that,  on  paved  roads,  which  give  rise  to  constant  concus- 
sion, the  resistance  increases  with  the  speed;  whilst  it  is  diminished  by  an 
enlargement  of  the  tyre  up  to  a  certain  limit. 

M.  Debauve2  has  deduced  from  experiment,  that  the  advantage  of  a  pave- 
ment over  a  metalled  road  is  considerable  for  waggons,  is  less  for  stage- 
coaches, and  is  nearly  nothing  for  voitures  de  luxe,  or  private  carriages.  He 
summarizes  the  resistances  as  follows: — 

V_HI_.  „  RESISTANCE  TO  TRACTION. 

On  Metalled  Roads.  On  Paved  Roads. 

Waggon, 67  Ibs.  per  ton 38  Ibs.  per  ton. 

Stage-coach,...     67  „  45  „ 

Cabriolet, 81  „  76  to  83          „ 

M.  Tresca  tested  the  resistance  of  a  tramway  omnibus  on  Loubat's 
system,  adapted  with  wheels  for  running  on  a  common  road.  The  experi- 
ments were  made  on  an  inclined  street  in  Paris,  in  good  condition,  having 
ascending  gradients  of  i  in  55,  one  part  of  which  was  paved,  and  another 
part  macadamized.  The  frictional  resistance,  after  the  gravitation  on  the 
incline  was  eliminated,  was  as  follows : — 

i-  Gross  Weight.  Speed.  Frictional  Resistance, 

tons.  miles  per  hour.  Ibs.  per  ton. 

Macadam, 5.67         10.7         83 

Pavement, 5.67         10.1         66 

1  The  above  data  are  derived  from  Mr.  Westmacott's  paper  on  "  Corn- Warehousing 
Machinery,"  Proceedings  of  the  Institution  of  Mechanical  Engineers,  1869,  page  208. 

2  Manuel  de  Flngenieur  des  Fonts  et  Chaussees,  1873 ;  9me  fascicule,  page  32. 

61 


962 


FRICTIONAL  RESISTANCES. 


RESISTANCE  OF  CARTS  AND  WAGGONS   ON   COMMON 
ROADS  AND   ON   FIELDS. 

The  resistance  to  traction  of  agricultural  carts  and  waggons  was  tested  at 
Bedford  in  July,  1874,  by  means  of  a  horse-dynamometer  designed  by 
Messrs.  Eastons  &  Anderson.1  The  first  course  was  a  piece  of  hard  road 
200  yards  in  length,  rising  i  in  430;  it  was  dry  and  in  fair  condition,  largely 
made  of  gravel.  The  surface  was  in  many  places  somewhat  loose.  The 
second  course  was  along  an  arable  field,  growing  oats,  on  a  rising  gradient 
of  i  in  1000;  it  was  very  dry,  and  was  harder  than  in  average  condition. 

The  fore-wheels  of  the  waggons  averaged  3  feet  3  inches,  and  the  hind- 
wheels  4  feet  9  inches  in  diameter;  the  width  of  tyres  was  from  2^  to  4 
inches.  The  weight,  empty,  averaged  about  a  ton,  and  it  was  nearly  equally 
divided  between  the  front  and  hind  wheels.  The  cart  wheels  were,  say, 
4  feet  6  inches  high,  with  tyres  3}^  and  4  inches  wide.  The  weight  of  the 
empty  carts  averaged  10  cwt. 

The  following  results  arise  out  of  the  published  data : — 


ON  ROAD. 

Pair-horse 
Waggon,  with- 
out Springs, 
Loaded  with 
Roots. 

Waggon  with- 
out Springs, 
Loaded  with 
Maize. 

Waggon  with 
Springs, 
Loaded  with 
Roots. 

Cart  without 
Springs, 
Loaded  with 
Roots. 

Load,  

44  cwt. 

80  cwt. 

44  cwt. 

20  cwt. 

Gross  weight  drawn,  about 
Average  speed  per  hour,... 
Maximum  draft,  

64    „ 
•2^/2  miles 
320  Ibs 

100     „ 

2.60  miles 
400  Ibs 

64  „ 

2.47  miles 
300  Ibs 

5°  *4 

2.65  miles 
1  80  Ibs 

Average  draft,  

ICQ      , 

2C.I 

1  3^    , 

AQ  A 

Horse-power    developed,  ") 
at    33,000  foot-pounds  > 
per  minute...                ..  \ 

1.06  H.P. 

1.74  H.P. 

.88  H.P. 

.35  H.P. 

Draft  per  ton  gross,  on  level, 

ON  FIELD. 

Load,  

(  43-5  Ibs., 
\    orI/52 

A  A    CWt 

44-5  Ibs., 
or  1/50 

80  cwt 

34.7  Ibs., 
or  1/65 

AA   CWt. 

28  Ibs., 

or  1/80 

/ 

20  cwt 

Gross  weight  drawn,  about 
Average  speed  per  hour,... 
Maximum  draft,  

64    „ 
2.35  miles 
1000  Ibs 

100     „ 

2.52  miles 
1  200  Ibs 

64  „ 

2.35  miles 
1000  Ibs 

30  „ 

2.  6  1  miles 
400  Ibs 

Average  draft  

7OO 

QQ7 

7IO     . 

212 

Horse-power    developed,  "i 
at   33,000  foot-pounds  > 
per  minute,  ) 

4.36  H.P. 

6.70  H.P. 

445  H.P. 

1.48  H.P. 

Draft  per  ton  gross,  on  level, 

(  210  Ibs., 
\    or  i/i  i 

194  Ibs., 
or  i/I2 

210  Ibs., 
or  !/„ 

140  Ibs., 
or  i/z6 

From  these  data  it  appears  that,  on  the  hard  road,  the  resistance  is  only 
from  i^th  to  '/eth  of  the  resistance  on  the  field.  The  lowest  resistance  is 
that  of  the  cart  on  the  road — 28  Ibs.  per  ton;  due,  no  doubt,  as  observed 
in  Engineering,  to  the  absence  of  small  wheels  like  those  of  the  waggons. 


1  See  a  report  of  the  trials  in  Engineering,  July  10,  1874,  page  23. 


RESISTANCE  TO  TRACTION   ON  ROADS  AND  FIELDS.        963 

The  highest  resistance  is  210  Ibs.  per  ton — on  the  field.  The  addition  of 
springs  reduced  the  resistance  26  per  cent,  on  the  road;  but,  on  the  field, 
the  resistance  was  not  reduced  by  the  addition  of  springs. 

Also,  that  the  horse-power,  of  33,000  foot-pounds  per  minute,  developed, 
varied  from  y$  H.P.  to  7  H.P.  Allowing  a  pair  of  horses  for  the  first  and 
third  columns  above,  two  pairs  for  the  second  column,  and  one  horse  for 
the  last  column,  the  following  are  the  total  works  done  per  horse,  in  tech- 
nical horse-power: — 

HORSE-POWER  PER  HORSE. 
On  Road.  On  Field. 

Pair-horse  waggon,  without  springs, 53  H.P 2.18  H.P. 

Two  pair-horse  waggon,  without  springs,     .44     „        1.68     „ 

Pair-horse  waggon,  with  springs, 44     „        2.22     „ 

One-horse  cart,  without  springs, 35     „        1.48     „ 

Total  averages, 44     „        I. 

Averages,  without  springs, 44     „        1.7 

Taking  the  average  power  exerted  without  springs,  .44  horse-power,  on  the 
road,  as  the  average  for  a  day's  work,  it  represents  "44  x  33,000  =  14,520, 
say  15,000,  foot-pounds  per  minute,  for  the  power  of  a  horse  on  a  hard 
road. 

The  resistance  of  a  smooth  well-made  granite  tramway,  like  the  tram- 
ways in  the  City  of  London  and  Commercial  Road,  made  with  stones  5  or  6 
feet  in  length,  is  from  12^  Ibs.  to  13  Ibs.  per  ton  of  weight. 

Experiments  on  the  tractional  resistance  for  a  loaded  omnibus,  on  various 
kinds  of  roads,  were  made  by  a  committee  of  the  Society  of  Arts  i1 — 

Ibs. 

Weight  of  omnibus, 2480 

Load,  22  sacks  of  oats,  at  149  Ibs., 3278 

Total  weight,  2.57  tons,  or 5758 

The  loaded  omnibus  was  drawn  to  and  fro  over  each  trial  surface,  and 
the  mean  result  was  taken  as  the  resistance  for  an  exact  level : — 

RESISTANCE. 

Average  Speed.        Total.  Per  Ton. 

miles  per  hour.  Ibs.  Ibs. 

Granite  pavement,  sets  3  to  4  inches  wide,  2.87  44-75  17A1 

Asphalte  roadway, 3.56  69.75  27.14 

Wood  pavement, 3.34  106.88  41.60 

Good  gravelly  Macadam  road, 3.45  114.32  44-4$ 

Granite  Macadam,  newly  laid, 3.51  259.80  101.09 

There  is  a  want  of  consistency  here,  in  the  excessive  resistance  on  an 
asphalte  pavement,  compared  with  that  on  a  granite  pavement.  There 
can  be  no  doubt  that  asphalte  pavement,  properly  made,  is,  of  all  pave- 
ments, the  least  resistant;  and  that  its  resistance  cannot  be  greater  than 
the  resistance  of  a  granite  tramway. 

1  Report  of  the  Committee,  Journal  of  the  Society  of  Arts,  June  25,  1875. 


964  FRICTIONAL  RESISTANCES. 

Sir  John  Macneil  gives  the  tractive  force  necessary  to  move  a  waggon 
weighing  21  cwt,  at  2^  miles  per  hour,  on  roads  of  the  following  descrip- 
tions : — 

Total        Resistance 
Resistance,     per  Ton. 
Ibs.  Ibs. 

Well-made  pavement, 33  ...     31.2 

Road  made  with  6  inches  of  broken  stone  of  great  hardness, " 
on  a  foundation  of  large  stones  set  in  the  form  of  a 
pavement,  or  upon  a  bottoming  of  concrete, 

Old  flint  road,  or  a  road  made  with  a  thick  coating  of  broken 
stone,  laid  on  earth, 

Made  with  a  thick  coating  of  gravel,  laid  on  earth, 147  ...  140 


46  ...     44 
65  ...     62 


Sir  John  Macneil  made  a  series  of  experiments  on  the  tractive  resistance 
of  a  stage-coach,  on  a  section  of  the  Holyhead  Road.  The  weight  of  the 
coach,  empty,  was  18  cwt,  and  the  weight  of  seven  passengers  in  addition, 
allowing  i^  cwt.  for  each  passenger,  was  10^  cwt.;  total  weight  28^ 
cwt.  The  experimental  gradients  ranged  from  i  in  20  to  i  in  600,  and  the 
speeds  were  6,  8,  and  10  miles  per  hour.  It  was  found  that,  by  some  un- 
explained cause,  the  net  frictional  resistance  at  equal  speeds  varied 
considerably  according  to  the  gradient.  The  resistances  were  a  maximum 
for  the  steepest  gradient,  and  a  minimum  for  gradients  of  i  in  30  to  i  in 
40;  for  these  they  are  less  than  for  i  in  600.  The  mode  of  action  of  the 
horses  on  the  carriage  may  have  been  an  influential  element.  The  averages 
show,  — 

FOR  A  STAGE-COACH.  ON  A  METALLED  ROAD. 

At    6  miles  per  hour,  ...............     62  Ibs.  per  ton,  frictional  resistance. 

At    8     „          „  ...............     73      „  „ 

At  10    „          „  ...............     79      „  „ 

With  these  may  be  associated  the  resistance,  by  Sir  John  Macneil's 
experiments,  of  a  waggon  on  a  good  road,  namely,  44  Ibs.  per  ton,  at 
2^  miles  per  hour.  Plotting  the  resistances  for  the  above  four  speeds,  the 
following  approximate  formula  is  deduced:  — 

Frictional  Resistance  to  Traction  of  a  Stage-coach  on  a  Metalled  Road 
in  good  condition. 


R  —  the  frictional  resistance  to  traction  per  ton. 
v  =  the  speed  in  miles  per  hour. 

Note.  —  The  formula  is  applicable  to  waggons  at  low  speeds.  It  is 
simpler  than  the  formulas  deduced  by  Sir  John  Macneil.1 

M.  Charie-Marsaines  made  observations  of  a  general  character,  on  the 
performances  of  Flemish  horses  drawing  loads  upon  the  paved  and  the 
macadamized  roads  in  the  north  of  France,  where  the  country  is  flat,  and 
the  loads  are  considerable. 

1  Sir  John  Macneil's  formulas  are  given  in  Sir  Henry  Parnell  on  Roads,  page  464. 


RESISTANCE  ON  RAILWAYS. 


965 


Table  No.  320. — PERFORMANCE  OF  HORSES  ON  ROADS  IN  FRANCE. 

(M.  Charie-Marsaines.) 


Season  of  the 
Year. 

Description 
of  Road. 

Weight 
Horse. 

Speed  in 
Miles 
per  Hour. 

Work  Done 
per  Hour  in 
Tons  Drawr 
One  Mile. 

Ratio  of 
Paved  Road  to 
Macadamized 
Road. 

Winter,  

|  Pavement 

tons. 
1.306 

miles. 
2.05 

ton-miles. 
2.677 

i  644  to  i 

Summer 

/  Macadam 
1  Pavement 

.851 
1-395 

I.9I 
2.17 

1.625 
3.027 

I  22Q  tO  I 

\  Macadam 

I.I4I 

2.l6 

2.464 

The  average  daily  work  of  a  Flemish  horse  in  the  north  of  France  is,  on 
the  same  authority,  as  follows: — 

Winter, 21.82  ton-miles  per  day  in  winter. 

Summer, 27.82        „  „        in  summer. 


Mean  for  the  year,  say, ...  25.00        „  „ 

It  has  already  been  stated,  page  720,  that  a  good  horse  can  draw  a  weight 
of  i  ton  at  2^  miles  per  hour,  for  from  10  to  12  hours  a  day — equivalent  to 
(1x2^x10  =  )  2 5  tons  drawn  one  mile  per  day.  This  is  the  same  amount 
of  performance  as  is  above  given  from  M.  Charie-Marsaines. 

Conclusion. — With  the  exception  of  Messrs.  Eastons  and  Anderson  at 
Bedford,  the  authorities  on  the  tractional  resistance  to  vehicles  on  common 
roads,  ignore,  with  remarkable  unanimity,  the  influence  of  sizes  of  the 
wheels  and  other  essential  particulars.  It  is  better,  therefore,  to  refrain 
from  attempting  to  draw  general  conclusions,  and  to  leave  the  figures 
"  to  speak  for  themselves." 


RESISTANCE   ON   RAILWAYS. 

The  Author1  deduced  from  experimental  data,  the  following  formulas  for 
the  resistance  of  locomotives  and  trains,  under  these  conditions : — the  per- 
manent way  in  good  order;  the  engine,  tender,  and  train  in  good  order; 
a  straight  line  of  rails;  fair  weather,  and  dry  and  clean  rails;  an  average 
side  wind,  of  average  strength,  varying  (in  the  experiments)  from  slight  to 
VERY  strong: — 


Resistance  of  Engine, 
Tender,  and  Train. 


171 


Resistance  of  Train 
alone. 


24O 


R  =  total  resistance  of  engine,  tender,  and  train,  in  Ibs.  per  ton  gross; 
R' =  resistance  of  train  alone,  in  Ibs.  per  ton;  v  =  speed,  in  miles  per 
hour. 


1  Railway  Machinery,  1855,  pages  297,  298. 


966  FRICTIONAL  RESISTANCES. 

For  ordinary  practice,  to  meet  the  unfavourable  conditions,  which  may 
occur  in  combination,  of  frequent  quick  curves  under  one  mile  radius,  and 
strong  side  and  head  winds,  the  Author  estimated  from  his  own  observations 
that  the  resistance,  as  calculated  by  means  of  the  foregoing  rules,  should 
be  increased  50  per  cent.,  or  one-half  more.  On  this  basis,  for  speeds  of 

5,          10,       15,       20,       30,       40,       50,       60  miles  per  hour, 
the  frictional  resistances  per  ton  of  engine,  tender,  and  train,  are, 

12.2,       13,       14,     15.5,      20,       26,       34,     43.5  Ibs., 
and  the  frictional  resistances  per  ton  of  the  train  alone  are, 

9.15,      9.6,     10.5,    11.4,     14.6,    19.0,     24,     31.5  Ibs. 


RESISTANCE   ON   STREET  TRAMWAYS. 

The  rails  of  street  tramways  are  rolled  with  a  groove  for  the  guidance  of 
the  wheels  by  the  flanges.  The  wheels  of  cars,  therefore,  do  not  run  so 
freely  as  those  of  carriages  or  waggons  on  railways.  The  average  frictional 
resistance  of  vehicles  on  tramways  is  30  Ibs.  per  ton,  although  an  occasional 
maximum  of  60  Ibs.  per  ton  may  be  reached,  and,  on  the  contrary,  a  mini- 
mum of,  say,  15  Ibs.  per  ton,  when  the  rails  are  wet  and  clean,  straight  and 
new.  The  resistance  due  to  clogging  of  the  grooves  of  rails  was  brought 
into  evidence  by  Mr.  J.  Arthur  Wright,  who  found  that,  on  a  dusty  day,  on 
the  steam  lines  of  the  Rouen  tramways,  when,  despite  every  effort  to  keep 
the  rails  clear,  the  grooves  became  filled  with  dust  and  dirt,  the  engines 
consumed  about  2^  Ibs.  of  coke  per  mile  more  than  they  did  under  more 
favourable  circumstances,  when  the  consumption  averaged  about  1 2  Ibs.  per 
mile.  The  excess  is  nearly  20  per  cent,  over.1 

1  These  data  are  derived  from  Tramways,  their  Construction  and  Working,  1882,  by 
D.  Kinnear  Clark,  p.  180. 


APPENDIX. 


DR.    SIEMENS'    WATER-PYROMETER. 

[Appended  to  "  Pyrometers"  page  326.] 

This  apparatus  belongs  to  the  second  class  of  pyrometers  described  at 
page  327.  In  it,  Dr.  Siemens  has  reduced  the  water-pyrometer  to  a  form 
complete  and  exact  for  purposes  of  scientific  observation.  The  water 
is  contained  in  a  copper  vessel  capable  of  holding  rather  more  than  a  pint. 
The  sides  and  bottom  of  the  vessel  are  fitted  with  an  outer  casing  or 
jacket,  the  interspace  of  which  is  filled  with  felt,  by  which  any  radia- 
tion of  heat  from  the  vessel  is  prevented.  A  mercurial  thermometer  is 
fixed  in  the  vessel,  and  immersed  in  the  water.  It  is  fitted  with  a  small 
sliding  scale  in  addition  to  the  ordinary  scale,  which  is  graduated  and 
figured  with  50  degrees  to  i  degree  of  the  ordinary  scale.  Six  solid  copper 
cylinders  are  supplied  with  the  pyrometer,  each  of  which  is  accurately 
adjusted  so  that  its  capacity  for  absorbing  heat  shall  be  one-fiftieth  of  that 
of  a  pint  of  water.  In  using  the  pyrometer,  a  pint  of  water  is  measured 
into  the  copper  vessel,  and  the  sliding  scale  is  set  with  its  zero  at  the 
temperature  of  the  water  as  indicated  by  the  ordinary  scale  of  the  ther- 
mometer. A  copper  cylinder  is  put  into  the  furnace  or  the  hot-blast  current, 
of  which  the  temperature  is  to  be  measured,  where  it  is  left  for  a  space  of 
time  of  from  2  to  10  minutes,  according  to  the  intensity  of  heat  to  be 
measured.  Having  been  thus  raised  to  the  temperature  of  the  furnace  or 
current,  the  cylinder  is  withdrawn  and  quickly  dropped  into  the  water;  the 
temperature  of  the  water  is  raised  at  the  rate  of  i°  for  each  50°  of  the 
temperature  of  the  copper  cylinder.  The  rise  of  the  temperature  may  then 
be  read  off  direct  on  the  pyrometer  scale.  Add  to  the  temperature  thus 
noted,  the  observed  initial  temperature  of  the  water,  and  the  sum  is  the 
exact  temperature  required.  For  very  high  temperatures,  cylinders  of 
platinum  may  be  employed. 


ATMOSPHERIC     HAMMERS. 

{Appended  to  "Air  Machinery"  page  915.] 

In  the  hammers  designed  by  M.  Chenot  Aine,1  atmospheric  air  is 
employed  as  a  spring,  for  the  purpose  of  accumulating  and  of  applying  the 
motive  power  to  the  hammer.  The  hammer  is  a  cylinder  turned  from  end 
to  end,  and  bored  out  to  two  different  diameters.  It  is  divided  into  two 

1  Revue  Industrielle,  December,  1876,  page  521 


968  APPENDIX. 

chambers  by  a  diaphragm  at  the  middle.  The  lower  end  is  thus  completely 
inclosed,  whilst  the  upper  end  is  open.  Two  pistons  fixed  to  one  rod, 
passing  through  the  diaphragm,  play  in  the  upper  and  lower  chambers; 
and  they  receive  a  reciprocating  motion  from  a  crank  overhead,  driven  by 
a  band  passed  over  a  pulley  fixed  on  the  crank-shaft.  The  cylinder-hammer 
is  thus  floated  on  the  pistons  by  means  of  air-cushions,  of  which  there  is 
one  above  the  diaphragm,  one  below  it,  and  a  third  below  the  lower  piston ; 
and  it  is  impressed  with  a  reciprocating  movement  following  the  reciproca- 
tions of  the  pistons,  by  the  agency  of  these  cushions  of  air.  The  height  of 
the  fall  and  the  force  of  the  blow,  are  regulated  by  the  speed  at  which  the 
machine  is  driven.  There  is  no  sensible  heating  or  cooling  of  the  working 
parts,  and  M.  Chenot  estimates  that  the  efficiency  of  the  machine  amounts 
to  75  per  cent,  of  the  power  communicated  to  the  driving  pulley. 

The  chief  feature  of  interest  in  this  machine  is  the  employment  of  air 
compressed  and  expanded  to  two  or  three  times  its  normal  volume,  without 
any  inconvenience  by  either  heat  or  cold.  It  is  obvious  that  during  the 
momentary  actions  and  reactions,  time  is  not  afforded  for  the  heating  and 
cooling  effects  of  changes  of  temperature  in  the  air  to  take  place.  Hence 
the  high  efficiency. 


BERNAYS'    CENTRIFUGAL    PUMP. 

{Appended  to  "  Pumps"  page  944.] 

Mr.  Joseph  Bernays  constructs  the  discs  of  his  pump  with  a  double  joint, 
the  inner  one  being  the  joint  universally  employed  around  the  suction- 
openings,  by  which  the  water  is  admitted  from  the  suction-passages  into  the 
disc.  The  second,  or  outer,  joint  is  at  the  extreme  diameter  of  the  disc, 
and  it  prevents  the  pressure  of  the  water  in  the  delivery-pipe  from  reacting 
on  the  outer  faces  of  the  revolving  disc.  A  saving  of  power  is  thus  effected, 
in  reducing  the  loss  by  friction  on  the  disc. 

The  form  of  the  vanes  of  Mr.  Bernays'  encased  pump,  may  be  roughly 
described  as  semi-elliptical,  or  the  half  of  a  flat  ellipse  divided  at  its  longest 
diameter;  the  concave  surface  being  presented  to  the  water  in  the  direction 
of  the  motion.  By  the  adoption  of  such  a  form,  it  is  designed  that  the 
blade  should  scoop  up  the  water  arriving  at  the  centre  of  the  pump  by  its 
inner  edge,  and  should  project  the  water  forward  in  a  direction  as  nearly 
tangential  as  possible  by  its  outer  edge,  at  the  circumference.  When  the 
pump  is  not  encased,  and  the  water  is  delivered  into  an  open  well  or 
reservoir,  the  outer  end  of  the  vane  is  curved  backwards  for  the  purpose  of 
facilitating  the  discharge  radially.  In  this  case,  the  blades  acquire  an 
ogee  form,  like  Rankine's  fan. 

Mr.  Bernays  has  supplied  the  following  note  on  centrifugal  pumps : — 

"  The  only  parts  of  a  centrifugal  pump  which,  when  at  work,  absorb 
more  power  by  friction  than  is  due  to  the  mere  velocity  of  the  water  passing 
through  the  pump,  are  the  outer  faces  of  the  revolving  disc.  These  outer 
faces  are  surrounded  by  water  quite  or  nearly  stationary,  and  as  they  them- 
selves revolve  at  a  speed  proportionate  to  the  height  to  which  the  water  is 
to  be  lifted,  more  or  less  independent  of  the  quantity  that  passes  through 
the  pump,  they  tend  to  carry  the  surrounding  water  round  with  them. 


STEAM-VACUUM  PUMP.  969 

According  to  the  greater  or  less  pressure  under  which  the  pump  works,  the 
friction  produced  will  be  greater  or  less,  just  as  there  is  greater  skin  resist- 
ance in  a  vessel  of  greater  draft  than  in  one  of  light  draft,  both  having  an 
equal  extent  of  surface.  The  saving  of  power  effected  by  the  removal  of 
the  pressure  of  the  water  in  the  delivery  pipe  of  Mr.  Bernays'  pump,  as 
above  explained,  is  all  the  more  necessary,  since,  with  a  centrifugal  pump, 
the  power  required  for  driving  it  increases  rapidly,  and  in  a  greater  ratio 
than  the  heights  of  delivery  to  which  such  pump  may  have  to  be  applied. 
This  is  a  point  to  which  the  attention  of  engineers  and  users  of  centrifugal 
pumps  has  never  been  called,  or,  if  it  has  ever  been  mentioned,  it  has  been, 
and  is,  constantly  lost  sight  of.  Nevertheless  the  fact  is  clear,  and  the 
explanation  very  simple  indeed.  The  only  working  parts  of  a  centrifugal 
pump  which,  irrespective  of  the  friction  previously  mentioned,  actually 
propel  the  water  and  absorb  the  power  applied  to  it,  are  the  arms  or  vanes 
radiating  from  the  centre  to  the  outside  of  the  disc.  The  shape  of  these 
arms  may  for  a  moment  be  left  out  of  consideration,  as  their  more  or  less 
perfect  form  accounts  for  a  mere  percentage  only  of  the  whole  power  used 
for  driving  a  centrifugal  pump.  The  main  power  is  used  in  driving  or 
pushing  the  arms  against  the  water  at  a  speed  calculated  to  produce  a 
pressure  equal  to  or  rather  in  proportion  to  the  height  to  which  the  water 
is  raised.  Now,  the  speed  at  the  outside  diameter  of  the  pump  disc  is 
approximately  equal  to  that  of  a  body  falling  from  the  height  to  which  the 
water  is  lifted,  or  it  is  directly  proportionate  to  the  square  root  of  that 
height.  And  as  the  direct  resistance  which  the  arms  meet  with  in  their 
rotation,  is  simply  proportional  to  the  height  to  which  the  water  is  to  be 
lifted  (the  same  as  in  common  reciprocating  pumps),  it  follows  that  the 
amount  of  power  necessary  for  working  the  pump,  is  a  function  of  the  height 
multiplied  by  its  square  root,  or  h  ^/  h  =  >J  h\ 

Thus,  a  pump  requiring  20  H.P.  to  raise  water  10  feet  high,  will,  if  the 
height  be  increased  to  40  feet,  not  merely  require  4  times  the  power  for  the 
same  quantity  delivered,  but  4  multiplied  by  *J  4,  or  8  times,  that  is, 
1 60  H.P.  And  it  evolves  from  this,  that  although  centrifugal  pumps  are 
an  exceedingly  useful  and  simple  mechanical  appliance  for  raising  large 
quantities  of  liquids  to  moderate  height, — and  it  may  here  be  added,  variable 
quantities  to  variable  heights, — they  should  not  be  made  use  of  for  great 
heads  of  delivery,  where  the  cost  of  the  power  employed  to  work  them  is 
any  consideration." 


STEAM- VACUUM     PUMP. 

[Appended  to  ' '  Pumps, "  page  944.] 

The  steam-vacuum  pump  belongs  to  the  class  of  pumps  on  Savary's  old 
system,  in  which  steam  from  the  boiler  is  admitted  into  direct  contact  with 
the  surface  of  the  water  to  be  forced.  Experiments  were  conducted  by 
Mr.  J.  F.  Flagg,  at  the  Cincinnati  Exposition  of  1875,  on  such  a  pump.1 
The  water  was  drawn  directly  from  a  canal,  through  a  3-inch  pipe,  155  feet 

1  Journal  of  the  American  Society  of  Civil  Engineers,  December,  1876,  page  381. 


970  APPENDIX. 


long,  with  a  lift  of  10.83  feet«  The  head  of  pressure  was  regulated  by 
means  of  a  cock  applied  to  the  discharge  pipe.  The  proportion  of  primed 
water  in  mixture  with  the  steam  was  ascertained,  and  allowed  for  :  — 

ist  trial.  ad  trial. 

Effective  pressure  in  boilers,  Ibs.  per  sq.  inch,  .........  72  Ibs  ..........  57  Ibs. 

Temperature  of  water  in  canal,  Fahrenheit,          .........  60°         .........  6i°.5 

Do.  effluent  water,  „  .........  86°.9      .........  73°.! 

Pressure  at  the  gauge,  Ibs.  per  sq.  inch,  .........  35.3  Ibs  .......   16.3  Ibs. 

Do.  feet  of  head,  .........  81.8  ft  ..........  37.7  ft. 

Steam  consumed  per  horse-power,  per  hour,        .........  477.5  Ibs  .......  390.7^3. 

Coal        do.  do.  (allowing  9  Ibs.  water  per 

pound  of  coal),  .........  53.1  Ibs  .......  43.4  Ibs. 

Duty  per  100  Ibs.  of  coal,  foot-pounds,  .........  3,732,260  ......  4,561,200 


INDEX. 


—    ACCELERATED    — 

Accelerated  and  retarded  motion,  282,  286. 

Accelerating  forces,  282. 

Acre,  137;  equivalent  value  in  French  measures, 
i53,  .156. 

Adhesion  of  leather  belts,  744,  748. 

Air;  as  a  standard  for  weight  and  measure,  127; 
pressure  of  air,  127;  measures  of  pressure  of  air, 
127;  weight  and  volume  of  air,  127;  weight  of  air 
compared  with  that  of  water,  or  its  specific  gravity, 
128;  specific  heat  of  air,  128. 

Air  and  aqueous  vapours,  mixture  of,  394,  396. 

Air,  expansion  of,  343,  345;  relations  of  the  pressure, 
volume,  and  temperature  of  air  and  other  gases, 
346,  351;  special  rules  for  one  pound  weight  of  a 
gas,  349;  table  of  the  volume,  density,  and  pres- 
sure of  air  at  various  temperatures,  351;  specific 
heat  of  air,  354,  358,  363 ;  comparative  density 
and  volume  of  air  and  saturated  steam,  391. 

Air,  ascension  of,  by  difference  of  temperature,  897. 

Air  consumed  in  the  combustion  of  fuels,  400,  405: 
coals,  427;  coke,  435;  wood,  443;  wood-charcoal, 
452- 

Air  and  other  gases,  flow  of,  891.  See  Flow  of  Air, 
page  974. 

Air,  dry,  or  other  gas,  work  of,  compressed  or  ex- 
panded, 898.  See  Work  of  Dry  Air,  page  984. 

Air,  hot,  and  stoves,  heating  by,  488. 

Air,  resistance  of,  to  the  motion  of  flat  surfaces,  897. 

Air,  work  of  compression  of,  at  constant  tempera- 
ture, 899;  adiabatically,  903. 

Air,  work  of  expansion  of,  at  constant  temperature, 
899;  adiabatically,  904. 

Air  machinery,  915;  machinery  for  compressing  air, 
and  for  working  by  compressed  air,  915;  compres- 
sion of  air  by  water,  915;  by  direct-action  steam- 
Bimps,  915;  compressed-air  machinery  at  Powell- 
uffryn  Collieries,  916. 

Hot-air  engines,  917;  Rider's  hot-air  engine, 
917;  Belou's  hot-air  engine,  918. 

Gas-engines,  918  ;  Lenoir's  double-acting  gas- 
engine,  919;  Otto  and  Langen's  atmospheric 
gas-engine,  919;  Otto  engine,  921;  Clerk's  engine, 
922. 

Fans  or  ventilators,  924 ;  Common  centrifugal 
fan,  924;  mine- ventilators:  —  Guibal's  fan,  925; 
Cook's  ventilator,  926;  blowing -engines,  926; 
Root's  rotary  pressure-blower,  927. 

Air-pyrometer,  327. 

Air-thermometers,  325;  Regnault's,  326. 

Alloys,  melting  points  of,  363;  table,  366. 

Alloys,  specific  gravity  of  alloys  of  copper,  200,  201, 
626,  627;  of  gold  and  other  metals,  201. 

Aluminium-bronze,  tensile  strength  of,  627. 

American  coals,  418.     See  Coal,  page  972. 

Anemometer,  892. 

Animal  substances,  weight  and  specific  gravity  of, 

212. 

Annealed  and  unannealed  wrought-iron  plates,  com- 
parative strength  of:  —  Krupp  and  Yorkshire 
plates,  583-585;  Prussian  plates,  586. 

Annealed  and  unannealed  steel  plates,  comparative 
strength  of : — Fagersta,6o6, 607,  609-611;  Siemens' 
steel,  613. 

Annealed  and  unannealed  wire,  phosphor  bronze, 
copper,  brass,  steel,  iron,  tensile  strength  of,  629. 


—    BOLTS    — 

Anthracite: — British,  409,  413;  American,  418,  419; 
French,  421,  422;  Russian,  422;  manufacture  of 
coke  with,  432. 

Anthracitic  coke,  432. 

Applications  of  heat,  459:— Transmission  of  heat 
through  solid  bodies,  459;  warming  and  ventila- 
tion, 477;  heating  of  water  by  steam  in  direct  con- 
tact, 490;  spontaneous  evaporation  in  open  air, 
491;  desiccation,  493;  heating  of  solids,  497. 

Aqueous  vapour,  mixture  of,  with  air,  properties 
of,  394,  396. 

Are,  149;  equivalent  value  in  English  measures, 
i53- 

Armstrong's  hydraulic  machines,  950. 

Asphalte: — composition,  437;  heat  of  combustion, 
437,  438;  weight  and  specific  gravity,  207,  437. 

Ass,  work  of,  in  carrying  loads,  721. 

Atmospheric  gas-engine,  Otto  and  Langen's,  922. 

Atmospheric  hammers,  by  M.  Chenot  Aine,  967. 

Australian  coal,  423,  424.     See  Coal,  page  972. 

Axles,  railway,  proportions  of,  767. 


B 


Balls,  cast-iron,  weight  of;  multipliers  for  other 
metals,  258 ;  diameters  of,  for  given  weights,  258. 

Barker's  water-mill,  939. 

Beams,  flanged,  transverse  strength  of: — cast-iron, 
647;  wrought-iron,  653. 

Beams,  forms  of,  of  uniform  strength,  517. 

Beams,  uniform,  supported  at  three  or  more  points, 
533;  distribution  of  weight  on  the  points  of  sup- 
port, 533;  deflection,  534. 

Beams,  homogeneous,  transverse  strength  of,  503. 
See  Transverse  Strength,  page  981. 

Belt-pulleys  and  belts,  742: — Speeds,  742;  tensile 
strength  of  belts,  742;  horse-power  transmitted 
by  leather  belts,  743;  adhesion  and  power  of  belts, 
744;  M.  Morin's  experiments;  M.  Claudel's  data, 


es 

....  table 

of  the  driving  power  of  belts,  749. 
India-rubber  belting,  750. 
Weight  of  belt-pulleys,  750. 
Belts,  742.     See  Belt-pulleys  and  Belts. 
Belou's  hot-air  engine,  918. 
Bending  strength  of  wrought-iron  plates,  586. 
Bernays'  centrifugal  pump,  968. 
Bevil-wheels.     See  Toothed  Wheels,  page  981. 
Birmingham  wire-gauges,  130,  131;  metal-gauge  or 

plate-gauge,  131. 

Bitumen,  weight  and  specific  gravity  of,  207. 
Blower,  Root's,  927. 
Blowing  engines,  926 
Boghead  coal,  417. 

Boilers,  strength  of  stayed  surfaces  of,  685.  See  Eva- 
porative Efficiency,  Evaporative  Performance. 
Boiling  points  of  liquids,  368;  of  saturated  solutions 
of  salts,  369;  of  sea- water,  370;  boiling  points  at 
various  pressures,  370. 
Bolts  and  nuts,   standard  sizes  of: — Whitworth's 


972 


INDEX. 


—    BOLTS    — 


—    COALS    — 


system,  681,  682;  American  system,  683;  Armen- 
gaud's  French  system,  683,  684. 

Bolts  and  nuts,  screwed,  tensile  strength  of,  680. 
See  Tensile  Strength,  page  980. 

Boyden  turbine,  940. 

Brass,  tensile  strength  of,  627,  628;  brass  tubes,  627; 
brass  wire,  627,  629. 

Brass,  weight  of: — tabulated  weights,  219-221; 
rule  for  the  length  of  brass  wire,  224;  multiplier 
for  the  weight  of  brass  bars,  plates,  sheets,  &c., 
226;  special  tables  of  the  weight  of  brass  tubes 
and  sheets,  252,  266-268;  multiplier  for  the  weight 
of  brass  balls,  258.  See  Weight  of  Iron  and  other 
Metals,  page  983. 

Breakage  of  coal,  409,  410. 

Breast  water-wheel,  938. 

Bricks,  cemented,  adhesion  of,  630. 

Bricks,  crushing  strength  of,  631. 

Brickwork,  crushing  strength  of,  631. 

British  coals,  412.     See  Coal. 

Bronze,  tensile  strength  of,  627,  628;  aluminium- 
bronze,  627. 

Buckled  iron  plates,  strength  of,  660. 

Builders' measurement: — superficial,  137;  cubic,  137. 

Building,  measures  relating  to,  144. 

Bulging  strength  of  wrought  iron: — Sir  Wm.  Fair- 
bairn's  experiments,  569;  Mr.  Kirkaldy's  experi- 
ments on  Krupp  and  Yorkshire  iron,  585;  on 
Prussian  iron,  586. 

Bulging  strength  of  steel: — Fagersta  steel,  611 ;  Sie- 
mens' steel,  612. 

Bulk  of  coal,  &c.  See  Weight  and  Bulk  of  Coal, 
&c.,  page  983. 

Buoyancy,  277. 

Bursting  pressure,  resistance  to,  687.  See  Strength 
of  Hollow  Cylinders,  page  979. 

Bushel  measures:— Standard  bushel,  139;  sundry 
bushel  measures  for  coal,  142;  equivalent  value  in 
French  measures,  154. 


Camel,  work  of,  in  carrying  loads,  721. 

Carbon,  constituent,  influence  of,  on  the  tensile 
strength  of  steel,  621;  on  the  transverse  strength 
of  steel  rails,  664. 

Carbon,  process  of  combustion  of,  399. 

Carree's  cooling  apparatus,  373- 

Cast  iron,  strength  of,  553.  See  Strength  of  Cast 
Iron,  page  979. 

Cast  iron,  weight  of:— Data  for  the  weight,  217, 
218;  rules  for  the  weight,  223;  tabulated  weights, 
219-221 ;  multiplier  for  the  weight  of  cast-iron 
bars,  plates,  &c.,  226;  special  tables  of  weight  of 
cast-iron  pipes,  cylinders,  and  balls,  251,  253-258; 
and  of  weight  of  cast-iron  water-pipes,  936;  and 
gas-pipes,  936.  See  Weight  of  Iron  and  other 
Metals,  page  983. 

Cast-iron  columns,  strength  of,  643-645. 

Cast-iron  flanged  beams,  transverse  strength  of, 
647: — Experiments  by  Mr.  Hodgkinson,  by  Mr. 
Berkley,  by  Mr.  Cubitt,  and  others,  647;  tabu- 
lated results,  649;  formulas  and  rules,  651. 

Elastic  strength  and  deflection,  formulas  and 
rules,  652. 

Catenary,  273. 

Cement,  strength  of: — Tensile,  630;  crushing,  632. 

Central  forces,  294. 

Centres,  mechanical,  287.  See  Mechanical  Prin- 
ciples, page  977. 

Centrifugal  force,  274,  294;  rules,  295. 

Centrifugal  pumps,  946;  centrifugal  pump  by  Mr. 
J.  Bernays,  968. 

Chain,  endless,  pump,  947. 

Chains,  tensile  strength  of,  677.  See  Tensile 
Strength,  page  980. 

Chains,  weight  of,  678,  679. 


Channels,  flow  of  water  in,  932;  limits  of  velocity,  934. 
Charbon  de  Paris,  449. 
Charcoal,  brown,  manufacture  of,  449. 
Circles,  properties  of,  21;  mensuration  of,  24. 
Circular  arcs,  35;  tables,  95,  97. 
Circumferences  of  circles,  &c.,  35;  tables,  66,  87. 
Clerk's  gas-engine,  922. 
Cloth  measure,  130. 

Coal,  409;  classification  of  coals,  409;  small  coal, 
409;  utilization  of  small  coal,  410;  deterioration  of 
coal  by  exposure,  412. 

British  coals,  412;  composition  of  bituminous 
coals — Dr.-  Richardson's  analyses,  412;  weight 
and  composition  of  British  and  foreign  coals  by 
Messrs.  Delabeche  and  Playfair,  413;  variations 
of  chemical  composition,  415;  average  composi- 
tion, 415;  Welsh  coals,  analysis  by  Mr.  G.  J. 
Snelus;  patent  fuels,  416;  weight  and  bulk  of 
British  coals,  416;  hygroscopic  water  in  British 
coals,  416;  Torbanehill  or  Boghead  coal,  417;  its 
composition,  417;  air  chemically  consumed  in  the 
combustion  of  coal,  428. 

American  and  foreign  coals,  418;  Professor  W. 
R.  Johnson's  analyses,  418;  composition,  418; 
weight  and  bulk,  418,  419. 

French  coals,  420;  classification  according  to 
behaviour  in  furnace,  and  according  to  size,  420; 
utilization  of  small  coal,  420;  composition  and 
heating  power,  420,  422;  weight  and  volume,  420; 
lignites,  422. 

Indian  coals,  423;  comparative  composition  of 
Australian,  Nerbudda,  Nagpore,  and  English 
coals,  424;  composition  of  Indian  coals,  425. 

Combustion  of  coal,  426.     See  Combustion  of 
Coal,  page  973. 
Coal,  best,  and  inferior  fuels,  equivalent  weights  of, 

820. 

Coal,  brown.     See  Lignite,  page  976. 
Coals,  volume,  weight,  and  specific  gravity  of,  206, 

207. 

Coals:  evaporative  performance  of  English  coals  by 
Messrs.  Delabeche  and  Playfair,  770;  of  Hindley 
Yard,  Lancashire,  coal,  in  stationary  boilers,  771; 
of  South  Lancashire  and  Cheshire  coals,  in  a 
marine  boiler  at  Wigan,  781;  of  Newcastle  and 
Welsh  coals  in  the  Wigan  marine  boiler,  784;  of 
Newcastle  coals,  in  a  marine  boiler  at  Newcastle- 
on-Tyne,  785;  of  Newcastle  and  Welsh  coals  in 
the  Newcastle  boiler,  787;  of  Welsh  and  New- 
castle coals,  in  a  marine  boiler  at  Keyham,  790; 
of  American  coals  in  a  stationary  boiler,  791 ;  and 
in  a  marine  boiler,  795;  coal  in  locomotives,  800; 
Llangennech  coal  in  portable-engine  boilers,  801. 
Grate-area  and  heating  surface,  relation  of,  to 
evaporative  performance  in  steam-boilers,  802  ; 
experiments  by  Mr.  Graham,  802 ;  by  Messrs. 
Woods  &  Dewrance,  803 ;  by  M.  Paul  Havrez, 
803. 

Formulas,  deduced  from  the  results  of  experi- 
ments, 804-821 ;  table  of  evaporative  performance 
by  formulas,  819 ;  table  of  equivalent  weights  of 
best  coal  and  inferior  fuels,  820. 

Heating  surface  and  grate-area,  relations  of,  to 
evaporative  performance  in  steam-boilers,  802. 

Evaporative  performance  of  steam-boilers,  ex- 
perimental, influence  of  various  circumstances  and 
various  treatment  on: — proportion  of  oxygen  in 
coals,  771;  area  of  grate,  772,  781,  796;  coking 
fires  and  spreading  fires,  773,  779,  780,  782  ;  thick- 
ness of  fire,  773,  779,  780,  782 ;  admission  of  air 
above  the  grate,  772-774,  781,  794. 

Green's  economizer,  772,  775,  778,  779 ;  water- 
tubes,  775,  776  ;  volume  of  air  supply,  778  ;  steam 
of  higher  pressure,  779  ;  D.  K.  Clark's  steam- 
induction  apparatus,  779 ;  self-feeding  firegrates 
(Vicars'),  780;  calm  and  windy  weather,  780; 
forcing  the  draught,  781,  784;  inverted  bridge, 
782 ;  reduction  of  the  flue  surface,  775,  783,  795, 
796;  prolonged  firing,  782;  C.  W.  Williams' 


INDEX. 


—    COALS    — 


—    CRANES    — 


973 


smoke  preventer,  788 ;  soot  in  the  flues,  794 ;  level 
of  the  grate,  795. 

American  marine  boiler,  varying  rate  of  com- 
bustion, varying  area  of  grate,  reduction  of  heat- 
ing surface,  796. 
Coal-gas,  its  composition,  457  ;  heat  of  combustion, 

Coal  measure,  139 ;  coal  weight,  142. 

Coal-pits,  waggons  in,  resistance  of,  956. 

Cockle-stove,  488. 

Coins,  current  weight  of,  English,  141, 190;  French, 
190;  German,  191;  Austrian,  192;  Egyptian, 
194 ;  Indian,  195  ;  Japanese,  195. 

Coke,  430 ;  residuary  coke  in  coals,  by  laboratory 
analysis,  430;  quality  dependent  on  proportion 
of  hydrogen  in  the  coal,  431  ;  anthracitic  coke, 
432;  proportion  of  coke  yielded  by  coal,  432; 
weight  and  bulk,  432  ;  composition,  433;  moisture 
in  coke,  434 ;  loss  of  combustible  matter  in  the 
conversion  of  coal  into  coke,  434 ;  air  chemically 
consumed  in  combustion  of  coke,  435  ;  gaseous 
products  of  combustion,  435 ;  heating  power  of 
coke,  436 ;  temperature  of  combustion,  436 ;  gas- 
coke,  439.  See  also  Coke,  Proportion  of,  in 
Coals. 

Coke,  American,  419. 

Coke,  proportion  of,  in  coals,  British,  414,  415,  417, 
425  ;  American,  418,  419 ;  French,  421 ;  Indian, 
423-425 ;  Australian,  424 ;  sundry,  430,  432. 

Coke  of  lignite,  436 ;  proportion  of,  437,  438  ;  quality 
according  to  constituent  hydrogen  m  lignite,  437, 
438. 

Cold,  greatest  degree  of,  373,  377  ;  sources  of,  373  ; 
Siebe's  ice-making  machine,  373;  Carree's  cooling 
apparatus,  373 ;  frigorific  mixtures,  373 ;  cold  by 
evaporation,  376. 

Cold  rolling,  influence  of,  on  the  density  of  wrought 
iron,  578. 

Collapsing  pressure,  resistance  to,  694.  See  Strength 
of  Holloiu  Cylinders,  page  979. 

Colliery  winding  engines,  resistance  of,  956. 

Columns,  strength  of,  643 ;  leading  principles,  Mr. 
Hodgkinson's  investigations,  643 ;  short  flexible 
columns,  long  columns,  644;  Mr.  F.  W.  Shields 
on  hollow  cast-iron  columns ;  rules  by  Mr.  Gordon, 
Mr.  Stoney,  Mr.  Unwin,  Mr.  Baker,  645 ;  timber 
columns,  646 ;  Mr.  Brereton  on  timber  piles,  646 ; 
Mr.  Laslett  on  columns  of  wood,  647. 

Combustible  elements  of  fuel,  398 ;  gases  concerned 
in  the  combustion  of  fuel,  398 ;  process  of  com- 
bustion, 399 ;  air  consumed  in  combustion,  400, 
405 ;  gaseous  products,  400 ;  heat  evolved  by 
combustion,  402  ;  heating  powers  of  combustibles, 
404—406 ;  temperature  of  combustion,  407. 

Combustibles,  chemical  composition  of,  403.  See 
Combustible  Elements  of  Fi4el. 

Combustion,  398. 

Combustible  elements,  398 ;  gases  concerned  in 
combustion,  398  ;  process  of  combustion,  399. 

Air  consumed  in  the  combustion  of  fuels,  400, 
405. 

Quantity  of  gaseous  products  of  combustion, 
400. 

Surplus  air,  402,  407. 

Heat  evolved  by  the  combustion  of  fuel,  402. 
Table  of  the  heating  powers  of  combustibles, 
404,  405. 

Temperature  of  combustion,  407. 

Combustion,  heat  of,  402,  404,  405 ;  English  coals, 
414,  428,  430;  French  coals,  421,  422;  coke,  436. 

Combustion,  heat  of,  gas-coke,  402 ;  lignites,  423, 
437,  438  ;  asphalte,  437,  438  ;  of  wood,  444 ;  of 
wood-charcoal,  452 ;  of  peat,  455 ;  of  peat-char- 
coal, 455  ;  of  tan,  455  ;  liquid  fuels,  456 ;  coal-gas, 
457- 

Combustion  of  coal,  426 ;  process,  426  ;  summary  of 
the  products  of  decomposition  in  the  furnace,  426  ; 
quantity  of  air  chemically  consumed  in  the  com- 
plete combustion  of  coal,  427;  table,  showing 


composition,  heat  of  combustion,  and  air  consumed 
by  British  coals,  428 ;  gaseous  products  of  the 
complete  combustion  of  coal,  428;  surplus  air, 
429 ;  total  heat  of  combustion,  430. 

Compass,  points  of,  37,  117. 

Composition   of  coals,   British,    412,  413,  415-417, 

$24,    428 ;    American,    418 ;     French,    420,    422 ; 
ndian,  423-425  ;  Australian,  423,  424 ;  coke,  433 ; 
lignite,  422,  436,  438 ;  asphalte,  437  ;  wood,  440. 

Composition  of  coke,  433. 

Composition,  chemical,  of  steel,  603. 

Compound  steam  engine,  849 ;  Woolf  engine,  ideal 
diagrams,  849 ;  receiver-engine,  ideal  diagrams, 
852  ;  intermediate  expansion  in  the  Woolf  engine, 
855 ;  and  in  the  receiver-engine,  857 ;  work  of  the 
Woolf  engine,  with  clearance,  859 ;  and  of  the 
receiver-engine,  862 ;  comparative  work  of  steam 
in  the  Woolf  engine  and  the  receiver-engine,  867. 
Formulas  and  rules  for  calculating  the  expansion 
and  the  work  of  steam  in  compound  engines,  869. 
To  find  the  work  done  in  the  two  cylinders  of 
compound  engines,  875. 

Compound  units,  English,  comparison  of: — velocity, 
144;  volume  and  time,  pressure  and  weight, 
weight  and  volume,  power,  145. 

Compressed  air,  flow  of,  through  pipes,  896. 

Compressed-air  engines,  efficiency  of,  909 ;  table  of 
corresponding  ratios  of  temperatures  and  pressures, 
when  the  air  is  admitted  for  the  whole  of  the 
stroke,  908 ;  table  of  comparative  final  temper- 
atures and  efficiencies,  when  the  air  is  expanded 
adiabatically,  and  when  it  is  admitted  for  the 
whole  of  the  stroke,  908. 

Machinery  for  working  by  compressed  air  at 
Powell  Duffryn  collieries,  916. 

Compressed  steel,  614. 

Compressibility  of  water,  126. 

Compressing  air,  machinery  for,  915 ;  by  water,  915; 
by  a  direct-action  steam-pump,  915,  916. 

Compression  of  gases,  345. 

Compression,  of  a  gas,  work  of,  at  constant  temper- 
ature, 899 ;  adiabatically,  903. 

Compression  of  steam  in  the  cylinder,  878. 

Compressive  strength  of  cast  iron.  See  Tensile 
Strength,  page  980. 

Compressive  strength  of  steel,  595,  596,  599,  602, 
605,  609. 

Compressive  strength  of  timber : — Mr.  Laslett's  ex- 
periments, 539,  541 ;  Mr.  Kirkaldy's,  546,  547. 

Compressive  strength  of  wrought  iron : — Mr.  Edwin 
Clark's  experiments,  570 ;  the  Steel  Committee's, 
579 ;  Swedish  hammered  bars,  581 ;  Mr.  J. 
Tangye's  experiments,  582. 

Concrete,  crushing  strength  of,  632. 

Condensation  of  steam  and  vapours,  462,  472,  475. 

Constructions,  elementary,  strength  of,  633.  See 
Strength  of  Elementary  Constructions,  page  979. 

Contraction  of  wrought  iron  under  tensile  stress : — 
bars,  572,  580;  notched  bars,  574;  plates,  578; 
wire,  587. 

Cooke's  ventilator,  926. 

Cooling  apparatus,  Carree's,  373. 

Copper,  alloys  of,  specific  gravity,  200,  201,  626, 
627;  tensile  strength  of,  626,  627.  See  Tensile 
Strength  of  A  Hoys  of  Copper,  page  980. 

Copper,  tensile  strength  of,  626;  wire,  628,  629. 

Copper,  weight  of: — tabulated  weights,  219-221  ; 
multipliers  for  the  weight  of  copper  bars,  plates, 
sheets,  &c.,  220,  226 ;  rule  for  the  length  of 
copper  wire,  224;  special  tables  of  the  weight 
of  copper  in  sheets,  pipes,  and  cylinders,  251, 
261-265.  See  Weight  of  Iron  and  other  Metals, 
page  983. 

Cord  of  wood,  186. 

Corde  of  wood,  154. 

Cotton  ropes  for  transmission  of  power,  by  Mr. 
Ramsbottom,  755. 

Cranes,  stress  in,  697;  power  of  men  at,  718,  719. 


974 


INDEX. 


—    CRUSHING    — 


—    FRENCH    — 


Crushing  resistance.  See  Compressive  Strength, 
page  973. 

Crushing  strength  of  bricks  and  brickwork,  631. 

Crushing  strength  of  concrete,  632. 

Curvilineal  figures,  mensuration  of,  25. 

Cycloid  and  epicycloid,  problems  on,  19 ;  mensura- 
tion of,  25. 

Cylinders,  hollow,  strength  of,  687.  See  Strength 
of  Hollow  Cylinders,  page  979. 

D 

Daniell's  pyrometer,  327. 

Deflection  of  beams  and  girders,  527.  See  Trans- 
verse Deflection,  page  981. 

Deflection,  torsional,  536.  See  Torsional  Deflec- 
tion, page  981. 

Deflection,  transverse,  of  shafts,  756 ;  formulas  for 
deflection,  756 ;  for  diameter  and  side,  757 ;  for 
distributed  weight,  757  ;  over-hung  shafts,  757. 

Formulas  for  gross  distributed  weight,  length 
of  span,  757. 

Deflection  of  cast  iron.  See  Transverse  Strength, 
page  981;  and  Torsional  Strength,  page  981. 

Deflection,  transverse,  of  flanged  beams: — cast- 
iron,  652 ;  wrought-iron,  657,  660. 

Deflection  of  steel  bars.  See  Transverse  Strength, 
page  981;  and  Torsional  Strength,  page  981. 

Deflection  of  timber.  See  Transverse  Strength, 
page  982. 

Deflection  of  wrought  iron.  See  Transverse 
Strength,  page  982;  and  Torsional  Strength, 
page  981. 

Density,  specific,  of  steam,  384,  385. 

Desiccation,  493  ;  drying  chambers,  494  ;  drying  by 
contact  with  heated  metallic  surfaces,  496 ;  drying 
grain,  496 ;  drying  wood,  496. 

Deterioration  of  coal  by  exposure,  412,  425. 

Dew-point,  392. 

Diagonal  rivet-joints  in  iron  plates,  strength  of,  638. 

Diamond  weight,  141. 

Distillation  of  wood,  449. 

Drilled  holes,  influence  of,  on  strength  of  iron 
plates,  584. 

Drilled  wrought-iron  plates  and  punched  plates, 
tensile  strength  of: — Krupp  and  Yorkshire  iron, 
584  ;  Staffordshire  bar,  633. 

Drilled  steel  plates  and  punched  steel  plates,  tensile 
strength  of,  610,  611 ;  elongation,  611. 

Dry  measure,  English,  139 ;  French,  149. 

Ductility.     See  Elongation. 


Elasticity,  coefficient  of,  503. 

Elements,  mechanical,  296.  See  Mechanical  Prin- 
ciples, page  977. 

Ellipse,  problems  on,  13  ;  mensuration  of,  25. 

Elongation  of  cast  iron,  under  tensile  stress,  558, 
560. 

Elongation  of  steel,  under  tensile  stress: — bars, 
593-596,  598,  601,  605-611,  613-615,  624. 

Elongation  of  timber,  under  tensile  stress,  545,  546. 

Elongation  of  wrought  iron,  under  tensile  stress : — 
bars,  572-577»  580,  581,  624;  plates,  577,  578,  583; 
holes  in  plates,  584,  615. 

Engines,  pumping,  water-works,  947. 

Evaporation,  cold  by,  376. 

Evaporation,  spontaneous,  491. 

Evaporative  efficiency  of  steam-boilers,  768. 

Evaporative  performance  of  steam  boilers,  768;  nor- 
mal standards,  768;  heating  power  of  fuels,  769; 


stationary  boilers  at  Wigan,  771,  811;  performance 
of  a  marine-boiler  at  Wigan,  781,  809,  816-821; 


performance  of  a  marine-boiler  at  Newcastle-on- 
Tyne,  785,  807,  816-821;  performance  of  a  marine- 
boiler  at  Keyham  Factory,  790;  performance  of 
American  coals  in  a  stationary  boiler,  791;  and  in 
a  marine-boiler,  795,  810;  stationary  boilers  in 
France,  796,  812. 

Locomotive  boilers,  798,  805,  813,  817-821; 
portable  steam-engine  boilers,  801,  814,  817-821. 

Relations  of  grate-area  and  heating-surface  to 
evaporative  performance,  802;  general  formulas 
for  practical  use,  816. 

Evaporative  power  of  coal— English,  414. 

Expansion  by  heat,  335;  linear  expansion  of  solids, 
with  table,  335;  expansion  of  liquids,  338;  expan- 
sion of  water,  with  table,  338-341;  Rankine's  for- 
mula for  expansion  of  water,  340;  table  of  the  ex- 
pansion of  liquids,  342;  expansion  of  gases,  342; 
expansion  of  air,  343-345;  table  of  the  expansion 
of  air  and  other  gases,  343. 

Expansion  of  air,  work  of,  at  constant  temperature, 
899;  adiabatically,  904. 

Explosive  force,  resistance  of  steel  and  iron  to,  622. 

Extension  of  iron  under  stress.     See  Elongation. 

Extension  of  timber  under  stress,  545,  546. 

Extension  of  steel  under  stress.     See  Elongation. 


Factors  of  safety  for  cast  iron,  wrought  iron,  steel, 
625;  iron  chains,  678,  679;  timber,  foundations, 
mason-work,  625;  ropes,  626,  674;  dead  load,  live 
load,  626;  screwed  bolts  and  nuts,  68 1;  cast-iron 
water-pipes,  936. 

Fans  or  ventilators,  924.  See  Air-machinery,  page 
971. 

Fires,  open,  heating  by,  488. 

Fire-wood: — French  wood-measure,  149,  154,  443; 
American  measure,  186,  443;  moisture  in  fire- 
wood, 439,  441 ;  composition,  441,  442. 

Flax-mills,  machinery  of,  resistance  of,  957;  horse- 
power required,  959. 

Floatation,  axis  of,  277;  plane  of,  277. 

Flow  of  air  and  other  gases,  891;  discharge  of  gases 
through  orifices,  891;  anemometer,  892;  outflow 
of  steam  through  an  orifice,  893. 

Flow  of  air  through  pipes  and  other  conduits, 
894;  flow  of  compressed  air  through  pipes,  896. 

Ascension  of  air  by  difference  of  temperature, 
897. 

Flow  of  water,  929;  flow  through  orifices,  929;  co- 
efficients of  discharge,  930;  Mr.  Bateman's  experi- 
ments, 930. 

Flow  through  a  submerged  nozzle,  Mr.  Brown- 
lee's  experiments,  931. 

Flow  over  waste-boards,  weirs,  &c.,  932. 
Flow  in  channels,  pipes,  and  rivers,  932;  limits 
of  velocity  at  the  bottom  of  a  channel,  934. 
Cast-iron  water-pipes,  934. 
Cast-iron  gas-pipes,  936. 

Flue-tubes,  large,  resistance  of,  to  collapsing  pres- 
sure, 696. 

Fluid  bodies,  276. 

Fluid-compressed  steel,  614. 

Fluids,  pressure  of,  276. 

Fontaine's  turbine,  942. 

Foot  and  its  multiples,  129;  equivalent  value  in 
French  measures,  1511,  153,  154,  156. 

Foot,  square: — decimal  parts,  in  square  inches,  138. 

Forces  in  equilibrium,  271.  See  Mechanical  Prin- 
ciples, page  976. 

Form  of  specimen,  influence  of,  on  the  tensile 
strength  of  iron,  574,  584. 

Fourneyron  turbine,  940. 

Framed  work,  strength  of,  697.  See  Strength  o} 
Elementary  Constructions,  page  979. 

French  coals,  420.     See  Coal,  page  972. 


INDEX. 


—    FRICTION    — 


—    HEAT    — 


975 


Friction  of  solid  bodies,  722:— journals,  722;  flat 
surfaces  in  contact,  723;  friction  on  rails,  724; 
work  absorbed  by  friction,  725;  horse-power 
absorbed  by  friction,  726. 

Friction  of  leather  belts,  744,  748. 

Frictional  resistance  of  rivetted  plates,  570. 

Frictional  resistance  of  shafting,  763: — resistance  of 
journals  of  horizontal  shafting,  722-726,  763;  up- 
right shafting,  763;  table  based  on  Mr.  Webber's 
data,  764. 

Ordinary  data:— by  Mr.  Tweddell,  763;  by  Mr. 
Westmacott,  Mr.  Walker,  M.  Cornut,  Mr.  R. 
Davison,  Mr.  Webber,  766. 

Frictional  resistance,  951: — steam-engines,  951; 
tools,  951;  wood-cutting  machines,  954;  grind- 
stones, 955 ;  colliery  winding-engines,  956;  wag- 
gons in  coal  pits,  956;  machinery  of  flax-mills, 
957;  machinery  of  woollen  mills,  959;  conveyance 
of  grain,  960;  traction  on  common  roads  and  on 
fields,  961 ;  resistance  on  railways,  965 ;  on  street 
tramways,  966. 

Frictional  wheel-gearing,  741. 

Frigorific  mixtures,  373. 

Fuel,  artificial,  411,  416,  420,  432,  449. 

Fuel,  combustible  elements  of,  398.  See  Combustible 
Elements,  page  973. 

Fuels,  409: — coal,  409;  coke,  418,  430;  lignite  and 
asphalte,  436;  wood,  439;  wood-charcoal,  444; 
peat,  452;  peat- charcoal,  455;  tan,  455;  straw,  456; 
liquid  fuels,  456;  coal-gas,  457;  gaseous  fuel,  922. 

Fuels,  heating  power  of,  269. 

Fuels,  inferior,  equivalent  weights  of  best  coal  and, 
820. 

Fuels  in  France,  weight  and  specific  gravity  of, 
207. 

Fuels,  liquid,  456.     See  Liquid  Fuels,  page  976. 

Fuels  patent,  411,  416;  Warlich's,  411;  Wylam's, 
411;  Mezaline's,  411;  Barker's,  411;  Holland's, 
411. 

Fusibility  of  solids,  363. 

Fusion  of  solid  bodies,  latent  heat  of,  367. 

Fuss,  German,  values  of,  161. 


Gallon:  definition  of,  volume  of,  relative  weight  of 
water  and  its  volume  in  gallons,  125;  imperial 
standard  measure  of  capacity,  128;  French  equiv- 
alent, 154,  157;  American  gallon,  186. 

Galloway  boiler,  trials  of,  771-777. 

Gaseous  fuel,  922;— Wilson's,  922;  Dowson's,  92.3. 

Gaseous  products  of  combustion,  quantity  of,  by 
weight,  400;  by  volume,  401;  surplus  air,  402, 
407;  specific  heat  of  products,  408;  products  for 
coal,  426,  428;  for  coke,  435;  for  wood,  443;  for 
wood-charcoal,  452. 

Gaseous  steam,  specific  heat  of,  353,  384;  total  heat 
of,  384;  density,  384,  385. 

Gases,  air  and  other,  flow  of,  891.  See  Flow  of  A  ir, 
page  974. 

Gases,  expansion  of,  342;  of  air,  343-345;  table,  343; 
compression  of,  345;  relations  of  the  pressure, 
volume,  and  temperature  of  gases,  346,  351; 
special  rules  for  one  pound  weight  of  a  gas  349  • 
specific  heat  of,  354,  358,  363. 

Gases  concerned  in  the  combustion  of  fuel,  composi- 
tion and  combining  equivalents  of,  398,  399;  vol- 
ume of,  399. 

Gases  and  vapours,  weight  and  specific  gravity  of, 
216. 

Gases  and  vapours,  mixture  of,  392: — hygrometry, 
392;  hygrometers,  393;  properties  of  saturated 
mixtures  of  air  and  aqueous  vapour,  with  table, 

„  394,  396. 

Gases,  liquefaction  and  solidification  of,  372. 

Gas-coke,  heat  of  combustion,  402. 

Gas-engines,  918.    See  Air  Machinery,  page  971. 

Gas-pipes,  cast-iron,  thickness  of,  936. 

Gas-thermometers,  325. 


|    Gearing.     See  Mill-gearing,  page  977. 

Gearing  by  ropes,  753.  See  Rope-gearing,  page 
978. 

Geometrical  problems,  i :— on  straight  lines,  i; 
straight  lines  and  circles,  5;  circles  and  rectilineal 
figures,  8;  ellipse,  13;  parabola,  17;  hyperbola, 
18;  cycloid  and  epicycloid,  19;  catenary,  20; 
circles,  21;  plane  trigonometry,  21;  mensuration 
of  surfaces,  23;  mensuration  of  solids,  27;  men- 
suration of  heights  and  distances,  30; 

Girders,  strength  of: — warren-girder,  699;  lattice- 
girder,  708;  strut-girder,  708. 

Glass,  tensile  strength  of,  629;  crushing  strength, 
632. 

Gold,  weight  of,  219-221;  rule  for  the  weight  of 
gold  wire,  224. 

Gold  wire,  tensile  strength  of,  628. 

Goods  carried  on  the  Bombay,  Baroda,  and  Central 
India  Railway,  weight  and  volume  of,  213. 

Grain  (weight),  140;  equivalent  value  in  French 
measures,  155-157. 

Gram,  apparatus  for  conveyance  of,  resistance  of, 
960. 

Gramme,  150;  equivalent  values  in  English  mea- 
sures, 155,  157. 

Graphite: — weight  and  specific  gravity,  207;  heat 
of  combustion,  402. 

Gravity,  277;  action  on  inclined  planes,  285;  centre 
of,  287;  work  done  by  it,  315.  See  Mechanical 
Principles,  page  977. 

Guibal's  fan,  925. 

Gun-metal,  weight  of:— tabulated  weights,  219-221; 
multiplier  for  the  weight  of  gun-metal  balls,  258,- 

Gun-metal,  tensile  strength  of,  626,  627. 

Gyration,  centre  of,  radius  of,  288. 

H 

Hammers,  atmospheric,  967. 

Hardened  or  tempered  steel,  tensile  strength  of, 

594,  602,  603,  613. 
Head  of  pressure,  276. 
Heat,  371:— 

Thermometers,  317;  air-thermometers,  325. 

Pyrometers,  326,  966;  air-pyrometer,  327. 

Luminosity  at  high  temperatures,  328. 

Radiation  of  heat,  329. 

Conduction  of  heat,  331. 

Convection  of  heat,  331. 

Mechanical  theory  of  heat,  332. 

Mechanical  equivalent  of  heat,  718. 

Expansion  by  heat,  335;  solids,  335;  liquids, 
338;  gases,  342. 

Compression  of  gases,  345. 

Relations  of  the  pressure,  volume,  and  tempera- 
ture of  air  and  other  gases,  346. 

Special  rules  for  one  pound  weight  of  a  gas, 
349- 

Table  of  the  volume,  density,  and  pressure  of 
air  at  various  temperatures,  351. 

Specific  heat,  352;  of  solids,  358,  359;  of  water, 
353,  354;  of  air  and  other  gases,  354,  363. 

Fusibility  or  melting  points  of  solids,  363. 

Latent  heat  of  fusion  of  solid  bodies,  367. 

Boiling  points  of  liquids,  368. 

Latent  heat  and  total  heat  of  evaporation  of 
liquids,  370. 

Liquefaction  and  solidification  of  gases,  372. 

Sources  of  cold,  373;  frigorific  mixtures,  374. 

Cold  by  evaporation,  376. 

Heat  evolved  by  the  combustion  of  fuel,  402, 
404,  405. 

Temperature  of  combustion,  407. 

Heat,  transmission  of,  through  solid  bodies — from 

water  to  water  through  plates,  459;  heating  and 

evaporation  of  liquids  by  steam  through  metallic 

surfaces,  461;  cooling  of  hot  water  in  pipes,  469; 


9/6 


INDEX. 


—    HEAT    — 


—    MECHANICAL    — 


cooling  of  hot  wort  on  metal  plates  in  air,  470; 

cooling  of  hot  wort   by  cold  water  in   metallic 

refrigerators,  471;  condensation  of  steam  in  pipes 

exposed  to  air,  472;  condensation  of  vapours  in 

pipes  or  tubes  by  water,  475. 
Heat,  applications  of,  459.      See  Applications  of 

Heat,  page  971. 
Heating  of  solids,  497: — cupola  furnace,  497;  plaster 

ovens,   497 ;   metallurgical  furnaces,   497 ;    blast 

furnace,  498. 

Heating  of  water  by  steam  in  direct  contact,  490. 
Heating  power.     See  Combustion,  Heat  of,  page 

973- 

Heating  power  of  fuels,  769. 
Hectare,  149;  equivalent  value  in  English  measures, 

J?3' 
Heights  and  distances,  mensuration  of,  30. 

Hemp  ropes  for  gearing,  753. 

Horses,  labour  of,  720;  work  of,  in  carrying  loads, 

721. 

Horses,  performances  of,  on  roads  in  France,  964. 
Horse-power,  718;  absorbed  by  friction,  726. 
Horse-power  of  round  shafting,  760,  762;  absorbed 

by  friction,  763. 

Holtzapffel's  Birmingham  wire-gauge,  131;  his  Bir- 
mingham metal-gauge,  or  plate-gauge,  131 ;  his 

Lancashire  gauge,  132  ;  values  in  parts  of  an  inch 

of  his  wire-gauge,  134. 
Horse-power  of  toothed  wheels,  737. 
Horse-power  transmitted  through  leather  belts,  743- 

747- 
Horse-power  required  for  a  flax-spinning  mill,  959 ; 

for  a  woollen  mill,  960. 
Hot-air  engines,    917 ;    (first-class),    Rider's,   917 ; 

(second-class),  Belou's  hot-air  engine  at  Cusset, 

918. 

Hydraulic  machines,  Armstrong's,  950. 
Hydraulic  press,  950 ;  strength  of,  687. 
Hydraulic  rams,  949. 
Hygrometers,  393  ;  Daniell's,  393  ;  Regnault's,  393 ; 

Mason's  wet  and  dry  thermometers,  394. 
Hygrometry,  392 ;  dew-point,  392. 
Hygroscopic  water  in  British  coals,  416. 
Hyperbola,  problems  on,  18 ;  mensuration  of,  25. 


Ice,  its  volume,  weight,  melting  point,  specific  heat, 
127  ;  specific  heat  of,  127,  353  ;  specific  gravity  of, 
127 ;  melting  point  of,  364  ;  latent  heat^  of  fusion 
of,  367;  frigorific  mixtures  with,  375. 

Ice-making  machine,  Siebe's,  373. 

Inches,  their  multiples,  129;  their  decimal  values 
in  parts  of  a  foot,  135  ;  fractional  parts  and  deci- 
mal equivalents,  135  ;  equivalent  value  in  French 
measures,  151,  153,  154,  156. 

Inclined  plane,  action  of  gravity  upon,  285 ;  prin- 
ciple of,  306 ;  identity  of  it  with  the  lever,  308 ; 
work  done  with  it,  314. 

India-rubber  belts,  strength  of,  680,  750. 

Indian  coals,  423.     See  Coal,  page  972. 

Inertia,  moment  of,  288. 

Iron,  weight  and  specific  gravity: — of  wrought  iron, 
202,  217,  218 ;  of  cast  iron,  203,  217,  218. 

Iron  and  other  metals,  tables  of  weight  of,  217. 
See  Weight  of  Iron  and  Other  Metals,  page  983. 

Iron,  wrought,  weight  of;  data  for  the  weight,  217, 
218  ;  rules  for  the  weight,  223  ;  tabulated  weights, 
219-221 ;  weight  of  French  galvanized  wire,  225  ; 
special  tables  of  weight  of  wrought-iron  bars, 
plates,  sheets,  hoops,  wire,  >and  tubes,  226-250 ; 
multipliers  for  wrought  -  iron  balls,  258.  See 
Weight  of  Iron  and  Other  Metals,  page  983. 

Iron,  hammered ;  multipliers  for  weight  of  bars, 
plates,  &c.,  220. 

Iron  wire  ropes,  strength  and  weight  of,  674-677. 


j 

Jonval  turbine,  942. 
Joule's  equivalent,  332. 

Journals  of  shafts,  766 ;  of  railway  axles,  767. 
Journals,  friction  of,  in  their  bearings,  722  ;  coeffi- 
cients, 724 ;  work  of,  725  ;  horse-power,  726. 

K 

Kilogramme,  standard,  146,  150;  equivalent  value  in 
English  measures,  155,  157. 

Kilometre,  147;  equivalent  value  in  English  mea- 
sures, 150,  156. 

L 

Labour,  718.     See  Work,  page  984. 

Lama,  work  of,  in  carrying  loads,  721. 

Lancashire  wire-gauge,  132. 

Land  measure,  English,  lineal,  130 ;  superficial,  137; 
French,  superficial,  149. 

Latent  heat  of  fusion  of  solid  bodies,  367 ;  of  eva- 
poration of  liquids,  370,  373 ;  of  steam,  380. 

Lattice-girder,  parallel,  strength  of,  708. 

Lead,  weight  of;  tabulated  weights,  219-221 ;  multi- 
plier for  the  weight  of  lead  bars,  plates,  &c.,  220; 
special  table  of  the  weight  of  lead  pipes,  269. 
See  Weight  of  Iron  and  OtJter  Metals,  page  983. 

Lead,  tensile  strength  of,  627. 

Leather-belting,  tensile  strength  of,  679.  See  Ten- 
sile Strength,  page  980. 

Lenoir's  gas-engine,  919. 

Lever,  296;  work  done  with  it,  313. 

Lignite : — composition,  422,  436,  438  ;  heat  of  com- 
bustion, 423,  437,  438  ;  hygrometric  moisture,  436; 
weight  and  specific  gravity,  207,  436,  437. 

Lineal  measure,  English,  129  ;  French,  147. 

Liquefaction  and  solidification  of  gases,  372. 

Liquid  fuels,  456 ;  petroleum,  456 ;  petroleum-oils, 
456 ;  schist-oil,  457  ;  pine-wood  oil,  457. 

Liquid  measure,  English,  138 ;  French,  149. 

Liquids,  expansion  of,  338;  of  water,  338-341;  table, 
342 ;  specific  heat  of,  362  ;  boiling  points  of,  368 ; 
latent  heat  and  total  heat  of  evaporation  of,  370, 372. 

Liquids,  weight  arid  specific  gravity,  215. 

Litre,  defined,  147,  149  ;  equivalent  value  in  English 
measures,  154,  157. 

Logarithms  of  numbers,  32 ;  table,  38. 

Logarithms,  hyperbolic,  of  numbers,  35  ;  table,  60. 

Lubrication,  957. 

Luminosity  at  high  temperatures,  328. 

M 

Machines  for  raising  water,  944.  See  Water,  page 
982. 

Malleable  cast  iron,  strength  of,  561. 

Mass,  287. 

Materials,  strength  of,  500.  See  Strength  of  Mater- 
ials, page  979. 

Mathematical  tables,  32. 

Measurement,  compound  units  of,  equivalents  of 
French  and  English,  157 ;  weight,  pressure,  and 
measure,  157 ;  volume,  area,  and  length,  158; 
work,  158 ;  heat,  speed,  money,  159. 

Measurement,  principal  units  of,  129. 

Measures,  English  and  French,  approximate  equi- 
valents of,  156. 

Mechanical  centres,  287.  See  Mechanical  Prin- 
ciples, page  977. 

Mechanical  elements,  296.  See  Mechanical  Prin- 
ciples,^^ 977. 

Mechanical  equivalent  of  heat,  332. 

Mechanical  theory  of  heat,  332;  Joule's  equivalent, 
332;  illustrations,  333. 

Mechanical  principles,  fundamental,  271. 

Forces  in  equilibrium: — solid  bodies,  parallel- 
ogram of  forces,  polygon  of  forces,  moments  of 
forces,  271;  the  catenary,  273;  centrifugal  forces, 


INDEX. 


—    MELTING    — 


—    PRESSURE    — 


977 


274;  parallel  forces,  275;  parallelepiped  of  forces, 
276. 

Fluid  bodies: — Pressure  of  fluids,  head  of  pres- 
sure, buoyancy,  axis  of  floatation,  plane  of  float- 
ation, stability,  metacentre,  276. 

Motion: — uniform  motion,  velocity,  accelerated 
and  retarded  motion,  277. 

Gravity: — falling  bodies,  time,  velocity,  and 
height  of  fall,  278;  rules  for  the  action  of  gravity, 
279;  table  of  velocity  due  to  height,  280;  table  of 
height  due  to  the  velocity,  281;  table  of  height  of 
fall  and  velocity,  for  the  time,  282. 

Accelerated  and  retarded  motion  in  general: — 
general  rules  for  accelerating  forces,  282 ; 

Action  of  gravity  on  inclined  planes,  special 
rules  for  the  descent  on  inclined  planes,  285. 

Average  velocity  of  a  moving  body  uniformly 
accelerated  or  retarded,  286. 
Mass,  287. 

Mechanical  centres: — centre  of  gravity,  287; 
centres  of  gyration,  radius  of  gyration,  moment 
of  inertia,  288;  centre  of  oscillation,  290;  the  pen- 
dulum, 291 ;  centre  of  percussion,  294. 

Central  forces:— centripetal  force,  centrifugal 
force,  rules,  294. 

Mechanical  elements: — lever,  296;  pulley,  302; 
wheel  and  axle,  305;  inclined  plane,  306;  identity 
of  the  inclined  plane  and  the  lever,  308 ;  wedge, 
309;  screw,  311. 

Work: — definitions,  312;  work  done  with  the 
lever,  313;  with  the  pulley,  313;  with  the  wheel 
and  axle,  314;  with  the  inclined  plane,  314;  with 
the  wedge,  315;  with  the  screw,  315;  by  gravity, 
315;  work  accumulated  in  solid  bodies,  315;  work 
done  by  percussive  force,  316. 
Melting  points  of  solids,  363;  table,  364. 
Men,  labour  of,  718;  work  of,  in  carrying  loads,  720. 
Mensuration  of  surfaces,  23;  solids,  27;  heights  and 
distances,  30;  circle,  24;  ellipse,  parabola,  hyper- 
bola, cycloid,  and  epicycloid,  curvilineal  figures,  25. 
Metacentre,  277. 

Metals,  melting  points  of,  363;  table,  365. 
Metals,  weight  and  specific  gravity  of,  202,  626,  627. 
Metre,    standard,    146,    147;    equivalent    value    in 

English  measure,  150,  156. 

Metric  system,  French;  countries  in  which  it  is 
legalized,  146;  mile,  129;  equivalent  value  in 
French  measures,  151,  156. 

Mill-gearing,    727;    toothed    gear,    727;    frictional- 
wheel- gearing,  741;  belt-pulleys  and  belts,  742; 
rope-gearing,  753;  shafting,  756. 
Mine  ventilators,  925.     See  Air  Machinery,  page 

971. 
Mineral   substances,    sundry,    weight  and  specific 

gravity  of,  205. 

Mitre  wheels.     See  Toothed  WJieels,  page  981. 
Moisture  in  coal,  416;  in  coke,  434;  in  wood,  439; 

in  wood-charcoal,  451. 
Moments  of  forces,  271. 
Money,  190: — 

Great  Britain  and  Ireland: — value,  weight,  and 
composition  of  coins,  190. 

France:  —  value,  weight,  and  composition  of 
coins,  and  equivalent  value  in  English  money, 
190. 

Germany: — currency  established  in  1872,  old 
currency,  191. 

Hanse  towns: — Hamburg,  Bremen,  Lubeckjigi. 
Austria,  192. 
Russia,  192. 

Holland,  Belgium,  Denmark,  Sweden,  Nor- 
way, 192. 

Switzerland,  193. 

Spain,  Portugal,  Italy,  Turkey,  Greece  and 
Ionian  Islands,  Malta,  193. 

Egypt,  Morocco,  Tunis,  Arabia,  Cape  of  Good 
Hope,  194. 


Indian  Empire,  195. 

China,  Cochin-china,  Persia,  Japan,  Java,  195. 
United  States  of  America,  195. 
Canada — British  North  America,  196. 
Mexico,  196. 

Central  America  and  West  Indies : — British  West 
Indies,  Cuba,  Guatemala,  Honduras,  Costa  Rica, 
St.  Domingo,  196. 

South  America : — Colombia,  Venezuela,  Ecua- 
dor, Guiana,  Brazil,  Peru,  Chili,  Bolivia,  Argen- 
tine Confederation,  Uruguay,  Paraguay,  196. 

Australasia,  197. 

Mortar,  strength  of: — tensile,  629;  crushing,  632. 
Motion,  277;  accelerated  and  retarded  motion,  282. 
Mules,  work  of,  in  carrying  loads,  721. 
Muntz's  metal,  tensile  strength  of,  627. 
Music-wire  gauge,  132. 

N 

Nautical  measure,  130. 
Needle-gauge,  132. 
Noria,  947. 

North  Moor  Foundry  turbine,  942. 
Notched  form   of  specimen   of  iron,   comparative 
strength  of,  574;  of  steel,  622. 

o 

Oscillation,  centre  of,  290. 

Otto  &  Langen's  gas-engine,  919;  Otto  engine,  921. 
Outflow  of  steam  through  an  orifice,  893. 
Overshot  water-wheel,  938. 

P 

Paddle  water-wheel,  938. 

Palladium  wire,  tensile  strength  of,  628. 

Parabola,  problems  on,  17;  mensuration  of,  25. 

Parallel  forces,  275. 

Parallelogram  of  forces,  271. 

Parallelepiped  of  forces,  276. 

Peat,  volume,  weight,  and  specific  gravity  of,  207. 

Peat,  453: — its  origin,  453;  composition,  453,  454; 
moisture,  453;  air-dried  peat,  454;  products  of 
distillation,  454;  heat  of  combustion,  455. 

Peat-charcoal,  455 ;  heat  of  combustion,  455. 

Pendulum,  291. 

Percussion,  centre  of,  294. 

Percussive  force,  work  done  by,  316. 

Phosphor-bronze,  tensile  strength  of,  628,  629. 

Piles,  timber,  strength  of,  646. 

Pipes,  flow  of  air  through,  894,  896. 

Pipes,  flow  of  water  in,  932. 

Pipes,  heating  of  water  by  steam- in,  463;  cooling 
of  water  in,  469;  condensation  of  steam  in,  472, 
475;  heating  of  rooms  by  hot  water  in,  481. 

Pipes,  gas,  cast-iron,  thickness  of,  936. 

Pipes,  water,  cast-iron,  dimensions,  weight,  and 
strength  of,  934. 

Pipes,  weight  of,  251,  253-258,  262,  269,  936. 

Piping,  screwed  iron,  Whitworth's  standard  pitches 
for,  683. 

Pitch  of  toothed  wheels,  728. 

Pivots,  friction  of,  in  their  bearings,  work  of,  725; 
horse-power,  726. 

Plaster  of  Paris,  tensile  strength  of,  629,  630. 

Platinum  wire,  rule  for  weight  of,  224. 

Platinum  wire,  tensile  strength  of,  628. 

Poncelet's  undershot  water-wheel,  937. 

Pound  (weight),  140, 141;  avoirdupois  and  troy  com- 
pared, 140;  equivalent  in  French  measures,  155. 

Power  transmitted  by  shafting,  760;  work  for  one 
turn,  horse-power,  760,  762. 

Power.     See  Work,  984. 

Press,  hydraulic,  949. 

Pressure  of  fluids,  head  of  pressure,  276. 


INDEX. 


—    PRESSURE    — 


—    STEAM    — 


Pressure  of  vapours  at  212°  F.,  370. 

Pressure  of  water,  measures  of  pressure,  126. 

Products  of  combustion,  400.  See  Gaseous  Pro- 
ducts, page  975. 

Pulley,  302;  work  done  with  it,  313. 

Pulleys,  belt,  742.     See  Belt-pulleys,  page  971. 

Pumps,  reciprocating,  944;  centrifugal,  946;  end- 
less-chain, 947;  steam- vacuum,  969. 

Pumping  engines,  water-works,  948. 

Punched  holes,  iron  plates  and  bars  with,  strength 
of,  584,  633. 

Punched  steel  plates,  tensile  strength  of,  610,  611, 
642. 

Punching  strength  of  wrought  iron,  587. 

Pyrometers,  326;  Wedgewood's,  327;  DanielPs,  327; 
air-pyrometer,  327;  Wilson's  pyrometer,  327;  Sie- 
mens' pyrometer,  967. 

R 

Rails,  railway,  transverse  strength  and  deflection  of, 
661.  See  Transverse  Strength,  page  981. 

Rails,  friction  on,  724. 

Railway  axles,  proportions  of,  767. 

Railway  rails,  transverse  strength  and  deflection  of, 
661.  See  Transverse  Strength,  page  981. 

Railways,  resistance  on,  965. 

Rams,  hydraulic,  948. 

Reciprocals  of  numbers,  37;  table,  118. 

Reciprocating  pumps,  944. 

Regnault's  air-thermometer,  326. 

Resistance,  frictional,  763,  951.  See  Frictional 
Resistance,  page  975. 

Resistance  of  air  to  the  motion  of  flat  surfaces,  897. 

Resistance  of  materials.  See  Strength  of  Materials, 
page  979. 

Resistance  on  railways,  965. 

Resistance  on  street  tramways,  966. 

Rider's  hot-air  engine,  917. 

Rivers,  flow  of  water  in,  932. 

Rivets  and  rivet  iron,  shearing  strength  of,  570,  588. 

Rivet-joints,  tensile  strength  of,  633. 

Rivet-joints,  proportions  of,  641. 

Rivetted  plates,  frictional  resistance  of,  570. 

Rivetted  wrought-iron  joists,  transverse  strength  of, 
653- 

Roads,  resistance  to  traction  on,  961,  962. 

Roofs,  strength  of,  713. 

Root's  blower,  927. 

Rope-gearing,  753: — transmission  of  power  by  hemp- 
ropes,  753;  use  of  wire-ropes,  by  M.  Him,  754; 
cotton-ropes,  by  Mr.  Ramsbottom,  755. 

Ropes,  tensile  strength  and  weight  of,  673.  See 
Tensile  Strength,  page  980. 

Rylands  Brothers,  wire-gauge,  133,  247. 


Safety,  factors  of,  625.    See  Factors  of  Safety,  p.  974. 

Salts,  saturated  solutions  of,  boiling  points  of,  369. 

Screw,  311;  work  done  with  it,  315. 

Screw-threads,  68 1. 

Sea-water;  its  volume,  weight  and  composition,  126. 

Sea-water,  composition  of,  126;  weight  of,  126; 
specific  gravity  of,  126;  boiling  point  of,  369,  370. 

Section  of  land,  189. 

Ser,  1 80. 

Shafting,  756;  transverse  deflection,  756;  overhung 
shafts,  757;  torsional  strength  and  deflection  of 
round  shafts,  758;  power  transmitted,  760;  net 
weight/of  shafting,  761;  table  of  the  strength  of 
round  Vrought-iron  shafting,  with  multipliers  for 
cast-iron  and  steel,  762. 

Frictional  resistance  of  shafting,  763. 

Shearing  strength  of  cast  iron,  561. 

Shearing  strength  of  steel,  595,  605;  deduction,  617. 

Shearing  strength  of  timber,  551. 


gases,  372. 


Shearing  strength  of  wrought  iron,  587:— Mr.  Edwin 
Clark,  570;  Swedish  iron,  581;  Mr.  Little's  ex- 
periments, 587;  chief-engineer  Shock's,  587;  con- 
clusion, 588. 

Shearing  stress,  definition  of,  500;  stress  in  beams 
and  plate-girders,  525. 

Shearing  stress,  torsional,  of  steel  bars,  620. 

Ships,  measure  for,  144. 

Siebe's  ice-making  machine,  373. 

Siemens'  pyrometer,  967. 

Silver,  weight  of,  219-221  ;  rule  for  the  weight  of 
silver  wire,  224. 

Silver  wire,  tensile  strength  of,  628. 

Sines,  cosines,  &c.,  of  angles,  30,  36;  tables,  103, 
no. 

Small  coal,  409,  420;  utilization  of,  410,  420;  wash- 
ing of,  411. 

Snow,  its  volume  and  weight,  127. 

Snow,  frigorific  mixtures  with,  375. 

Solder,  soft,  tensile  strength,  627. 

Solidification  and  liquefaction  of  gases, 

Solids,  mensuration  of,  27. 

Solids,  weight  and  specific  gravity  of,  199-214. 

Solids,  expansion  of,  335;  specific  heat  of,  359;  fusi- 
bility of,  363;  latent  heat  of  fusion  of,  367. 

Solids,  heating  of,  497.  See  Heating  of  Solids, 
page  976. 

Specific  gravity,  198.  See  Weight  and  Specific 
Gravity,  page  982. 

Specific  gravity  of  coals: — English,  414,  416,  417, 
424;  American,  418,  419;  French,  421;  Indian, 
424;  Australian,  424. 

Specific  gravity  of  timber,  539-543,  545~547- 

Specific  gravity  of  cast  iron,  554,  557. 

Specific  gravity  of  wrought  iron,  568,  578,  603. 

Specific  gravity  of  wrought  iron,  influence  of  wire- 
drawing on  specific  density,  247;  influence  of  cold- 
rolling,  578;  influence  of  stretching,  578. 

Specific  gravity  of  steel,  603. 

Specific  heat,  352:  —  specific  heat  of  water,  ice, 
steam,  127,  353;  of  water  at  various  temperatures, 
354;  of  air  and  other  gases,  128,  354;  of  gases  for 
equal  volumes,  358;  table  of  specific  heat  of  solids, 
359;  of  liquids,  362;  of  gases,  363;  specific  heat  of 
gaseous  steam,  384. 

Speeds  of  toothed  wheels,  727. 

Spontaneous  evaporation  in  open  air,  491. 

Springs,  steel,  strength  of,  671.  See  Transverse 
Strength,  page  981. 

Spur  wheels.     See  Toothed  Wheels,  page  981. 

Stability  of  floating  bodies,  277. 

Staybolts,  screwed,  and  flat  surfaces,  strength  of, 
685. 

Steam,  378. 

Physical  properties  of  steam,  378. 
Saturated  steam: — relation  of  the  temperature 
and  pressure,  378;  total  heat,  379;  latent  heat, 
380;  appropriation  of  its  constituent  heat  at  212°, 
380;  volume  and  density,  381;  relative  volume, 
382. 

Gaseous  steam,  383;  its  total  heat,  384;  specific 
heat,  384. 

Specific  density  of  gaseous  steam,  384;  of  satu- 
rated steam,  384. 

Density  of  gaseous  steam,  385. 
Table  of  the  properties  of  saturated  steam,  from 
32°  to  212°,  385,  386. 

Table  of  the  properties  of  saturated  steam  for 
high  pressures,  385,  387. 

Table  of  comparative  density  and  volume  of  air 
and  saturated  steam,  391. 

Steam,  expansive  working  of,  principles  of,  822- 
877. 

Practice  of,  879: — actual  performance  of  steam 
in  the  steam-engine,  879;  data  of  the  practical 
performance  of  steam  in  single-cylinder  condens- 
ing engines,  880;  in  compound  condensing  engines, 
88 1 ;  in  single-cylinder  non-condensing  engines, 


INDEX. 


—    STEAM    — 


—    STRENGTH    — 


979 


French 


883;  in  American  marine  engines,  £ 
stationary  engines,  886. 

General  deductions  from  the  data  of  the  actual 
performance  of  steam,  886: — single-cylinders  with 
steam  jackets,  condensing,  886;  non-condensing, 
886;  without  steam  jackets,  condensing,  887;  non- 
condensing,  887 ;  compound-cylinders  with  steam 
jackets,  condensing,  887;  proportional  ratios  of 
expansion  in  the  first  and  second  cylinders,  887; 
compound-engines  without  steam  in  jackets,  con- 
densing, 888. 

Conclusions  on  the  actual  performance  of  steam, 
888 ;  table  of  the  practical  performance  of  steam- 
engines,  890. 

Steam,  gaseous.     See  Gaseous  Steam,  page  975. 

Steam  in  the  cylinder,  compression  of,  878. 

Steam,  outflow  of,  through  an  orifice,  893. 

Steam-boilers,  evaporative  performance  of,  768. 

Steam-engine:  — 

Action  of  steam  in  a  single-cylinder,  822: — pres- 
sure of  steam  during  expansion  in  a  cylinder,  822; 
work  of  steam  by  expansion,  824;  clearance  in 
steam-cylinders,  827;  formulas  for  the  work  of 
steam  in  the  cylinder,  828 ;  initial  pressure  in  the 
cylinder,  829;  average  total  pressure  in  the  cylin- 
der, 830;  average  effective  pressure  in  the  cylin- 
der, 830;  period  of  admission  and  the  actual  ratio 
of  expansion,  830;  relative  performance  of  equal 
weights  of  steam  worked  expansively,  832;  pro- 
portional work  done  by  admission  and  expansion, 
833;  influence  of  clearance  in  reducing  the  per- 
formance of  steam  in  the  cylinder,  834. 

Table  of  ratios  of  expansion  of  steam,  with  rela- 
tive periods  of  admission,  pressure,  and  total  per- 
formance, 835,  836;  total  work  done  by  one  pound 
of  steam  expanded  in  a  cylinder,  838;  consump- 
tion of  steam  worked  expansively  per  horse-power 
of  total  work  per  hour,  840;  table  of  the  total 
work  done  by  one  pound  of  steam  of  100  Ibs. 
total  pressure  per  square  inch,  840,  841 ;  net  cylin- 
der capacity  relative  to  the  steam  expended  and 
work  done  in  one  stroke,  843;  table  of  relations  of 
net  capacity  of  cylinder  to  steam  admitted  and 
work  done,  844,  846. 

Compression  of  steam  in  the  cylinder,  878. 

Steam-engines,  internal  resistance  of,  951,  957. 

Steam-engine,  compound.  See  Compound  Steam- 
engine,  page  973. 

Steam-vacuum  pump,  969. 

Steel,  weight  and  specific  gravity  of,  202,  217. 

Steel,  weight  of: — data  for  the  weight,  217;  rules 
for  the  weight,  223;  tabulated  weights,  219-221; 
multiplier  for  weight  of  steel  bars,  plates,  &c., 
226;  special  tables  of  weight  of  steel  bars  and 
chisel-steel,  251,  259-261;  multiplier  for  steel  balls, 
258.  See  Weight  of  Iron  and  other  Metals,  page 
983- 

Steel,  strength  of,  593.    See  Strength  of  Steel. 

Steel  columns,  strength  of,  644,  646. 

Steel  springs,  strength  of,  671.  See  Transverse 
Strength,  page  981. 

Steel  wire,  tensile  strength  of,  617,  629. 

Steel  wire-ropes,  strength  and  weight  of,  674,  675, 
677. 

Stere,  149;  equivalent  value  in  English  measures, 
i54- 

Sterro-metal,  tensile  strength  of,  627. 

Stones,  specific  gravity  of,  203;  of  precious  stones, 
202. 

Stones,  strength  of: — tensile  strength,  629;  crushing 
strength,  631. 

Stoves  and  hot  air,  heating  by,  488. 

Strain,  definition  of,  500. 

Straw,  composition  of,  456. 

Strength  of  materials,  500: — kinds  of  stress,  500; 
elastic  strength,  ultimate  or  absolute  strength, 
500;  work  of  resistance  of  material,  501;  coeffi- 
cient of  elasticity,  503;  transverse  strength  of 


homogeneous  beams,  504 ;  forms  of  beams  of  uni- 
form strength,  517;  shearing-stress  in  beams  and 
plate  -  girders,  525;  deflection  of  homogeneous 
beams  and  girders,  527;  uniform  beams  supported 
at  three  or  more  points,  533;  torsional  strength 
and  deflection  of  shafts,  534. 

Strength  of  timber,  537;  strength  of  cast  iron, 
553;  strength  of  wrought  iron,  567;  strength  of 
steel,  593;  recapitulation  of  data  on  the  direct 
strength  of  iron  and  steel,  623. 

Working  strength  of  materials — factors  of  safety, 
625. 

Strength  of  copper  and  other  metals,  626;  tensile 
strength  of  wire  of  various  metals,  628. 

Strength  of  stone,  bricks,  &c.,  629. 
Strength,  transverse,  of  homogeneous  beams,  503. 

See  Transverse  Strength,  page  981. 
Strength,  uniform,  forms  of  beams  of,  517:  rectan- 
gular semi-beams,  518—520;  flanged  semi-beams, 
519,  521;  semi-beams  of  circular  or  elliptical  sec- 
tions, 519, 521;  rectangular  beams,  52 1-523;  flanged 
beams,  523,  524;  stress  in  curved  flange,  525. 
Strength  of  framed  work,  697:  illustrations  of  stress 
in  framed  work,  697 ;  elementary  truss,  698 ;  framed 
girders — the  Warren-girder,  699-708;  parallel  lat- 
tice-girder, 708;  parallel  strut-girder,  708-713; 
roofs,  713-717. 

Strength  of  cast  iron,  553: — Mr.  Hodgkinson's  ex- 
periments on  tensile  and  compressive  strength, 
553;  Dr.  Anderson's,  555;  strength  as  affected  by 
the  mass  of  the  metal,  555;  as  affected  by  cold- 
blast  and  hot-blast,  555;  increased  by  remelting, 
Mr.  Bramwell's  experiments,  556;  Sir  Wm.  Fair- 
bairn's  557;  Major  Wade's,  557. 

Elastic  tensile  and  compressive  strength  of  cast 
iron,  558. 

Shearing  strength  of  cast  iron,  561. 

Malleable  cast  iron,  561. 

Transverse  strength  of  cast  iron: — Mr.  Barlow's 
experiments,  561;  Mr.  Edwin  Clark's  data,  562; 
Mr.  Hodgkinson's  data,  563;  test  bars,  564. 

Transverse  deflection  and  elastic  strength  of 
cast  iron;  formulas,  564. 

Torsional  strength  and  deflection  of  cast  iron; 
Mr.  Dunlop's  data;  formulas,  565. 

Recapitulation  of  data  on  the  direct  strength  of 
cast  iron,  623. 

Factors  of  safety,  625. 
Strength  of  cement: — Portland,  630,  632;  Roman, 

630. 

Strength  of  elementary  constructions,  633 ;  rivet- 
joints,  633 ;  pillars  or  columns,  643 ;  cast-iron 
flanged  beams,  647 ;  wrought-iron  flanged  beams 
or  joints,  buckled  iron  plates,  660 ;  railway  rails, 
661 ;  steel  springs,  671 ;  ropes,  673  ;  chains,  677  ; 
leather  belting,  679 ;  bolts  and  nuts,  680 ;  iron 
piping,  683 ;  screwed  stay-bolts  and  stayed  sur- 
faces, 685. 

Hollow  cylinders: — tubes,  pipes,  boilers,  &c., 
687.  Framed  work: — cranes,  girders,  roofs,  &c., 
697. 

Strength  of  glass: — tensile,  629;  crushing,  632. 
Strength  of  hollow  cylinders,  687: — resistance  to 
internal  or  bursting  pressure,  687;  hydraulic  press, 
687;  its  bursting  strength,  with  formulas,  689,  690; 
longitudinal  resistance,  692;  resistance  of  wrought- 
iron  tubes,  692,  695;  a  Lancashire  boiler,  693;  a 
cylindrical  marine  boiler,  693 ;  cast-iron  pipes,  694 ; 
lead  pipes,  696. 

Resistance  to  collapsing  pressure,  694: — solid- 
drawn  tubes,  with  formulas,  694;  large  flue-tubes, 
696. 

Strength  of  steel,  593: — Sir.  Wm.  Fairbairn's  experi- 
ments, 568;  Mr.  Kirkaldy's  early  experiments  on 
tensile  strength  of  bars  and  plates,  593;  strength 
of  hematite  steel,  594;  strength  of  Krupp  steel, 

595- 

Experiments  of  the  Steel  Committee  of  Civil 
Engineers,  596;  tensile  strength  of  tempered  steel; 
experiments  at  Woolwich  Dockyard,  602 ;  strength 


980 


INDEX. 


—    STRENGTH 


—    TENSILE    — 


of  Fagersta  steel,  604;  Siemens'-steel  plates  and 
tyres,  612;  Whit  worth's  fluid-compressed  steel, 
614. 

Chernoff's  experiments  on  the  influence  of  tem- 
perature on  the  structure  of  steel,  616. 

Tensile  strength  of  steel  wire,  617,  629. 

Shearing  strength  of  steel,  617. 

Transverse  strength  and  deflection  of  steel: — 
data  and  formulas,  617,  618. 

Torsional  strength  and  deflection  of  steel  bars, 
619. 

Strength  of  steel  relatively  to  the  proportion  of 
constituent  carbon,  621,  664. 

Resistance  of  steel  and  iron  to  explosive  force, 
622. 

Recapitulation  of  data  on  the  direct  strength  of 
steel,  623. 

Factors  of  safety,  625. 

Strength  of  stones: — tensile,  629;  crushing,  631. 
Strength  of  timber,  537: — general  conditions  of 
strength,  538;  Mr.  Laslett's  experiments  on  trans- 
verse, tensile,  and  compressive  strength,  538, 
647;  Mr.  Fincham's  experiments  on  the  transverse 
strength  of  soft  woods,  542;  Mr.  Barlow's  experi- 
ments on  transverse  strength,  547. 

Transverse  strength  of  beams  of  large  scantling, 
Mr.  Maclure's  experiments,  542 ;  Mr.  Edwin 
Clark's  experiments,  544;  Mr.  G.  Graham  Smith's 
experiments,  544;  Mr.  Baker's  data,  544;  MM. 
Chevandier  and  Wertheim's  experiments,  545. 

Elastic  strength  and  deflection  of  timber,  545; 
experiments  on  tensile  strength  by  MM.  Chevan- 
dier and  Wertheim,  546 ;  and  by  Mr.  Laslett,  546 ; 
experiments  on  compressive  strength  by  Mr. 
Kirkaldy,  546 ;  Mr.  Barlow's  experiments  on 
transverse  strength,  547. 

Rules  for  the  strength  and  deflection  of  timber, 
548;  analysis  of  Mr.  Laslett's  experimental  results, 
548;  and  of  Mr.  Fincham's  results,  549;  calculated 
tensile  strength  of  timber  of  large  scantling,  549; 
formulas  for  the  transverse  strength  of  timber  of 
large  scantling,  550,  551 ;  formulas  for  the  trans- 
verse deflection  of  timber  beams  of  uniform  rec- 
tangular section,  550;  shearing  strength  of  timber, 

551- 

Strength  of  toothed  wheels,  735. 

Strength  of  wrought  iron,  567;  Mr.  Telford's  experi- 
ments on  tensile  strength,  567;  Mr.  Barlow's,  567; 
Sir  William  Fairbairn's,  567;  Mr.  Thomas  Lloyd's, 
569;  Mr.  Edwin  Clark's,  570;  Mr.  Kirkaldy's 

571- 

Experiments  of  the  Steel  Committee  of  Civil 
Engineers,  579;  hammered  iron  bars  (Swedish), 
581 ;  Mr.  J.  Tangye's  experiments  on  compressive 
resistance;  Krupp  and  Yorkshire  iron  plates,  583; 
Prussian  iron  plates,  586;  Sir  Joseph  Whitworth's 
experiments,  615. 

Tensile  strength  of  iron  wire,  247,  586,  628,  629, 
676. 

Shearing  and  punching  strength  of  iron,  587. 
Transverse  strength  of  iron: — Swedish  bars,  582; 
Mr.  Barlow's  data,  588;  Mr.  Edwin  Clark's  data, 
588;   formulas;   transverse  deflection  and  elastic 
strength  of  wrought  iron,  590. 

Torsional  strength  and  deflection,  590. 
Resistance  of  iron  to  explosive  force,  622. 
Recapitulation  of  data  on  the  direct  strength  of 
iron,  623. 

Factors  of  safety,  625,  679. 
Strength  of  round  wrought-iron  shafting,  table  of, 

with  multipliers  for  cast  iron  and  for  steel,  762. 
Stress,  kinds  of,  500. 
Stress,  working,  for  screwed  bolts,  681. 
Stretching,  influence  of,  on  the  density  of  wrought- 
iron,  578. 

Strut-girder,  parallel,  strength  of,  708. 
Sulphur,  process  of  combustion  of,  399. 
Surfaces,  mensuration  of,  23. 


Surplus  air,  in  the  combustion  of  coal,  770,  778,  794. 
Swaine  turbine,  943. 


Tan,  heat  of  combustion  of,  455. 

Tangential  water-wheels, — Girard's,  943. 

Teeth  of  wheels.     See  Toothed  Wheels,  page  981. 

Temperature,  difference  of,  ascension  of  air  by,  897. 

Temperature  of  combustion,  407;  coal,  407,  408; 
coke,  408,  436;  wood,  444. 

Temperature,  influence  of,  on  the  strength  of  metallic 
wires,  628. 

Temperature,  influence  of,  on  the  structure  of  steel, 
616. 

Temperatures,  standard,  124,  198. 

Temperatures,  high,  luminosity  at,  328. 

Tempered  or  hardened  steel,  tensile  strength  of, 
594,  602,  603,  613. 

Tensile  strength  of  brass,  627,  628;  brass  tube,  627; 
brass  wire,  627,  629. 

Tensile  strength  of  bronze,  627,  628. 

Tensile  and  compressive  strength  of  cast  iron:  — Mr. 
Hodgkinson's  experiments,  553;  Dr.  Anderson's, 
555;  affected  by  mass  of  metal,  555;  by  cold  blast 
and  hot  blast,  555;  by  remelting,  556. 
Elastic  strength,  558. 

Tensile  strength  of  chains,  677: — stud-link  chain - 
cable,  678;  open-link  chains,  678. 

Tensile  strength  of  copper,  626;  copper  wire,  628, 
629. 

Tensile  strength  of  alloys  of  copper,  626: — alloyed 
with  phosphorus,  626;  gun-metal,  626,  627;  alloys 
of  copper  and  tin,  aluminium-bronze,  yellow  brass, 
brass  tube,  Muntz's  metal,  sterro-metal,  627;  phos- 
phor-bronze, bronze,  and  brass,  628;  brass  wire, 
629. 

Tensile  strength  of  gold  wire,  628. 

Tensile  strength  of  gun-metal,  626,  627. 

Tensile  strength  of  leather  belting,  679,  742: — Mr. 
Towne's  experiment,  679;  Messrs.  Norris  &  Co.'s 
belting,  Spill's  machinery  belting,  untanned 
leather  belts,  india-rubber  belts,  680,  750. 

Tensile  strength  of  lead,  627. 

Tensile  strength  of  Muntz's  metal,  627. 

Tensile  strength  of  palladium  wire,  628. 

Tensile  strength  of  phosphor-bronze,'628,  629. 

Tensile  strength  of  platinum  wire,  628. 

Tensile  strength  of  plaster  of  Paris,  629,  630. 

Tensile  strength  of  rivet  joints,  633: — 

In  iron  plates,  633: — perforated  iron  plates,  633; 
experiments  on  rivet-joints  by  Sir  Wm.  Fairbairn, 
633;  by  Mr.  Bertram,  634;  by  Mr.  J.  G.  Wright 
on  diagonal  joints,  637;  by  Mr.  L.  E.  Fletcher, 
638;  by  Messrs.  John  Elder  &  Co.,  638;  by  Mr. 
Brunei,  638;  shearing  strength  of  rivets,  640;  con- 
clusions on  the  strength  of  rivet-joints  in  iron 
plates,  640;  proportions  of  rivet-joints,  641. 

In  steel  plates,  642: — perforated  steel  plates, 
642;  rivetted  joints,  642. 

Tensile  strength  of  ropes:  — hemp  rope,  673- 
675;  influence  of  twist  and  of  moisture  on  the 
strength,  674;  American  hemp  rope,  676:  iron- 
wire  rope,  674,  675;  American  iron-wire  rope,  676; 
French  iron- wire  rope,  677;  steel-wire  rope,  674, 
675,  677;  cable  fencing  strands  and  solid  fencing 
wire,  676. 

Tensile  strength  of  screwed  bolts  and  nuts,  680: 
Mr.  Brunei's  experiments,  680;  working  stress, 
681,  684. 

Screwed  stay-bolts  and  flat  stayed  surfaces  of 
locomotive  boilers,  685;  of  marine  boilers,  686; 
rules,  686. 

Tensile  strength  of  silver  wire,  628. 

Tensile  strength  of  soft  solder,  627. 

Tensile  strength  of  steel: — Sir  Wm.  Fairbairn's  ex- 
periments on  steel  plates,  568;  Mr.  Kirkaldy's 
early  experiments  on  steel  bars,  593;  on  steel 


INDEX. 


—    TENSILE    — 


—    TRANSVERSE    — 


plates,   594;  on   hardened  steel;  hematite  steel,    I 
595;  crank  shafts  of  Krupp  steel,  595. 

Experiments  of  the  Steel  Committee,  596: — 
tables  of  results  for  tensile  strength,  first  series,  598; 
second  series,  601 ;  chemical  analysis  and  specific 
gravity  of  the  steel  bars,  603. 

Tensile  strength  of  tempered  steel,  603. 

Fagersta  steel  hammered  bars,  604;  ingots  and 
hammered  bars,  606 ;  hammered  and  rolled  bars,  607 ; 
bars  reduced  by  hammering  and  by  rolling,  608; 
plates  of  different  thicknesses,  608;  plates  of  dif- 
ferent forms,  608,  610;  plates  with  drilled  holes 
and  punched  holes,  610. 

Siemens'-steel  plates  and  tyres: — plates  of  dif- 
ferent thicknesses,  613;  hardened  plates,  613;  tyres, 
614. 

Whitworth's  fluid-compressed  steel,  614,  615. 

Steel  wire,  617,  629. 

Influence  of  constituent  carbon  on  tensile 
strength  of  steel,  621. 

Resistance  of  steel  to  explosive  force,  622. 

Notched  specimen,  comparative  strength  of, 
622. 

Tensile  strength  of  steel  wire,  617,  629. 
Tensile  strength  of  sterro-metal,  627. 
Tensile  strength  of  timber: — Mr.  Laslett,  538-540; 
MM.  Chevandier  and  Wertheim,  545,  546;  Mr. 
Laslett,  546;  calculated  from  Mr.  Laslett's  experi- 
ments on  transverse  strength,  548;  and  from  Mr. 
Fincham's  experiments,  549;  calculated  for  tim- 
ber of  large  scantling,  549. 
Tensile  strength  of  tin,  627. 

Tensile  strength  of  wrought  iron : — Mr.  Telford's 
experiments,  567;  Mr.  Barlow's  experiments,  567; 
Sir  Wm.  Fairbairn's  experiments  on  plates  and 
bars,  567;  influence  of  cold-rolling,  569;  Mr. 
Thomas  Lloyd's  experiments,  569;  successive 
fracture  of  the  same  bars,  569;  bars  of  different 
lengths,  570;  Mr.  Edwin  Clark  on  strength  of 
plates  and  bars,  570. 

Experiments  on  the  tensile  strength  and  elonga- 
tion of  wrought  iron  by  Mr.  Kirkaldy,  571 ;  speci- 
men bars,  571;  tensile  strength  and  elongation  of 
iron  bars,  571;  contraction  of  the  sectional  area 
of  fracture,  572;  strength  of  bars  as  affected  by 
the  diameter,  571 ;  by  rolling,  by  turning,  by  forg- 
ing, by  reheating,  by  intense  cold,  573,  576;  by 
notching,  by  screwing,  574;  by  welding,  by  sud- 
den stress,  575;  by  hardening,  by  case-hardening, 
576;  by  cold-rolling,  577;  hammered  iron,  576; 
strength  as  affected  by  additional  hammering,  576; 
by  removing  the  skin,  576. 

Tensile  strength  of  angle-iron,  ship-strap,  and 
beam  iron  (Mr.  Kirkaldy),  577. 

Tensile  strength  of  iron  plates  (Mr.  Kirkaldy), 
577;  fractured  sectional  area,  578;  strength  as 
affected  by  cold-rolling  and  by  galvanizing,  578. 

Specific  gravity  of  the  irons  tested  by  Mr. 
Kirkaldy,  578.  ' 

Sir  Joseph  Whitworth's  experiments  on  best 
irons,  615. 

Experiments  by  the  Steel  Committee,  580;  speci- 
fic gravity  of  iron  tested,  603;  Swedish  hammered 
bars,  581;  Krupp  and  Yorkshire  plates,  entire, 
drilled,  and  punched,  583;  Prussian  plates,  586, 

Iron  wire,  586,  628,  629,  676. 

Tensile  strength  and  elongation  of  wrought  iron,  in- 
fluence of  various  treatment  on,  Mr.  Kirkaldy's 
experiments,  573: — iron  bars,  rolling  down,  turning 
down,  forging,  reheating,  intense  cold,  573;  notch- 
ing, screwing,  574;  welding,  sudden  stress,  575; 
frost,  hardening,  case-hardening,  576;  cold-roll- 
ing* 577- 

Hammered  iron: — additional  hammering,  re- 
moving the  skin,  576. 

Iron    plates :  —  cold-rolling,   galvanizing,   578 ; 
annealing,  583-586;   drilled  holes,  584;  punched 
holes,  584. 
Tensile  strength  of  zinc,  627. 


Tensile  stress,  contraction  of  wrought  iron  under: — 
bars,  572;  notched  bars,  574;  plates,  578;  wire, 
587. 

Test-bars  of  cast  iron,  564. 

Thermometers,  317;  equivalent  temperatures  by 
Fahrenheit,  Centigrade,  and  Reaumur  scales,  318; 
tables  of  equivalent  temperatures  by  Fahrenheit 
and  Centigrade  scales,  319,  323;  air-  or  gas-ther- 
mometers, 325;  Regnault's,  326;  Mason's  wet  and 
dry  bulbs,  394. 

Timber,  strength  of,  537.  See  Strength  of  Tim- 
ber, page  980. 

Timber  columns,  strength  of,  644,  646,  647. 

Tin,  tensile  strength  of,  627. 

Tin,  weight  of: — tabulated  weights,  219-221;  multi- 
plier for  the  weight  of  tin  bars,  plates,  &c.,  220; 
special  tables  of  the  weight  of  tin  plates  and 
pipes,  252,  268,  269.  See  Weight  of  Iron  and 
other  Metals,  page  983. 

Ton,  140;  iron-ton,  141;  equivalent  weight  in  French 
measure,  155,  157;  New  York  ton,  187;  Canadian 
ton,  187. 

Tools,  resistance  of,  951;  work  of,  952;  resistance  of 
wood-cutting  machines,  954;  grindstones,  955. 

Toothed  wheels,  727;  speed,  727;  pitch,  728;  table  of 
multipliers  for  number  of  teeth  and  diameter,  729; 
table  of  diameters  of  toothed  wheels,  730;  form 
of  the  teeth  of  wheels,  731;  strength,  735;  work- 
ing strength,  736;  breadth,  737;  horse-power,  737; 
weight  of  toothed  wheels,  739. 

Torsional  deflection,  general  investigation  of,  536; 
round  shaft,  536;  square  shaft,  537;  hollow  shaft, 

Torsional  strength  of  shafts,  general  investigation 
of,  534;  solid  round  shafts,  534;  hollow  round 
shafts,  535 ;  square  shafts,  535 ;  torsional  deflection 
of  round  and  square  shafts,  536. 

Torsional  strength  and  deflection  of  cast  iron,  565. 

Torsional  strength  and  deflection  of  round  shafting, 
758,  759,  762. 

Torsional  strength  of  steel  bars,  595,  596,  600,  604  ; 
formulas,  619. 

Torsional  shearing  stress  and  deflection  of  steel 
bars,  formula,  620. 

Torsional  strength  and  deflection  of  wrought  iron : 
— Swedish  bar,  582;  formulas,  590. 

Torsional  stress,  definition  of,  500. 

Traction  on  common  roads,  resistance  to,  961. 

Tramways,  street,  resistance  on,  966. 

Transverse  deflection  of  homogeneous  beams  and 
girders: — general  investigation,  527;  beams  of 
rectangular  section,  529;  double-flanged  or  hollow 
rectangular  beams,  530;  table  of  relative  deflec- 
tions of  beams,  variously  proportioned  and  loaded, 
S32- 

Transverse  strength  of  homogeneous  beams : — gen- 
eral investigations,  503 ;  symmetrical  solid  beams, 
503;  formulas  for  rectangular  beams,  507;  gener- 
alized formula  for  solid  beams  (without  overhang) 
of  symmetrical  section,  509;  flanged  or  hollow 
beams  of  symmetrical  section,  510;  flanged  beams 
not  symmetrical  in  section,  513. 

Transverse  strength  and  deflection  of  cast  iron : — 

Mr.  Barlow's  experiments,  561;  Mr.  Edwin  Clark's 

data,  562;  Mr.  Hodgkinson's  data,  563;  test-bars, 

564. 

Transverse  deflection  and  elastic  strength,  564. 

Transverse  strength  of  flanged  beams : — cast  iron, 
647;  wrought  iron,  653. 

Transverse  deflection  of  shafts,  756.  See  Deflection, 
page  974. 

Transverse  strength  and  deflection  of  steel : — bars, 
S95»  596,  599,  604;  formulas,  617—619;  railway 
rails,  661,  665,  668. 

Transverse  strength  of  steel  springs,  671;  formulas 
and  rules  for  laminated  springs,  671;  for  helical 
springs,  672. 

Transverse  strength  and  deflection  of  railway  rails, 
661;  rails  of  symmetrical  section,  or  double-headed 


982 


INDEX. 


—    TRANSVERSE    — 


—    WEIGHTS    — 


rails,  661;  general  formula,  662;  Mr.  Price  Wil- 
liams' data,  662;  influence  of  constituent  carbon 
on  the  strength,  664;  Mr.  J.  T.  Smith's  data,  665. 
Rails  of  unsymmetrical  section,  665;  general 
formula  and  rule,  665;  steel  flange-rail  by  Mr. 
John  Fowler,  666;  Mr.  Kirkaldy's  test  of  Mr. 
Fowler's  rail,  667;  influence  of  holes  in  the  flanges, 
668;  wrought-iron  flange- rails,  668. 

Deflection  of  rails,  668;    double-headed  rails, 
668;  formulas,  669;  flange  rails,  669;  formulas,  670. 

Transverse  strength  and  deflection  of  timber : — Ex- 
periments described  by  M.  Morin,  537;  Mr. 
Laslett's  experiments,  538-540;  Mr.  Fincham's, 
542,  543;  Mr.  Maclure's,  542,  543;  Mr.  Edwin 
Clark's,  543,  544;  Mr.  G.  Graham  Smith's,  543, 
544;  Mr.  Baker's,  544;  MM.  Chevandier  and 
Wertheim's,  545;  Mr.  Barlow's,  547. 

Formulas  for  the  transverse  strength  and  de- 
flection of  timber  of  large  scantling,  550-552. 

Transverse  strength  and  deflection  of  wrought  iron: 
—hammered  iron  bars  (Swedish)  582,  589;  Mr. 
Barlow's  data,  588;  Mr.  Edwin  Clark's  data,  588; 
formulas,  589;  railway  rails,  661,  665. 

Transverse  deflection   and   elastic  strength: — 
data,  590;  formulas,  590;  railway  rails,  668. 

Traversers,  rope-gearing  for  working,  755. 

Trigonometry,  plane,  21;  tables,  103,  no. 

Tubes,  weight  of,  248,  250,  266. 

Tubes,  wrought-iron,  strength  of,  692,  695;  large 
flue-tubes,  677. 

Tub  water-wheel,  939. 

Turbines,  940;  Fourneyron's,  Boyden's,  940;  rules 
for  outward-flow  turbines,  941;  Fontaine's,  Jon- 
val's,  North  Moor  Foundry's,  942;  vortex  wheel, 
Swaine  turbine,  943. 


u 


let's,  937. 

of  heat,  mechanical  equivalents,  332. 


Undershot  water-wheels: — with  radial  floats,  Ponce- 
let's 
Units 

V 

Vapours.     See  Gases  and  Vapours,  page  975. 
Vapours  and  gases,  mixture  of,  392.     See  Gases  and 

Vapours,  page  975. 
Vapours,  pressure  of,  at  212°  F.,  370;  boiling-points 

at  various  pressures,  371. 
Vegetable  substances,  weight  and  specific  gravity 

of,  212. 

Velocity,  definition  of,  277. 

Ventilation,  477.     See  Warming  and  Ventilation. 
Ventilators,  or  fans,  924.     See  A  ir  Machinery,  page 

971. 
Vortex  turbine,  943. 

w 

Waggons  in  coal  pits,  resistance  of,  956. 
Warming   and   ventilation,    477 ;   ventilation,  477 ; 
ventilation  of  mines  by  heated  columns  of  air, 
479  ;  cooling  action  of  window  glass,  480  ;  heating 
rooms    by   hot- water,   481  ;     heating  rooms    by 
steam,  486  ;  heating  by  ordinary  open  fires,  488  ; 
heating  by  hot-air  and  stoves,  488. 
Warren-girder,  strength  of,  699. 
Warrington  wire-gauge,  133. 
Washing  small  coal,  411. 
Waste-boards,  flow  of  water  over,  932. 
Water,  as  a  standard  for  weight  and  measure,  124: — 
notable     temperatures,    weight    and    volume, 
124,  125. 

The  gallon  and  other  measures  of  water,  rela- 
tive weights  and  volumes  of  water,  125. 
Pressure  of  a  column  of  water,  126. 
Compressibility  of  water,  126. 
Sea-water,  its  volume,  weight,  and  composi- 
tion, 126. 


Ice  and  snow,  their  volume  and  weight,  127. 
French  and  English  measures  of  water,  127. 

Water,  expansion  of,  with  table,  338-341;  specific 
heat  of,  353,  354. 

Water,  flow  of,  929.    See  Flow  of  Water,  page  974. 

Water,  hygroscopic,  in  coals,  416. 

Water,  machines  for  raising,  944;  reciprocating 
pumps,  944;  centrifugal  pumps,  946;  endless-chain 
pump,  947;  noria,  947;  water -works  pumping 
engines,  947;  hydraulic  rams,  948. 

Water-mill:— Barker's,  Whitelaw's,  939. 

Water-pipes,  cast-iron,  dimensions,  weight,  and 
strength  of,  934. 

Water-tube  boiler,  trials  of,  771,  777. 

Water-wheels,  937 : — 

Wheels  on  a  horizontal  axis,  937;  undershot, 
937;  Poncelet's,  937;  paddle  water-wheel,  breast- 
wheel,  overshot-wheel,  938. 

Wheels    on   a   vertical   axis,    939;    tub-wheels, 
Whitelaw's    water-mill,    939;    turbines,  outward- 
flow,  940;   downward-flow,  942;  inward -flow,  943. 
Tangential  wheels,  Girard  turbine,  943. 

Water-works  pumping  engines,  947. 

Wedge,  309;  work  done  with  it,  315. 

Wedgewood's  pyrometer,  327. 

Weight  and  specific  gravity,  198. 

Standard  temperatures,  rules  for  specific  gravity, 
comparative  weights  of  various  solids,  liquids,  and 
gases,  198. 

Specific  gravity  of  allo}^  of  copper,  alloys 
having  a  greater  density  than  the  mean,  alloys 
having  a  less  density  than  the  mean,  200.  See 
also  626,  627. 

Weight  and  specific  gravity  of  solid  bodies: — 
metals,  202,  578,  626,  627 ;  precious  stones,  203 ; 
stones,  204 ;  sundry  mineral  substances,  205 ; 
coals,  206 ;  peat,  207 ;  fuel  in  France,  207  ;  woods, 
Indian  woods,  colonial  woods,  208 ;  wood-char- 
coal, 211  ;  animal  substances,  vegetable  sub- 
stances, 212. 

Weight  and  volume  of  various  substances,  by  Tred- 
gpld,  213. 

Weight  and  volume  of  goods  carried  on  the  Bombay, 
Baroda,  and  Central  Indian  Railway,  213. 

Weight  and  specific  gravity  of  liquids,  215. 

Weight  and  specific  gravity  of  gases  and  vapours, 
216. 

Weights  and  measures,  124. 

Water  as  a  standard  for  weight  and  measure, 
124  ;  air  as  a  standard,  127. 

Great  Britain  and  Ireland,  imperial  weights  and 
measures,  128. 

Measures  of  length,  129  ;  land  measure,  nautical 
measure,  cloth  measure,  130. 

Wire -gauges,  130;  Birmingham  wire -gauge 
(Holtzapffel's),  Birmingham  metal-gauge  or  plate- 
gauge  (Holtzapffel's)  for  sheet  metals,  brass,  gold, 
silver,  &c.,  131 ;  Lancashire  gauge  (Holtzapffel's) 
for  round  steel  wire  and  for  pinion  wire,  needle- 
gauge,  music  wire-gauge,  132 ;  Warrington  wire- 
gauge  (Rylands  Brothers),  Birmingham  wire- 
gauge  for  iron  sheets  chiefly  (South  Stafford- 
shire), 133;  Sir  Joseph  Whitworth  &  Co. 's  stand- 
ard wire-gauge ;  imperial  standard  wire-gauge, 
134- 

Inches,  their  equivalent  decimal  values  in  parts 
of  a  foot,  fractional  parts  of  an  inch,  and  their 
decimal  equivalents,  135;  sixteenths  and  thirty- 
seconds,  136. 

Measures  of  surface,  136 ;  superficial  measure, 
136 ;  builders'  measurement,  land  measure,  137 ; 
decimal  parts  of  a  square  foot  in  square  inches, 
138- 

Measures  of  volume,  137;  solid  or  cube  measure, 
builders'  measurement,  137. 

Measures  of  capacity,  138  ;  liquid  measure,  138; 
dry  measure,  standard  bushel,  coal  measure,  old 
wine  and  spirit  measure,  139 ;  old  ale  and  beer 
measure,  apothecaries'  fluid  measure,  140. 


INDEX. 


-    WEIGHTS    — 


—    WIRE    — 


Measures  of  weight,  140 ;  avoirdupois  weight, 
140 ;  troy  weight,  diamond  weight,  apothecaries' 
weight,  old  apothecaries'  weight,  weight  of  current 
coins,  141  ;  coal  weight,  sundry  bushel  measures, 
wool  weight,  hay  and  straw  weight,  corn  and 
flour  weight,  142. 

Miscellaneous  tables,  143;  drawing  papers, 
commercial  numbers  and  stationery,  measures 
relating  to  building,  143;  sundry  commercial 
measures,  measures  for  ships,  144. 

Compound  units,  comparison  of,  144  ;  measures 
of  velocity,  144 ;  volume  and  time,  pressure  and 
weight,  weight  and  volume,  power,  145. 

France,  the  metric  standards  of  weights  and 
measures,  146;  countries  in  which  they  are 
adopted,  147. 

Measures  of  length,  147 ;  old  measures,  147 ; 
French  wire-gauges,  148. 

Measures  of  surface,  land,  149. 

Measures  of  volume,  149 ;  cubic  measure,  wood 
measure,  149. 

Measures  of  capacity,  149  ;  liquid  measure,  dry 
measure,  149. 

Measures  of  weight,  150. 

Equivalents   of    British   imperial   and    French 

metric  weights  and  measures,  150: — length,  150; 

millimetres   and  inches,    151;  inches   and   milli- 

'  metres,   152;    surface,  153;   cubic  measure,  154; 

wood  measure,  154;  capacity,  154;  weight,  155. 

Approximate  equivalents  of  English  and  French 
measures,  156. 

Equivalents  of  French  and  English  compound 
units  of  measurement,  157;  weight,  pressure,  and 
measure,  157;  volume,  area,  length,  and  work, 
158  ;  heat,  speed,  money,  159. 

German  Empire,  weights  and  measures,  160. 

Old  weights  and  measures  of  the  German  States, 
161 ;  German  fuss,  161 ;  kingdom  of  Prussia,  162; 
Bavaria,  164;  Wiirtemberg,  165;  Saxony,  166; 
Baden,  167  ;  Hanse  Towns,  Hamburg,  168  ;  Bre- 
men, Lubec,  169 ;  German  Customs  Union,  169. 

Austrian  Empire,  170. 

Russia,  171. 

Holland,  Belgium,  173. 

Norway,  Denmark,  Sweden,  173. 

Switzerland,  175. 

Spain,  Portugal,  Italy,  Turkey,  Greece  and 
Ionian  Islands,  Malta,  176. 

Egypt,  Morocco,  Tunis,  Arabia,  Cape  of  Good 
Hope,  179. 

Indian  Empire,  Bengal,  Madras,  Bombay, 
Ceylon,  180. 

Burmah,  China,  Cochin-China,  Persia,  Japan, 
Java,  183. 

United  States  of  America,  186. 

British  North  America,  187. 

Mexico,  187. 

Central  America  and  West  Indies,  British  West 
Indies,  Cuba,  Guatemala  and  Honduras,  British 
Honduras,  Costa  Rica,  St.  Domingo,  187. 

South  America: — Colombia,  Venezuela,  Ecua- 
dor, Guiana,  Brazil,  Peru,  Chili,  Bolivia,  Argen- 
tine Confederation,  Uruguay,  Paraguay,  188. 

Australasia,  189. 

Weight  of  iron  and  other  metals,  table  of,  217 ; 
data  for  wrought  iron,  217,  218 ;  data  for  steel, 
217 ;  data  for  cast  iron,  217,  218 ;  notice  of  the 
tables,  218. 

Table,  weight  of  given  volumes  of  metals,  219. 

Table,  volume  of  given  weights  of  metals  for 
given  weights,  219. 

Table,  weight  of  i  square  foot  of  metals,  220. 

Table,  weight  of  metals  of  a  given  sectional 
area,  per  lineal  foot  and  per  lineal  yard,  221. 

Rules  for  the  weight  of  wrought  iron,  cast  iron, 
and  steel,  223. 


Rule  for  the  length  of  one  hundredweight  of 
wire  of  different  metals  of  a  given  thickness,  224. 
Table,  weight  of  French  galvanized  iron  wire, 
225. 

Weight  of  wrought-iron  bars,  plates,  &c.,  special 
tables,  226 ;  multipliers  for  other  metals,  226. 
Table,  weight  of  flat  bar  iron,  227. 
Table,  weight  of  square  iron,  239. 
Table,  weight  of  round  iron,  240. 
Table,  weight  of  angle-iron  and  tee-iron,  242. 
Table,  weight  of  wrought-iron  plates,  243. 
Table,  weight  of  sheet  iron,  244. 
Table,   weight  of  black  and  galvanized  iron 
sheets,  245. 

Table,  weight  of  hoop  iron,  246. 
Table,  weight  and  strength  of  Warrington  iron 
wire  (Rylands  Brothers),  247. 

Table,  weight  of  wrought-iron  tubes,  by  internal 
diameter,  248. 

Table,  weight  of  wrought-iron  tubes,  by  exter- 
nal diameter,  250. 

Weight  of  cast  iron,  steel,  copper,  brass,  tin,  lead, 
and  zinc, — special  tables,  251. 

Table,  weight  of  cast-iron  cylinders,  by  internal 
diameter,  253. 

Table,  weight  of  cast-iron  cylinders,  by  external 
diameter,  255. 

Table,  volume  and  weight  of  cast-iron  balls, 
for  given  diameters,  multipliers  for  other  metals, 
258. 

Table,  diameter  of  cast-iron  balls,  for  given 
weights,  258. 

Table,  weight  of  flat  bar  steel,  259. 
Table,  weight  of  square  steel,  260. 
Table,  weight  of  round  steel,  260. 
Table,    weight    of   chisel  steel: — hexagonal, 
octagonal,  and  oval-flat,  261. 

Table,  weight  of  one  square  foot  of  sheet 
copper,  261. 

Table,  weight  of  copper  pipes  and  cylinders, 
by  internal  diameter,  262. 

Table,  weight  of  brass  tubes,  by  external  dia- 
meter, 266. 

Table,  weight  of  one  square  foot  of  sheet 
brass,  268. 

Table,  size  and  weight  of  tin  plates,  268. 
Table,  weight  of  tin  pipes,  269. 
Table,  weight  of  lead  pipes,  269. 
Table,  dimensions  and  weight  of  sheet  zinc,  270. 
Weight  of  belt  pulleys,  750. 
Weight  of  chains,  678,  679. 
Weight  and  bulk  of  coal: — British,  206,  414,  416; 

American,  418,  419;  French,  422. 
Weight  and  bulk  of  coke,  206,  432;  of  lignite,  207; 

of  wood,  442. 
Weight  of  ropes:— hemp,  674,  675;  iron,  674,  675, 

677;  steel,  674,  675,  677. 
Weight  of  round  wrought-iron  shafting: — net  and 

gross,  761. 
Weight  of  toothed  wheels: — spur,  739;  mortise,  741, 

bevel  and  mitre,  741. 
Weight  of  pure  water,  124. 
Of  sea-water,  126. 
Of  ice  and  snow,  127. 
Weirs,  flow  of  water  over,  932. 
Welded  joints  in  iron  plates,  tensile  strength  of,  634, 

635- 

Wheel  and  axle,  305;  work  done  with  it,  314. 
Wheels,  toothed,  727.    See  Toothed  Wheels,  p.  981. 
Whitelaw's  water-mill,  939. 
Whit  worth's  standard  wire-gauge,  134;   bolts  and 

nuts,  682;  screwed  piping,  683. 
Williams'  system  of  smoke  prevention,  785. 
Wilson's  pyrometer,  327. 

Wire,  weight  of: — rule  for  the  length  of  one  cwt.  of 
wire  of  various  metals  of  a  given  thickness,  224; 


INDEX. 


—    WIRE    — 


—    ZINC    — 


table    of  the    weight    of  galvanized    iron  wire 

(French),  225;  table  of  the  weight  and  strength 

of  Warrington  iron  wire,  247. 
Wire,  tensile  strength  of: — iron,  247,  586,  628,  629, 

676;  steel,  617,  629;  copper,  628,  629;  brass,  627, 

629;  phosphor-bronze,  629;  gold,  628;  silver,  628; 

platinum,  628;  palladium,  628. 
Wire -gauges,    English,    130;    French,   148.      See 

Weights  and  Measures,  page  982. 
Wire-ropes,  iron,  strength  and  weight  of,  674-677. 
Wire-ropes,  steel,  strength  and  weight  of,  674,  675, 

677. 
Wire-ropes  for  transmission  of  power,  by  M.  Him. 

Wire-drawing,  influence  of,  on  the  density  of  iron 
wire,  247. 

Woods,  weight  and  specific  gravity  of,  208;  Indian 
woods,  209;  colonial,  209. 

Wood,  439 : — classification,  439  ;  constituent  mois- 
ture, 439;  composition,  440,  441 ;  weight  and  bulk, 
442;  quantity  of  air  consumed  in  combustion,  443; 
gaseous  products  of  combustion,  443 ;  heat  of  com- 
bustion, 444;  temperature  of  combustion,  444;  dis- 
tillation of,  449. 

Wood  measure,  French,  149. 

Wood,  strength  of.     See  Strength  of  Timber,  page 


/ood-charcoal,  weight  and  specific  gravity  of,  211. 

Wood-charcoal: — wood  for  making  it,  443;  process 
of  carbonization,  444,  448;  yield  of  charcoal,  445, 
447,  448 ;  composition,  446,  447 ;  charbon  de  Paris, 
449;  weight  and  bulk  of  charcoal,  450;  moisture, 
451;  air  consumed  in  combustion,  452;  gaseous 
products  of  combustion,  452;  heat  of  combustion, 
452. 

Woollen  mills,  machinery  of,  resistance  of,  959; 
horse-power  required,  960. 

Work,  definition  of,  312;  accumulated  work  in  solid 
bodies,  315.  See  Mechanical  Principles,  page 

Work,  or  labour,  718; — units  of  work,  718;  labour  of 
men,  718;  labour  of  horses,  720;  work  of  animals 
in  carrying  loads,  720;  work  absorbed  by  friction, 

Work  of  animals  in  carrying  loads,  720. 
Work  of  tools,  in  metal,  952. 

Work  that  may  be  done  for  one  turn  of  a  shaft  760, 
762;  absorbed  by  friction,  763. 


Work  of  resistance  of  material,  501. 

Work  of  dry  air  or  other  gas,  compressed  or  ex- 
panded, 898: — general  formulas,  898;  work  of 
compression  of  air  at  constant  temperature, 
isothermally,  without  clearance,  899;  with  clear- 
ance, 900;  work  of  expansion  of  air  .at  constant 
temperature,  900. 

Work  of  dry  air  in  a  non-conducting  cylinder, 
adiabatically,  901;  adiabatic  compression  of  a  gas, 
901 ;  table  of  compression  or  expansion  of  air  with- 
out receiving  or  giving  out  heat,  902;  work  ex- 
pended in  compressing  dry  gas,  903,  904;  adia- 
batic expansion  of  gases,  904-909;  table  of  corre- 
sponding ratios  of  pressures  and  temperatures, 
when  air  is  admitted  for  the  whole  stroke,  908; 
table  of  comparative  final  temperatures  and  effici- 
encies of  air  expanded  adiabatically  and  air  ad- 
mitted for  the  whole  of  the  stroke,  908. 

Efficiency  of  compressed-air  engines,  909;  com- 
pression and  expansion  of  moist  air,  912;  work  in 
expansion,  913;  temperature  in  expansion,  914. 

Wrought  iron,  strength  of,  567.  See  Strength  of 
Wrought  Iron,  page  980. 

Wrought-iron  columns,  strength  of,  644,  645. 

Wrought-iron  flanged  beams,  transverse  strength 
of,  653. 

Solid  wrought-iron  joists: — tables  of  dimensions, 
weight,  and  strength,  653,  654;  experiments  by 
Mr.  Kirkaldy,  655;  rules  and  formulas,  656; 
elastic  strength  and  deflection,  formulas  and 
rules,  657. 

Rivetted  wrought-iron  joists,  657;  Mr.  Davies' 
experiments,  658;  rules,  659,  660. 

Wrought  iron,  resistance  of,  to  explosive  force,  622. 


Yard,  imperial  standard,  128. 


Zinc,  weight  of: — tabulated  weights,  219-221;  multi- 
plier for  the  weight  of  zinc  bars,  plates,  &c.,  220; 
special  table  of  the  size  and  weight  of  sheet 
zinc,  270.  See  Weight  of  Iron  and  other  Metals, 
page  983. 

Zinc,  tensile  strength  of,  627. 


THE    END. 


GLASGOW :  W.  G.  BIACKIB  AND  CO.,  PRINTERS,  VILLAFIEtD. 


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